Finite Group Databasebeta

← C2×#22610939 #33112780

Subgroups — C2×#22610939 #33112780

Order 18688.

2182 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11924018C211yesmaximal, characteristic, complemented
2C22×#14556570C211yesmaximal, characteristic
3C22×#14556515C211yesmaximal, characteristic, complemented
4C22×#14556514C211yesmaximal, characteristic, complemented
5C22×C22⋊#10596831C211yesmaximal, characteristic
6#22610939C218yesmaximal, direct product factor
7C2×#17988349C211yesmaximal, characteristic, complemented
8C22×C22⋊Q8×C73C211yesmaximal, characteristic
9C23×Dic146C2211yescharacteristic
10C22×#11924018—24no—
11C22×#11924018—24no—
12C22×#11924018—24no—
13C22×#11924018C2214yescomplemented
14C23×#10596831C2211yescharacteristic
15C23×#10596831C2211yescharacteristic
16C2×#14556570—24no—
17C2×#14556570—24no—
18C2×#14556570—22no—
19C2×#14556570C2214yes—
20C22×C4⋊Dic73C2211yescharacteristic
21C22×#11924007—21no—
22C22×C22⋊Dic73—21no—
23C2×#14556515—24no—
24C2×#14556515C2214yescomplemented
25C2×#14556514—24no—
26C2×#14556514C2214yescomplemented
27C2×C22⋊#10596831—24no—
28C2×C22⋊#10596831C2214yes—
29#17988349C2214yescomplemented
30C23×Q8×C73C2211yescharacteristic
31C24×C4×C73C2211yescharacteristic
32C2×C22⋊Q8×C73—24no—
33C2×C22⋊Q8×C73—22no—
34C2×C22⋊Q8×C73—24no—
35C2×C22⋊Q8×C73C2214yes—
36C22×C4⋊C4×C73—21no—
37C22×C4⋊C4×C73C2211yescharacteristic
38C22×C22⋊C4×C73—21no—
39C22×Dic146—22no—
40C22×Dic146—21no—
41C22×Dic146—22no—
42C22×Dic146—22no—
43C22×Dic146—22no—
44C22×Dic146—22no—
45C22×Dic146D414yes—
46C22×Dic146C2312yes—
47C23×Dic73—21no—
48C23×Dic73—21no—
49C2×#11924018—232no—
50C2×#11924018—28no—
51C2×#11924018—28no—
52C2×#11924018—28no—
53C22×#10596831—22no—
54C22×#10596831—22no—
55C22×#10596831—22no—
56C22×#10596831—22no—
57C22×#10596831—22no—
58C22×#10596831D414yes—
59C22×#10596831C2312yes—
60C22×#10596831C2312yes—
61#14556570—216no—
62#14556570—28no—
63#14556570—28no—
64C2×C4⋊Dic73—22no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73D414yes—
67C2×C4⋊Dic73C2312yes—
68C2×#11924007—28no—
69C2×#11924007—24no—
70C2×C22⋊Dic73—28no—
71C2×C22⋊Dic73—24no—
72#14556515—28no—
73#14556514—28no—
74C22⋊#10596831—28no—
75C25×C73D411yescharacteristic
76C22×Q8×C73—22no—
77C22×Q8×C73—21no—
78C22×Q8×C73—22no—
79C22×Q8×C73—22no—
80C22×Q8×C73—22no—
81C22×Q8×C73—22no—
82C22×Q8×C73D414yes—
83C22×Q8×C73C2312yes—
84C23×C4×C73—21no—
85C23×C4×C73—21no—
86C23×C4×C73—22no—
87C23×C4×C73—22no—
88C23×C4×C73—22no—
89C23×C4×C73—22no—
90C23×C4×C73—22no—
91C23×C4×C73—22no—
92C23×C4×C73D411yescharacteristic
93C23×C4×C73D412yes—
94C23×C4×C73C2312yes—
95C23×C4×C73C2311yescharacteristic
96C22⋊Q8×C73—216no—
97C22⋊Q8×C73—28no—
98C22⋊Q8×C73—28no—
99C2×C4⋊C4×C73—28no—
100C2×C4⋊C4×C73—24no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73—22no—
103C2×C4⋊C4×C73D414yes—
104C2×C4⋊C4×C73C2312yes—
105C2×C22⋊C4×C73—28no—
