Finite Group Databasebeta

← C2×#22622864 #33966498

Subgroups — C2×#22622864 #33966498

Order 18688.

1778 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Dic146C211yesmaximal, characteristic
2C2×#17989313C211yesmaximal, characteristic
3#22622864C2116yesmaximal, direct product factor
4C2×#17988996C211yesmaximal, characteristic
5C2×#17988994C211yesmaximal, characteristic
6C2×#17988984C211yesmaximal, characteristic
7C2×#17988896C211yesmaximal, characteristic
8C2×#17988894C211yesmaximal, characteristic
9C2×#17988880C211yesmaximal, characteristic
10C2×#17988872C211yesmaximal, characteristic
11C2×#17988870C211yesmaximal, characteristic
12C2×#17988870C211yesmaximal, characteristic
13C2×#17988715C211yesmaximal, characteristic
14C2×#17988694C211yesmaximal, characteristic
15C2×Q16×Dic73C211yesmaximal, characteristic
16C2×#441×C73C211yesmaximal, characteristic
17Q8×Dic146—24no—
18Q8×Dic146C2218yes—
19#17989313—24no—
20#17989313C2218yes—
21C2×#14556591—21no—
22C2×#14556591C2211yescharacteristic
23C2×#14556590—21no—
24C2×#14556590—21no—
25C2×#14556590—21no—
26C2×#14556590C2211yescharacteristic
27C2×#14556590C2211yescharacteristic
28C2×#14556590C2211yescharacteristic
29C2×#14556583—21no—
30C2×#14556583C2211yescharacteristic
31C2×#14556582—21no—
32C2×#14556582C2211yescharacteristic
33C2×Q8×Dic73C2211yescharacteristic
34C2×Q8×Dic73C2211yescharacteristic
35C2×Q8×Dic73C2211yescharacteristic
36C2×Q8×Dic73C2211yescharacteristic
37C2×C4×Dic146—21no—
38C2×C4×Dic146—21no—
39C2×C4×Dic146C2211yescharacteristic
40#17988996C2218yes—
41#17988994C2218yes—
42#17988984C2218yes—
43#17988896C2218yes—
44#17988894C2218yes—
45#17988880C2218yes—
46#17988872C2218yes—
47#17988870C2218yes—
48#17988870C2218yes—
49#17988715C2218yes—
50#17988694C2218yes—
51Q16×Dic73C2218yes—
52C2×#14556525C2211yescharacteristic
53C2×#14556525C2211yescharacteristic
54C2×#14556524C2211yescharacteristic
55C2×#14556523C2211yescharacteristic
56C2×#14556523C2211yescharacteristic
57C2×#14556522C2211yescharacteristic
58C2×#14556522C2211yescharacteristic
59C2×#14556522C2211yescharacteristic
60C2×C4⋊#10596831C2211yescharacteristic
61C2×#14556520C2211yescharacteristic
62C2×#14556520C2211yescharacteristic
63C2×#14556515C2211yescharacteristic
64C2×#14556515C2211yescharacteristic
65C2×#14556515C2211yescharacteristic
66C2×#14556515C2211yescharacteristic
67C2×#14556501C2211yescharacteristic
68C2×C8×Dic73C2211yescharacteristic
69C22×Q16×C73C2211yescharacteristic
70C2×C4×Q8×C73C2211yescharacteristic
71C2×C4×Q8×C73C2211yescharacteristic
72#441×C73C2218yes—
73C2×#106×C73C2211yescharacteristic
