Finite Group Databasebeta

← C2×#22608354 #33052376

Subgroups — C2×#22608354 #33052376

Order 18688.

1714 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D4×D73C211yesmaximal, characteristic
2C2×#17988638C211yesmaximal, characteristic, complemented
3C2×#17988609C211yesmaximal, characteristic, complemented
4#22608354C218yesmaximal, direct product factor
5C2×#17988316C211yesmaximal, characteristic, complemented
6C2×D4⋊#10596831C211yesmaximal, characteristic, complemented
7C2×#17988314C211yesmaximal, characteristic, complemented
8C2×D4⋊C8×C73C211yesmaximal, characteristic, complemented
9C22×D4×D73C411yescharacteristic
10C23×C4×D73—21no—
11C22×#11924011—21no—
12C2×C4×D292C2211yescharacteristic
13C2×D4×Dic73C411yescharacteristic
14C2×#14556553C411yescharacteristic
15C2×C4×C22⋊D73—21no—
16C2×#14556551—21no—
17C4×D4×D73—22no—
18C4×D4×D73—22no—
19C4×D4×D73C414yes—
20C4×D4×D73C2214yes—
21C2×C4⋊C4×D73C411yescharacteristic
22C2×C22⋊C4×D73—21no—
23C2×C42×D73C2211yescharacteristic
24#17988638—22no—
25#17988638C2214yescomplemented
26#17988609—22no—
27#17988609C2214yescomplemented
28C2×C4⋊#10596831C2211yescharacteristic, complemented
29C2×#14556520—21no—
30C2×#14556520—21no—
31C2×#14556503—21no—
32C2×C4×#10596831C2211yescharacteristic, complemented
33#17988316C2214yescomplemented
34D4⋊#10596831C2214yescomplemented
35#17988314C2214yescomplemented
36C2×C4×D4×C73C2211yescharacteristic
37C2×C4⋊C8×C73C2211yescharacteristic, complemented
38C2×C4×C8×C73C2211yescharacteristic, complemented
39D4⋊C8×C73C2214yescomplemented
40C24×D73—21no—
41C22×D292C2×C411yescharacteristic
42C22×C22⋊D73—21no—
43C2×D4×D73—24no—
44C2×D4×D73—22no—
45C2×D4×D73—24no—
46C2×D4×D73C2×C414yes—
47C23×Dic73—21no—
48C22×C4×D73—21no—
49C22×C4×D73—21no—
50C22×C4×D73—24no—
51C22×C4×D73—24no—
52C22×C4×D73—21no—
53C22×C4×D73—44no—
54C22×C4×D73—44no—
55C22×C4×D73C2×C411yescharacteristic
56C22×C4×D73D411yescharacteristic
57C22×C4×D73D411yescharacteristic
58C2×#11924011—24no—
59C2×#11924011—44no—
60C22×#10596831—21no—
61C22×#10596831—21no—
62C4×D292—21no—
63C4×D292—21no—
64C4×D292C2×C412yes—
65C4×D292C2312yes—
66D4×Dic73—22no—
67D4×Dic73C2×C414yes—
68#14556553—22no—
69#14556553C2×C414yes—
70C4×C22⋊D73—24no—
71C4×C22⋊D73—24no—
72#14556551—24no—
73#14556551—24no—
74C2×C4⋊Dic73C2×C411yescharacteristic
75C2×#11924007—21no—
76C4⋊C4×D73—22no—
77C4⋊C4×D73C2×C414yes—
78C2×C22⋊Dic73—21no—
79C2×#11924005—21no—
80C2×#11924005—21no—
81C22⋊C4×D73—24no—
82C22⋊C4×D73—24no—
83C2×C4×Dic73C2×C411yescharacteristic
84C2×C4×Dic73D411yescharacteristic
85C2×C4×Dic73D411yescharacteristic
86C42×D73C2×C412yes—
87C42×D73D412yes—
88C42×D73D412yes—
89C42×D73C2312yes—
90C4⋊#10596831—21no—
91C4⋊#10596831C2312yescomplemented
92#14556520—24no—
93#14556520—24no—
94#14556503—24no—
95C4×#10596831—21no—
96C4×#10596831C2312yescomplemented
97C22×D4×C73C2×C411yescharacteristic
98C23×C4×C73—21no—
99C22×C8×C73—21no—
100C22×C8×C73—21no—
101C4×D4×C73—21no—