106C2×C22⋊C4×C73—24no—
107C2×Dic146—28no—
108C2×Dic146—28no—
109C2×Dic146—22no—
110C2×Dic146—22no—
111C2×Dic146—22no—
112C2×Dic146—22no—
113C2×Dic146—24no—
114C2×Dic146—22no—
115C2×Dic146—42no—
116C2×Dic146—44no—
117C2×Dic146—42no—
118C2×Dic146—44no—
119C2×Dic146C2×D414yes—
120C22×Dic73—24no—
121C22×Dic73—24no—
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—42no—
125C22×Dic73—42no—
126C22×Dic73—42no—
127C22×Dic73—42no—
128#11924018—232no—
129C2×#10596831—28no—
130C2×#10596831—22no—
131C2×#10596831—22no—
132C2×#10596831—22no—
133C2×#10596831—22no—
134C2×#10596831—22no—
135C2×#10596831—44no—
136C2×#10596831C2×D414yes—
137C4⋊Dic73—24no—
138C4⋊Dic73—22no—
139C4⋊Dic73C2×D414yes—
140#11924007—216no—
141C22⋊Dic73—216no—
142C24×C73—22no—
143C24×C73—22no—
144C24×C73—22no—
145C24×C73—22no—
146C24×C73—22no—
147C24×C73—22no—
148C24×C73Q1614yescomplemented
149C24×C73C2×D412yes—
150C24×C73C2×D411yescharacteristic
151C2×Q8×C73—24no—
152C2×Q8×C73—22no—
153C2×Q8×C73—28no—
154C2×Q8×C73—28no—
155C2×Q8×C73—22no—
156C2×Q8×C73—22no—
157C2×Q8×C73—22no—
158C2×Q8×C73—22no—
159C2×Q8×C73—42no—
160C2×Q8×C73—44no—
161C2×Q8×C73—42no—
162C2×Q8×C73—44no—
163C2×Q8×C73C2×D414yes—
164C22×C4×C73—24no—
165C22×C4×C73—24no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73—22no—
169C22×C4×C73—22no—
170C22×C4×C73—22no—
171C22×C4×C73—24no—
172C22×C4×C73—24no—
173C22×C4×C73—24no—
174C22×C4×C73—22no—
175C22×C4×C73—22no—
176C22×C4×C73—22no—
177C22×C4×C73—22no—
178C22×C4×C73—22no—
179C22×C4×C73—22no—
180C22×C4×C73—42no—
181C22×C4×C73—42no—
182C22×C4×C73—42no—
183C22×C4×C73—42no—
184C22×C4×C73—44no—
185C22×C4×C73—44no—
186C22×C4×C73—44no—
187C22×C4×C73—44no—
188C22×C4×C73C2×D412yes—
189C22×C4×C73C2×D412yes—
190C22×C4×C73C2×D412yes—
191C22×C4×C73C2×D412yes—
192C22×C4×C73C2×D411yescharacteristic
193C22×C4×C73C2×D412yes—
194C22×C4×C73C2×D412yes—
195C22×C4×C73#4112yes—
196C22×C4×C73C2411yescharacteristic
197C4⋊C4×C73—216no—
198C4⋊C4×C73—24no—
199C4⋊C4×C73—22no—
200C4⋊C4×C73C2×D414yes—
201C22⋊C4×C73—216no—
202C23×C73—22no—
203C23×C73—22no—
204C23×C73—22no—
205C23×C73—22no—
206C23×C73—22no—
207C23×C73—22no—
208C23×C73—28no—
209C23×C73—28no—
210C23×C73—28no—
211C23×C73—44no—
212C23×C73—44no—
213C23×C73—44no—
214C23×C73—44no—
215C23×C73#12012yes—
216C23×C73#12012yes—
217C23×C73#12112yes—
218C23×C73C2×Q1612yes—
219C23×C73C2×Q1614yescomplemented
220C23×C73#13512yes—
221C23×C73#13712yes—
222C23×C73#13712yes—
223C23×C73C22×D411yescharacteristic
224Q8×C73—24no—
225Q8×C73—24no—
226Q8×C73—42no—
227Q8×C73—42no—
228Q8×C73—48no—
229C2×C4×C73—24no—
230C2×C4×C73—24no—
231C2×C4×C73—22no—
232C2×C4×C73—22no—
233C2×C4×C73—22no—
234C2×C4×C73—42no—
235C2×C4×C73—42no—
236C2×C4×C73—42no—
237C2×C4×C73—42no—
238C2×C4×C73—44no—
239C2×C4×C73—48no—
240C2×C4×C73—48no—
241C2×C4×C73—42no—
242C2×C4×C73—42no—
243C2×C4×C73—42no—
244C2×C4×C73—42no—
245C2×C4×C73—48no—