74C2×#103×C73C2211yescharacteristic
75C2×#103×C73C2211yescharacteristic
76C2×#97×C73C2211yescharacteristic
77C22×Dic146—21no—
78C22×Dic146—21no—
79C22×Dic146—21no—
80C22×Dic146—21no—
81C22×Dic146D411yescharacteristic
82C22×#10596831C2311yescharacteristic
83C22×#10596831C2311yescharacteristic
84#14556591—28no—
85#14556591—22no—
86#14556591C2314yes—
87#14556590—28no—
88#14556590—28no—
89#14556590—28no—
90#14556590—22no—
91#14556590—22no—
92#14556590—22no—
93#14556590C2314yes—
94#14556590C2314yes—
95#14556590C2314yes—
96#14556583—28no—
97#14556583—22no—
98#14556583C2314yes—
99#14556582—28no—
100#14556582—22no—
101#14556582C2314yes—
102Q8×Dic73—22no—
103Q8×Dic73—22no—
104Q8×Dic73—22no—
105Q8×Dic73—22no—
106Q8×Dic73C2314yes—
107Q8×Dic73C2314yes—
108Q8×Dic73C2314yes—
109Q8×Dic73C2314yes—
110C4×Dic146—28no—
111C4×Dic146—28no—
112C4×Dic146D414yes—
113C4×Dic146C2314yes—
114C2×C4⋊Dic73—21no—
115C2×C4⋊Dic73—21no—
116C2×C4⋊Dic73—21no—
117C2×C4⋊Dic73—21no—
118C2×C4⋊Dic73—21no—
119C2×C4⋊Dic73—21no—
120C2×C4⋊Dic73D411yescharacteristic
121C2×C4⋊Dic73C2311yescharacteristic
122C2×C4⋊Dic73C2311yescharacteristic
123C2×C4⋊Dic73C2311yescharacteristic
124C2×C4⋊Dic73C2311yescharacteristic
125C2×#11924007—21no—
126C2×#11924007—21no—
127C2×#11924007—21no—
128C2×#11924007—21no—
129C2×#11924007—21no—
130C2×#11924007—21no—
131C2×#11924007D411yescharacteristic
132C2×#11924007D411yescharacteristic
133C2×C4×Dic73—21no—
134C2×C4×Dic73—21no—
135C2×C4×Dic73—21no—
136C2×C4×Dic73—21no—
137C2×C4×Dic73C2311yescharacteristic
138C2×C4×Dic73C2311yescharacteristic
139#14556525C2314yes—
140#14556525C2314yes—
141#14556524C2314yes—
142#14556523C2314yes—
143#14556523C2314yes—
144#14556522C2314yes—
145#14556522C2314yes—
146#14556522C2314yes—
147C4⋊#10596831C2314yes—
148#14556520C2314yes—
149#14556520C2314yes—
150#14556515C2314yes—
151#14556515C2314yes—
152#14556515C2314yes—
153#14556515C2314yes—
154#14556501C2314yes—
155C8×Dic73C2314yes—
156C22×Q8×C73C2311yescharacteristic
157C22×Q8×C73C2311yescharacteristic
158C2×Q16×C73Q818yescomplemented
159C2×Q16×C73C2314yes—
160C22×C8×C73C2311yescharacteristic
161C22×C8×C73C2311yescharacteristic
162C4×Q8×C73—22no—
163C4×Q8×C73—22no—
164C4×Q8×C73C2314yes—
165C4×Q8×C73C2314yes—
166C2×C4⋊C4×C73—21no—
167C2×C4⋊C4×C73—21no—
168C2×C4⋊C4×C73C2311yescharacteristic
169C2×C4⋊C4×C73C2311yescharacteristic
170C2×C42×C73—21no—
171C2×C42×C73—21no—
172C2×C42×C73C2311yescharacteristic
173#106×C73C2314yes—
174#103×C73C2314yes—
175#103×C73C2314yes—
176#97×C73C2314yes—
177C2×Dic146—24no—
178C2×Dic146—24no—
179C2×Dic146—24no—
180C2×Dic146—24no—