102C4×D4×C73—21no—
103C4×D4×C73C2×C412yes—
104C4×D4×C73C2312yes—
105C2×C4⋊C4×C73C2×C411yescharacteristic
106C2×C22⋊C4×C73—21no—
107C2×C42×C73C2311yescharacteristic
108C4⋊C8×C73—21no—
109C4⋊C8×C73C2312yescomplemented
110C4×C8×C73—21no—
111C4×C8×C73C2312yescomplemented
112C23×D73—28no—
113C23×D73—21no—
114C23×D73—24no—
115C23×D73—44no—
116C23×D73—44no—
117C23×D73C22⋊C411yescharacteristic
118C2×D292—21no—
119C2×D292—22no—
120C2×D292M1614yescomplemented
121C2×D292C22×C412yes—
122C2×C22⋊D73—28no—
123C2×C22⋊D73—24no—
124D4×D73—216no—
125D4×D73—28no—
126D4×D73—28no—
127C22×Dic73—21no—
128C22×Dic73—24no—
129C22×Dic73—44no—
130C22×Dic73C22⋊C411yescharacteristic
131C22×Dic73C22⋊C411yescharacteristic
132C22×Dic73C22⋊C411yescharacteristic
133C2×C4×D73—22no—
134C2×C4×D73—22no—
135C2×C4×D73—24no—
136C2×C4×D73—22no—
137C2×C4×D73—22no—
138C2×C4×D73—24no—
139C2×C4×D73—42no—
140C2×C4×D73—44no—
141C2×C4×D73—42no—
142C2×C4×D73—42no—
143C2×C4×D73—44no—
144C2×C4×D73—416no—
145C2×C4×D73—48no—
146C2×C4×D73—48no—
147C2×C4×D73C22⋊C412yes—
148C2×C4×D73C22⋊C412yes—
149C2×C4×D73C22⋊C412yes—
150C2×C4×D73D814yes—
151C2×C4×D73SD1614yes—
152C2×C4×D73C22×C412yes—
153C2×C4×D73C2×D412yes—
154C2×C4×D73C2×D412yes—
155#11924011—48no—
156C2×#10596831—22no—
157C2×#10596831—22no—
158C2×#10596831—42no—
159C2×#10596831—42no—
160C4⋊Dic73—21no—
161C4⋊Dic73C22×C412yes—
162#11924007—24no—
163C22⋊Dic73—24no—
164#11924005—22no—
165#11924005—22no—
166#11924005—24no—
167C4×Dic73C22⋊C412yes—
168C4×Dic73C22⋊C412yes—
169C4×Dic73C22⋊C412yes—
170C4×Dic73C22×C412yes—
171C4×Dic73C2×D412yes—
172C4×Dic73C2×D412yes—
173C24×C73—21no—
174C2×D4×C73—21no—
175C2×D4×C73—22no—
176C2×D4×C73M1614yescomplemented
177C2×D4×C73C22×C412yes—
178C22×C4×C73—21no—
179C22×C4×C73—22no—
180C22×C4×C73—22no—
181C22×C4×C73—42no—
182C22×C4×C73—42no—
183C22×C4×C73C22×C411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C2×C8×C73—22no—
187C2×C8×C73—22no—
188C2×C8×C73—42no—
189C2×C8×C73—42no—
190C4⋊C4×C73—21no—
191C4⋊C4×C73C22×C412yes—
192C22⋊C4×C73—22no—
193C22⋊C4×C73—22no—
194C42×C73C22×C411yescharacteristic
195C42×C73C2×D411yescharacteristic
196C42×C73C2×D411yescharacteristic
197C42×C73C2411yescharacteristic
198C23×C73—22no—
199C23×C73—24no—
200C23×C73—42no—
201C23×C73—42no—
202C23×C73C2×C22⋊C411yescharacteristic
203C22×D73—22no—
204C22×D73—22no—
205C22×D73—216no—
206C22×D73—22no—
207C22×D73—24no—
208C22×D73—48no—
209C22×D73—432no—
210C22×D73—42no—
211C22×D73—42no—
212C22×D73—48no—
213C22×D73#10214yes—
214C22×D73C2×C22⋊C412yes—
215D4×C73—22no—
216D4×C73—24no—
217D4×C73C2×M1614yescomplemented
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—42no—
221C2×C4×C73—42no—
222C2×C4×C73—42no—
223C2×C4×C73—44no—
224C2×C4×C73C2×C22⋊C411yescharacteristic
225C2×C4×C73C2×C22⋊C411yescharacteristic
226C2×C4×C73C2×C22⋊C411yescharacteristic
227C2×C4×C73C2×M1612yes—
228C2×C4×C73C2×M1612yes—
229C2×C4×C73C2×D812yes—
230C2×C4×C73C2×SD1612yes—