246C2×C4×C73#12014yes—
247C2×C4×C73#12012yes—
248C2×C4×C73#12012yes—
249C2×C4×C73#12114yes—
250C2×C4×C73#12114yes—
251C2×C4×C73#12112yes—
252C2×C4×C73#12714yes—
253C2×C4×C73C22×D411yescharacteristic
254C2×C4×C73C22×D411yescharacteristic
255C2×C4×C73C22×D411yescharacteristic
256C2×C4×C73C22×D411yescharacteristic
257C2×C4×C73C22×D411yescharacteristic
258C2×C4×C73C2×#4111yescharacteristic
259C2×Dic73—24no—
260C2×Dic73—24no—
261C2×Dic73—42no—
262C2×Dic73—42no—
263C2×Dic73—42no—
264C2×Dic73—42no—
265C2×Dic73—48no—
266C2×Dic73—48no—
267Dic146—24no—
268Dic146—24no—
269Dic146—42no—
270Dic146—42no—
271Dic146—48no—
272#10596831—24no—
273#10596831—42no—
274C22×C73—24no—
275C22×C73—24no—
276C22×C73—24no—
277C22×C73—42no—
278C22×C73—42no—
279C22×C73—42no—
280C22×C73—416no—
281C22×C73—42no—
282C22×C73#39214yes—
283C22×C73#45114yes—
284C22×C73#45114yes—
285C22×C73#45214yes—
286C22×C73#45214yes—
287C22×C73#46714yes—
288C22×C73#47014yes—
289C22×C73C2×#12011yescharacteristic
290C22×C73C2×#12011yescharacteristic
291C22×C73C2×#12111yescharacteristic
292C22×C73C22×Q1611yescharacteristic
293C22×C73C2×#13511yescharacteristic
294C22×C73C2×#13711yescharacteristic
295C22×C73C2×#13711yescharacteristic
296C4×C73—42no—
297C4×C73—44no—
298C4×C73—44no—
299C4×C73#39214yes—
300C4×C73C2×#12011yescharacteristic
301C4×C73C2×#12111yescharacteristic
302C4×C73C2×#12111yescharacteristic
303C4×C73C2×#12711yescharacteristic
304Dic73—44no—
305Dic73—44no—
306C2×#2001—731nomaximal, Sylow
307C2×C73—44no—
308C2×C73#200118yescomplemented
309C2×C73C2×#39211yescharacteristic
310C2×C73C2×#45111yescharacteristic
311C2×C73C2×#45111yescharacteristic
312C2×C73C2×#45211yescharacteristic
313C2×C73C2×#45211yescharacteristic
314C2×C73C2×#46711yescharacteristic
315C2×C73C2×#47011yescharacteristic
316C23×Q16—731no—
317C22×C22⋊Q8—731no—
318C22×C22⋊Q8D7311yescharacteristic
319C22×#103—731no—
320C22×#103—731no—
321C22×C22⋊C8—731no—
322#2001—738no—
323C2×#340—731no—
324C73C2×#200111yescharacteristic, complemented, Sylow
325C23×Q8—731no—
326C23×Q8C2×D7311yescharacteristic
327C24×C4C2×D7311yescharacteristic
328C22×Q16—734no—
329C22×Q16—1464no—
330C22×Q16—1464no—
331C22×Q16—1464no—
332C23×C8—731no—
333C23×C8—731no—
334C2×C22⋊Q8—24no—
335C2×C22⋊Q8—22no—
336C2×C22⋊Q8—24no—
337C2×C22⋊Q8—734no—
338C2×C22⋊Q8—1464no—
339C2×C22⋊Q8—1464no—
340C2×C22⋊Q8—1462no—
341C2×C22⋊Q8C2×D7314yes—
342C22×C4⋊C4—21no—
343C22×C4⋊C4—731no—
344C22×C4⋊C4—1461no—
345C22×C4⋊C4C2×D7311yescharacteristic
346C22×C22⋊C4—21no—
347C22×C22⋊C4—1461no—
348C2×#103—734no—
349C2×#103—734no—
350C2×#103—1464no—
351C2×#103—1464no—
352C2×C22⋊C8—734no—
353C2×C22⋊C8—1464no—
354#340—734no—
355C25C22⋊D7311yescharacteristic
356C22×Q8—22no—
357C22×Q8—21no—
358C22×Q8—22no—
359C22×Q8—22no—
360C22×Q8—22no—
361C22×Q8—22no—
362C22×Q8—734no—
363C22×Q8—732no—
364C22×Q8—1462no—
365C22×Q8—1462no—
366C22×Q8—1462no—
367C22×Q8—1462no—