181C2×Dic146—24no—
182C2×Dic146—44no—
183C2×Dic146—82no—
184C2×Dic146—82no—
185C2×Dic146—82no—
186C2×Dic146C2×D414yes—
187C22×Dic73—21no—
188C22×Dic73—21no—
189C22×Dic73—21no—
190C22×Dic73—21no—
191C22×Dic73C2×D411yescharacteristic
192C22×Dic73C2×D411yescharacteristic
193C22×Dic73C2×D411yescharacteristic
194C22×Dic73C2×D411yescharacteristic
195C2×#10596831—22no—
196C2×#10596831—22no—
197C2×#10596831C2412yes—
198C2×#10596831C2412yes—
199C4⋊Dic73—24no—
200C4⋊Dic73—21no—
201C4⋊Dic73—21no—
202C4⋊Dic73—21no—
203C4⋊Dic73—21no—
204C4⋊Dic73—24no—
205C4⋊Dic73—24no—
206C4⋊Dic73—24no—
207C4⋊Dic73—24no—
208C4⋊Dic73—24no—
209C4⋊Dic73C2×D414yes—
210C4⋊Dic73C2412yes—
211C4⋊Dic73C2412yes—
212C4⋊Dic73C2412yes—
213C4⋊Dic73C2412yes—
214#11924007—24no—
215#11924007—24no—
216#11924007—24no—
217#11924007—24no—
218#11924007—24no—
219#11924007—24no—
220#11924007C2×D414yes—
221#11924007C2×D414yes—
222C4×Dic73—24no—
223C4×Dic73—24no—
224C4×Dic73—24no—
225C4×Dic73—24no—
226C4×Dic73C2×D412yes—
227C4×Dic73C2×D412yes—
228C4×Dic73C2412yes—
229C4×Dic73C2412yes—
230C2×Q8×C73—21no—
231C2×Q8×C73—21no—
232C2×Q8×C73—22no—
233C2×Q8×C73—22no—
234C2×Q8×C73C2×Q814yes—
235C2×Q8×C73C2×Q814yes—
236C2×Q8×C73C2412yes—
237C2×Q8×C73C2412yes—
238C22×C4×C73—21no—
239C22×C4×C73—21no—
240C22×C4×C73—21no—
241C22×C4×C73—21no—
242C22×C4×C73C2×D411yescharacteristic
243C22×C4×C73C2×D411yescharacteristic
244C22×C4×C73C2411yescharacteristic
245Q16×C73C2×Q8116yescomplemented
246C2×C8×C73—22no—
247C2×C8×C73C2×Q814yes—
248C2×C8×C73C2412yes—
249C2×C8×C73C2412yes—
250C4⋊C4×C73—24no—
251C4⋊C4×C73—24no—
252C4⋊C4×C73—21no—
253C4⋊C4×C73—21no—
254C4⋊C4×C73C2412yes—
255C4⋊C4×C73C2412yes—
256C42×C73—24no—
257C42×C73—24no—
258C42×C73C2×D412yes—
259C42×C73C2412yes—
260C23×C73C22×D411yescharacteristic
261Q8×C73—24no—
262Q8×C73—24no—
263Q8×C73—22no—
264Q8×C73—22no—
265Q8×C73C22×Q814yes—
266Q8×C73C22×Q814yes—
267C2×C4×C73—24no—
268C2×C4×C73—24no—
269C2×C4×C73—22no—
270C2×C4×C73—22no—
271C2×C4×C73—22no—
272C2×C4×C73—22no—
273C2×C4×C73—22no—
274C2×C4×C73—22no—
275C2×C4×C73—42no—
276C2×C4×C73—42no—
277C2×C4×C73C22×D412yes—
278C2×C4×C73C22×D412yes—
279C2×C4×C73C22×D411yescharacteristic
280C2×C4×C73C22×Q812yes—
281C2×C4×C732−1+412yes—
282C2×C4×C73C2511yescharacteristic
283C2×Dic73—22no—
284C2×Dic73—22no—
285C2×Dic73—22no—
286C2×Dic73—22no—
287C2×Dic73—22no—
288C2×Dic73—22no—
289C2×Dic73—22no—
290C2×Dic73—42no—
291C2×Dic73—42no—
292C2×Dic73—42no—
293C2×Dic73—42no—
294C2×Dic73#13714yes—
295C2×Dic73C22×D412yes—
296C2×Dic73C22×D412yes—
297C2×Dic73C22×D412yes—
298C2×Dic73C22×D412yes—
299Dic146—28no—