231C2×C4×C73C23×C411yescharacteristic
232C2×C4×C73C22×D411yescharacteristic
233C2×C4×C73C22×D411yescharacteristic
234C22⋊D73—216no—
235C2×Dic73—22no—
236C2×Dic73—22no—
237C2×Dic73—22no—
238C2×Dic73—42no—
239C2×Dic73—48no—
240C2×Dic73#10214yes—
241C2×Dic73C2×C22⋊C412yes—
242C2×Dic73C2×C22⋊C412yes—
243C2×Dic73C2×C22⋊C412yes—
244D292—22no—
245D292—24no—
246D292C2×M1614yescomplemented
247C4×D73—22no—
248C4×D73—22no—
249C4×D73—24no—
250C4×D73—42no—
251C4×D73—42no—
252C4×D73—48no—
253C4×D73—42no—
254C4×D73—42no—
255C4×D73—44no—
256C4×D73—48no—
257C4×D73#10218yes—
258C4×D73C2×D814yes—
259C4×D73C2×SD1614yes—
260C8×C73—42no—
261C8×C73—42no—
262#10596831—42no—
263#10596831—42no—
264C22×C73—24no—
265C22×C73—42no—
266C22×C73—42no—
267C22×C73—48no—
268C22×C73#40712yes—
269C22×C73C2×#10212yes—
270C22×C73#42112yes—
271C22×C73C22×C22⋊C411yescharacteristic
272C2×D73—22no—
273C2×D73—22no—
274C2×D73—28no—
275C2×D73—24no—
276C2×D73—432no—
277C2×D73—42no—
278C2×D73—42no—
279C2×D73—48no—
280C2×D73C2×#10214yes—
281C4×C73—21no—
282C4×C73—21no—
283C4×C73—42no—
284C4×C73#40712yes—
285C4×C73C2×#10212yes—
286C4×C73C22×M1611yescharacteristic
287C4×C73C22×M1611yescharacteristic
288C4×C73C22×D811yescharacteristic
289C4×C73C22×SD1611yescharacteristic
290Dic73—22no—
291Dic73—22no—
292Dic73—42no—
293Dic73C2×#10214yes—
294C2×#1486—731nomaximal, Sylow
295C2×C73—44no—
296C2×C73#148614yescomplemented
297C2×C73C2×#40711yescharacteristic
298C2×C73C22×#10211yescharacteristic
299C2×C73C2×#42111yescharacteristic
300D73—24no—
301D73—44no—
302C22×C4×D4—731no—
303C2×#431—731no—
304C2×#423—731no—
305#1486—738no—
306C2×D4⋊C8—731no—
307C2×D4⋊C8—731no—
308C2×D4⋊C8—731no—
309C2×D4⋊C8D7311yescharacteristic, complemented
310C73C2×#148611yescharacteristic, complemented, Sylow
311C23×D4—731no—
312C24×C4—1461no—
313C22×M16—1461no—
314C2×C4×D4—731no—
315C2×C4×D4—734no—
316C2×C4×D4—734no—
317C2×C4×D4—731no—
318C2×C4×D4—731no—
319C2×C4×D4—1461no—
320C2×C4×D4—1462no—
321C2×C4×D4—1462no—
322C2×C4×D4—1461no—
323C2×C4×D4C2×D7311yescharacteristic
324C22×C4⋊C4—731no—
325C22×C22⋊C4—1461no—
326C22×C42—731no—
327#431—734no—
328#431—1462no—
329#423—734no—
330#423—1462no—
331C2×C4⋊C8—731no—
332C2×C4⋊C8—1461no—
333C2×C4⋊C8—1461no—
334C2×C4⋊C8C2×D7311yescharacteristic, complemented
335C2×C22⋊C8—1461no—
336C2×C4×C8—731no—
337C2×C4×C8C2×D7311yescharacteristic, complemented
338D4⋊C8—734no—
339D4⋊C8—734no—
340D4⋊C8—734no—
341D4⋊C8C2×D7314yescomplemented
342C25—1461no—
343C22×D4—734no—
344C22×D4—731no—
345C22×D4—1462no—
346C22×D4—1461no—
347C22×D4—1464no—
348C22×D4—1464no—
349C22×D4C4×D7311yescharacteristic
350C23×C4—21no—
351C23×C4—2924no—
352C23×C4—2924no—
353C23×C4—731no—
354C23×C4—731no—
355C23×C4—731no—
356C23×C4—1461no—
357C23×C4—1464no—
358C23×C4—1464no—
359C23×C4—1461no—
360C23×C4—1461no—
361C23×C4—1461no—
362C2×M16—2924no—
363C2×M16—1464no—
364C22×C8—21no—
365C22×C8—21no—
366C22×C8—1461no—
367C22×C8—1461no—
368C4×D4—21no—