300Dic146—48no—
301Dic146—84no—
302Dic146—84no—
303Dic146—84no—
304C8×C73—22no—
305C8×C73C22×Q814yes—
306#10596831—22no—
307#10596831—22no—
308C22×C73D4×Q812yes—
309C22×C73C2×#13712yes—
310C22×C73#57712yes—
311C22×C73C23×D411yescharacteristic
312C4×C73—24no—
313C4×C73—24no—
314C4×C73—22no—
315C4×C73—22no—
316C4×C73—42no—
317C4×C73—42no—
318C4×C73D4×Q812yes—
319C4×C73C23×Q811yescharacteristic
320C4×C73C2×2−1+411yescharacteristic
321Dic73—22no—
322Dic73—22no—
323Dic73—22no—
324Dic73—42no—
325Dic73—42no—
326Dic73—42no—
327Dic73—42no—
328Dic73C2×#13714yes—
329C2×#3387—731nomaximal, Sylow
330C2×C73#338714yescomplemented
331C2×C73C2×D4×Q811yescharacteristic
332C2×C73C22×#13711yescharacteristic
333C2×C73C2×#57711yescharacteristic
334C2×Q82—731no—
335C2×#557—731no—
336#3387—7316no—
337C2×#501—731no—
338C2×#501—731no—
339C2×#498—731no—
340C2×#479—731no—
341C2×#479—731no—
342C2×#477—731no—
343C2×#475—731no—
344C2×#475—731no—
345C2×#475—731no—
346C2×#446—731no—
347C2×#441—731no—
348C2×#441D7311yescharacteristic
349C2×C4×Q16—731no—
350C73C2×#338711yescharacteristic, complemented, Sylow
351C22×Q16C2×D7311yescharacteristic
352Q82—738no—
353Q82—1464no—
354#557—738no—
355#557—1464no—
356C2×C4⋊Q8—731no—
357C2×C4⋊Q8—731no—
358C2×C4⋊Q8—731no—
359C2×C4⋊Q8—731no—
360C2×C4⋊Q8—1461no—
361C2×C4⋊Q8—1461no—
362C2×C4⋊Q8—1461no—
363C2×C4⋊Q8—1461no—
364C2×#125—731no—
365C2×#125—731no—
366C2×#125—1461no—
367C2×#125—1461no—
368C2×C4×Q8—731no—
369C2×C4×Q8—731no—
370C2×C4×Q8—731no—
371C2×C4×Q8—731no—
372C2×C4×Q8—731no—
373C2×C4×Q8—1461no—
374C2×C4×Q8—1461no—
375C2×C4×Q8C2×D7311yescharacteristic
376C2×C4×Q8C2×D7311yescharacteristic
377#501—738no—
378#501—738no—
379#498—738no—
380#479—738no—
381#479—738no—
382#477—738no—
383#475—738no—
384#475—738no—
385#475—738no—
386#446—738no—
387#441—738no—
388#441C2×D7318yes—
389C4×Q16—738no—
390C2×#107—731no—
391C2×#107—731no—
392C2×#107—731no—
393C2×#106—731no—
394C2×#106—731no—
395C2×#106—731no—
396C2×#106—731no—
397C2×#106—731no—
398C2×#106C2×D7311yescharacteristic
399C2×C4⋊C8—731no—
400C2×C4⋊C8—731no—
401C2×C4⋊C8—731no—
402C2×#103—731no—
403C2×#103—731no—
404C2×#103—731no—
405C2×#103—731no—
406C2×#103C2×D7311yescharacteristic
407C2×#103C2×D7311yescharacteristic
408C2×#97—731no—
409C2×#97C2×D7311yescharacteristic
410C2×C4×C8—731no—
411C22×Q8—731no—
412C22×Q8—1461no—
413C22×Q8—1461no—
414C22×Q8—1461no—
415C22×Q8—1461no—
416C22×Q8C22×D7311yescharacteristic
417C22×Q8C22×D7311yescharacteristic
418C2×Q16Dic14618yescomplemented
419C2×Q16C22×D7314yes—
420C22×C8—731no—
421C22×C8—731no—
422C22×C8C22×D7311yescharacteristic
423C22×C8C22×D7311yescharacteristic
424C4⋊Q8—734no—
425C4⋊Q8—734no—
426C4⋊Q8—734no—
427C4⋊Q8—734no—
428C4⋊Q8—1468no—
429C4⋊Q8—1462no—
430C4⋊Q8—1462no—
431C4⋊Q8—1462no—
432C4⋊Q8—1468no—
433C4⋊Q8—1468no—
434C4⋊Q8—1468no—
435C4⋊Q8—1462no—
436#125—734no—
437#125—734no—
438#125—1468no—
439#125—1462no—
440#125—1468no—
441#125—1462no—
442C4×Q8—22no—
443C4×Q8—22no—
444C4×Q8—734no—
445C4×Q8—734no—
446C4×Q8—734no—
447C4×Q8—734no—
448C4×Q8—734no—
449C4×Q8—734no—
450C4×Q8—1468no—
451C4×Q8—1468no—
452C4×Q8—1462no—
453C4×Q8—1462no—
454C4×Q8—1462no—
455C4×Q8—1462no—
456C4×Q8C22×D7314yes—
457C4×Q8C22×D7314yes—
458C2×C4⋊C4—21no—
459C2×C4⋊C4—21no—
460C2×C4⋊C4—731no—
461C2×C4⋊C4—731no—
462C2×C4⋊C4—731no—
463C2×C4⋊C4—731no—
464C2×C4⋊C4—731no—
465C2×C4⋊C4—731no—
466C2×C4⋊C4—731no—
467C2×C4⋊C4—1461no—
468C2×C4⋊C4—1461no—
469C2×C4⋊C4—1461no—
470C2×C4⋊C4—1461no—
471C2×C4⋊C4—1461no—
472C2×C4⋊C4—1461no—
473C2×C4⋊C4—1461no—
474C2×C4⋊C4—1461no—
475C2×C4⋊C4—1461no—
476C2×C4⋊C4—1461no—
477C2×C4⋊C4—1461no—
478C2×C4⋊C4—1461no—
479C2×C4⋊C4C22×D7311yescharacteristic
480C2×C4⋊C4C22×D7311yescharacteristic
481C2×C42—21no—
482C2×C42—21no—
483C2×C42—731no—
484C2×C42—731no—
485C2×C42—1461no—
486C2×C42—1461no—
487C2×C42—1461no—
488C2×C42—1461no—
489C2×C42C22×D7311yescharacteristic
490#107—734no—
491#107—734no—
492#107—734no—
493#106—734no—
494#106—734no—
495#106—734no—
496#106—734no—
497#106—734no—
498#106C22×D7314yes—
499C4⋊C8—734no—
500C4⋊C8—734no—
501C4⋊C8—734no—
502#103—734no—
503#103—734no—
504#103—734no—
505#103—734no—
506#103C22×D7314yes—
507#103C22×D7314yes—
508#97—734no—
509#97C22×D7314yes—
510C4×C8—734no—
511C2×Q8—21no—
512C2×Q8—21no—
513C2×Q8—22no—
514C2×Q8—22no—
515C2×Q8—2924no—
516C2×Q8—5842no—
517C2×Q8—5842no—
518C2×Q8—5842no—
519C2×Q8—734no—
520C2×Q8—1464no—
521C2×Q8—1464no—
522C2×Q8—1464no—
523C2×Q8—1464no—
524C2×Q8—1464no—
525C2×Q8C2×Dic14614yes—
526C2×Q8C2×Dic14614yes—
527C2×Q8C23×D7312yes—
528C2×Q8C23×D7312yes—
529C22×C4—21no—
530C22×C4—21no—
531C22×C4—21no—
532C22×C4—21no—
533C22×C4—731no—
534C22×C4—731no—
535C22×C4—731no—
536C22×C4—731no—
537C22×C4—1461no—
538C22×C4—1461no—
539C22×C4—1461no—
540C22×C4—1461no—
541C22×C4D4×D7311yescharacteristic
542C22×C4D4×D7311yescharacteristic
543C22×C4C23×D7311yescharacteristic
544Q16C2×Dic146116yescomplemented
545C2×C8—22no—
546C2×C8—732no—
547C2×C8—732no—
548C2×C8—1462no—
549C2×C8—1462no—
550C2×C8C2×Dic14614yes—
551C2×C8C23×D7312yes—
552C2×C8C23×D7312yes—
553C4⋊C4—24no—
554C4⋊C4—24no—
555C4⋊C4—21no—
556C4⋊C4—21no—
557C4⋊C4—734no—
558C4⋊C4—734no—
559C4⋊C4—734no—
560C4⋊C4—732no—
561C4⋊C4—732no—