369C4×D4—21no—
370C4×D4—734no—
371C4×D4—734no—
372C4×D4—732no—
373C4×D4—732no—
374C4×D4—1464no—
375C4×D4—1464no—
376C4×D4—1462no—
377C4×D4—1464no—
378C4×D4—1464no—
379C4×D4—1462no—
380C4×D4—1461no—
381C4×D4—1461no—
382C4×D4C4×D7312yes—
383C4×D4C22×D7312yes—
384C2×C4⋊C4—731no—
385C2×C4⋊C4—734no—
386C2×C4⋊C4—1462no—
387C2×C4⋊C4—1461no—
388C2×C4⋊C4C4×D7311yescharacteristic
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1464no—
391C2×C22⋊C4—1464no—
392C2×C22⋊C4—1461no—
393C2×C22⋊C4—1461no—
394C2×C22⋊C4—1461no—
395C2×C42—731no—
396C2×C42—731no—
397C2×C42—731no—
398C2×C42—732no—
399C2×C42—732no—
400C2×C42—732no—
401C2×C42—732no—
402C2×C42C22×D7311yescharacteristic
403C4⋊C8—21no—
404C4⋊C8—732no—
405C4⋊C8—1464no—
406C4⋊C8—1464no—
407C4⋊C8—1461no—
408C4⋊C8C22×D7312yescomplemented
409C22⋊C8—1464no—
410C4×C8—21no—
411C4×C8—732no—
412C4×C8—1461no—
413C4×C8C22×D7312yescomplemented
414C24—21no—
415C24—2924no—
416C24—2924no—
417C24—731no—
418C24—1464no—
419C24—1468no—
420C24—1461no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—734no—
424C2×D4—732no—
425C2×D4—1464no—
426C2×D4—1468no—
427C2×D4—14616no—
428C2×D4—1468no—
429C2×D4—1468no—
430C2×D4—1461no—
431C2×D4—1462no—
432C2×D4#1192401114yescomplemented
433C2×D4C2×C4×D7312yes—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—2922no—
440C22×C4—2924no—
441C22×C4—2924no—
442C22×C4—29216no—
443C22×C4—2928no—
444C22×C4—2928no—
445C22×C4—2924no—
446C22×C4—2922no—
447C22×C4—2922no—
448C22×C4—732no—
449C22×C4—732no—
450C22×C4—731no—
451C22×C4—732no—
452C22×C4—734no—
453C22×C4—734no—
454C22×C4—732no—
455C22×C4—731no—
456C22×C4—731no—
457C22×C4—732no—
458C22×C4—732no—
459C22×C4—1464no—
460C22×C4—1464no—
461C22×C4—1464no—
462C22×C4—1461no—
463C22×C4—1462no—
464C22×C4—1462no—
465C22×C4—1462no—
466C22×C4—1462no—
467C22×C4C2×C4×D7311yescharacteristic
468C22×C4D4×D7311yescharacteristic
469C22×C4D4×D7311yescharacteristic
470M16—2928no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—2922no—
476C2×C8—2922no—
477C2×C8—1462no—
478C2×C8—1462no—
479C4⋊C4—21no—
480C4⋊C4—732no—
481C4⋊C4—1464no—
482C4⋊C4—1461no—
483C4⋊C4C2×C4×D7312yes—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—1464no—
487C22⋊C4—1464no—
488C22⋊C4—1462no—
489C22⋊C4—1462no—
490C42—732no—
491C42—732no—
492C42—732no—
493C42—732no—
494C42—732no—
495C42—732no—
496C42C2×C4×D7311yescharacteristic
497C42D4×D7311yescharacteristic
498C42D4×D7311yescharacteristic
499C42C23×D7311yescharacteristic
500C23—22no—
501C23—24no—
502C23—42no—
503C23—42no—
504C23—2928no—
505C23—2928no—
506C23—29232no—
507C23—2922no—
508C23—2922no—
509C23—732no—
510C23—734no—
511C23—14616no—
512C23—1462no—
513C23—1464no—
514C23—1462no—
515C23—1462no—
516C23C22⋊C4×D7311yescharacteristic
517D4—24no—
518D4—22no—
519D4—734no—
520D4—14616no—
521D4—1464no—
522D4—1462no—
523D4C2×#1192401114yescomplemented
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—2922no—
531C2×C4—2922no—