562C4⋊C4—732no—
563C4⋊C4—732no—
564C4⋊C4—1464no—
565C4⋊C4—1464no—
566C4⋊C4—1464no—
567C4⋊C4—1464no—
568C4⋊C4—1464no—
569C4⋊C4—1464no—
570C4⋊C4—1464no—
571C4⋊C4—1464no—
572C4⋊C4—1464no—
573C4⋊C4—1464no—
574C4⋊C4—1464no—
575C4⋊C4—1461no—
576C4⋊C4—1461no—
577C4⋊C4—1461no—
578C4⋊C4—1461no—
579C4⋊C4—1464no—
580C4⋊C4C23×D7312yes—
581C4⋊C4C23×D7312yes—
582C42—24no—
583C42—24no—
584C42—732no—
585C42—732no—
586C42—732no—
587C42—732no—
588C42—1464no—
589C42—1464no—
590C42—1464no—
591C42—1464no—
592C42D4×D7312yes—
593C42C23×D7312yes—
594C23C2×D4×D7311yescharacteristic
595Q8—22no—
596Q8—22no—
597Q8—24no—
598Q8—24no—
599Q8—2928no—
600Q8—5844no—
601Q8—5844no—
602Q8—5844no—
603Q8—1468no—
604Q8C22×Dic14614yes—
605Q8C22×Dic14614yes—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—2922no—
617C2×C4—2922no—
618C2×C4—2922no—
619C2×C4—2922no—
620C2×C4—734no—
621C2×C4—732no—
622C2×C4—732no—
623C2×C4—732no—
624C2×C4—732no—
625C2×C4—1462no—
626C2×C4—1462no—
627C2×C4—1462no—
628C2×C4—1462no—
629C2×C4—1462no—
630C2×C4—1462no—
631C2×C4—1462no—
632C2×C4C2×D4×D7312yes—
633C2×C4C2×D4×D7312yes—
634C2×C4C2×D4×D7311yescharacteristic
635C2×C4C22×Dic14612yes—
636C2×C4#1455663812yes—
637C2×C4C24×D7311yescharacteristic
638C8—22no—
639C8—1462no—
640C8—1462no—
641C8C22×Dic14614yes—
642C22D4×Dic14612yes—
643C22#137×D7312yes—
644C22#1798939612yes—
645C22C22×D4×D7311yescharacteristic
646C4—22no—
647C4—24no—
648C4—24no—
649C4—22no—
650C4—42no—
651C4—42no—
652C4—2922no—
653C4—2922no—
654C4—2922no—
655C4—2922no—
656C4—734no—
657C4—1462no—
658C4—1462no—
659C4—1462no—
660C4D4×Dic14612yes—
661C4C23×Dic14611yescharacteristic
662C4C2×#1455663811yescharacteristic
663C2#2262286414yesdirect product factor
664C2C2×D4×Dic14611yescharacteristic
665C2C2×#137×D7311yescharacteristic
666C2C2×#1798939611yescharacteristic
subgroupG/H
centerC23C2×D4×D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC2×#441×C73C2
socle, prime radicalC23×C73C22×D4
hypercenterC2×#441D73
hypocenterC73C2×#3387
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#441

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #22622864C2direct (N × H)
158. C2×Q16×C73Q8semidirect
245. Q16×C73C2×Q8semidirect
330. C2×C73#3387semidirect
350. C73C2×#3387semidirect
418. C2×Q16Dic146semidirect
544. Q16C2×Dic146semidirect
663. C2#22622864direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×Q8×Dic1462
C2×#179893132
#226228642
C2×#179889962
C2×#179889942
C2×#179889842
C2×#179888962
C2×#179888942
C2×#179888802
C2×#179888722
C2×#179888702
C2×#179887152
C2×#179886942
C2×Q16×Dic732
C2×#441×C732
C2×#338773