532C2×C4—2928no—
533C2×C4—2928no—
534C2×C4—2922no—
535C2×C4—2928no—
536C2×C4—2924no—
537C2×C4—2922no—
538C2×C4—2922no—
539C2×C4—732no—
540C2×C4—734no—
541C2×C4—738no—
542C2×C4—734no—
543C2×C4—734no—
544C2×C4—732no—
545C2×C4—732no—
546C2×C4—1462no—
547C2×C4—1462no—
548C2×C4—1462no—
549C2×C4—1464no—
550C2×C4—1462no—
551C2×C4—1462no—
552C2×C4C22⋊C4×D7311yescharacteristic
553C2×C4C22⋊C4×D7311yescharacteristic
554C2×C4C22⋊C4×D7311yescharacteristic
555C2×C4M16×D7312yes—
556C2×C4C2×#1192401112yes—
557C2×C4D8×D7312yes—
558C2×C4SD16×D7312yes—
559C2×C4C22×C4×D7311yescharacteristic
560C2×C4C2×D4×D7311yescharacteristic
561C2×C4C2×D4×D7311yescharacteristic
562C8—42no—
563C8—42no—
564C8—2922no—
565C8—2922no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—48no—
570C22—29232no—
571C22—2922no—
572C22—2922no—
573C22—2928no—
574C22—734no—
575C22—1464no—
576C22—1462no—
577C22—1468no—
578C22—1462no—
579C22#1798854912yes—
580C22#102×D7312yes—
581C22#1798860212yes—
582C22C2×C22⋊C4×D7311yescharacteristic
583C4—21no—
584C4—21no—
585C4—42no—
586C4—2922no—
587C4—734no—
588C4—1462no—
589C4—1462no—
590C4#1798854912yes—
591C4#102×D7312yes—
592C4C2×M16×D7311yescharacteristic
593C4C22×#1192401111yescharacteristic
594C4C2×D8×D7311yescharacteristic
595C4C2×SD16×D7311yescharacteristic
596C2—44no—
597C2—2924no—
598C2—1464no—
599C2#2260835414yesdirect product factor
600C2C2×#1798854911yescharacteristic
601C2C2×#102×D7311yescharacteristic
602C2C2×#1798860211yescharacteristic
subgroupG/H
centerC22×C4D4×D73
commutatorC2×C4×C73C23×C4
FrattiniC42C23×D73
FittingC2×D4⋊C8×C73C2
socleC23×C73C2×C22⋊C4
prime radicalC22×D4×D73C4
cosocleC42×C73C24
hypercenterC2×D4⋊C8D73
hypocenterC73C2×#1486
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#17988638C2semidirect
3. C2×#17988609C2semidirect
4. #22608354C2direct (N × H)
5. C2×#17988316C2semidirect
6. C2×D4⋊#10596831C2semidirect
7. C2×#17988314C2semidirect
8. C2×D4⋊C8×C73C2semidirect
25. #17988638C22semidirect
27. #17988609C22semidirect
28. C2×C4⋊#10596831C22semidirect
32. C2×C4×#10596831C22semidirect
33. #17988316C22semidirect
34. D4⋊#10596831C22semidirect
35. #17988314C22semidirect
37. C2×C4⋊C8×C73C22semidirect
38. C2×C4×C8×C73C22semidirect
39. D4⋊C8×C73C22semidirect
91. C4⋊#10596831C23semidirect
96. C4×#10596831C23semidirect
109. C4⋊C8×C73C23semidirect
111. C4×C8×C73C23semidirect
120. C2×D292M16semidirect
176. C2×D4×C73M16semidirect
217. D4×C73C2×M16semidirect
246. D292C2×M16semidirect
296. C2×C73#1486semidirect
309. C2×D4⋊C8D73semidirect
310. C73C2×#1486semidirect
334. C2×C4⋊C8C2×D73semidirect
337. C2×C4×C8C2×D73semidirect
341. D4⋊C8C2×D73semidirect
408. C4⋊C8C22×D73semidirect
413. C4×C8C22×D73semidirect
432. C2×D4#11924011semidirect
523. D4C2×#11924011semidirect
599. C2#22608354direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×D4×D732
C2×#179886382
C2×#179886092
#226083542
C2×#179883162
C2×D4⋊#105968312
C2×#179883142
C2×D4⋊C8×C732
C2×#148673