368C22×Q8—1462no—
369C22×Q8—1461no—
370C22×Q8C22⋊D7314yes—
371C22×Q8C22×D7312yes—
372C23×C4—21no—
373C23×C4—21no—
374C23×C4—22no—
375C23×C4—22no—
376C23×C4—22no—
377C23×C4—22no—
378C23×C4—22no—
379C23×C4—22no—
380C23×C4—1461no—
381C23×C4—1461no—
382C23×C4C22⋊D7312yes—
383C23×C4C22⋊D7311yescharacteristic
384C23×C4C22×D7312yes—
385C23×C4C22×D7311yescharacteristic
386C2×Q16—1468no—
387C2×Q16—1468no—
388C2×Q16—14632no—
389C2×Q16—1468no—
390C22×C8—734no—
391C22×C8—732no—
392C22×C8—732no—
393C22×C8—1462no—
394C22×C8—1462no—
395C22×C8—1462no—
396C22×C8—1462no—
397C22×C8—1462no—
398C22⋊Q8—216no—
399C22⋊Q8—28no—
400C22⋊Q8—28no—
401C22⋊Q8—1468no—
402C22⋊Q8—1468no—
403C22⋊Q8—14616no—
404C2×C4⋊C4—28no—
405C2×C4⋊C4—24no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—734no—
409C2×C4⋊C4—732no—
410C2×C4⋊C4—1468no—
411C2×C4⋊C4—1462no—
412C2×C4⋊C4—1461no—
413C2×C4⋊C4—1464no—
414C2×C4⋊C4C22⋊D7314yes—
415C2×C4⋊C4C22×D7312yes—
416C2×C22⋊C4—28no—
417C2×C22⋊C4—24no—
418C2×C22⋊C4—1468no—
419C2×C22⋊C4—1464no—
420#103—1468no—
421#103—1468no—
422C22⋊C8—1468no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24#1192401814yescomplemented
430C24C2×C22⋊D7312yes—
431C24C2×C22⋊D7311yescharacteristic
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—28no—
435C2×Q8—28no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×Q8—2922no—
445C2×Q8—2924no—
446C2×Q8—2922no—
447C2×Q8—2924no—
448C2×Q8—734no—
449C2×Q8—1468no—
450C2×Q8—1468no—
451C2×Q8—1462no—
452C2×Q8—1462no—
453C2×Q8—1462no—
454C2×Q8—1462no—
455C2×Q8—1464no—
456C2×Q8—1462no—
457C2×Q8C2×C22⋊D7314yes—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—2922no—
483C22×C4—2922no—
484C22×C4—2922no—
485C22×C4—2922no—
486C22×C4—1464no—
487C22×C4—1464no—
488C22×C4—1462no—
489C22×C4—1462no—
490C22×C4D4×D7312yes—
491C22×C4D4×D7312yes—
492C22×C4D4×D7312yes—
493C22×C4C2×C22⋊D7312yes—
494C22×C4C2×C22⋊D7311yescharacteristic
495C22×C4C2×C22⋊D7312yes—
496C22×C4C2×C22⋊D7312yes—
497C22×C4#1192402712yes—
498C22×C4C23×D7311yescharacteristic
499Q16—14632no—
500C2×C8—2924no—
501C2×C8—734no—
502C2×C8—1462no—
503C2×C8—1462no—
504C2×C8—1462no—
505C2×C8—1468no—
506C2×C8—1462no—
507C2×C8—1462no—
508C4⋊C4—216no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—734no—
512C4⋊C4—14616no—
513C4⋊C4—1464no—
514C4⋊C4—1462no—
515C4⋊C4C2×C22⋊D7314yes—
516C22⋊C4—216no—
517C22⋊C4—14616no—
518C23—22no—
519C23—22no—
520C23—28no—
521C23—28no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—28no—
526C23—22no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23#1455655912yes—
532C23#1455656012yes—
533C23#1455656312yes—
534C23C2×#1192401812yes—
535C23C2×#1192401814yescomplemented
536C23#1455661112yes—
537C23#1455662112yes—
538C23#1455662212yes—
539C23C22×C22⋊D7311yescharacteristic
540Q8—24no—
541Q8—24no—
542Q8—42no—
543Q8—48no—
544Q8—42no—
545Q8—2922no—
546Q8—2928no—
547Q8—2922no—
548Q8—1464no—
549Q8—1464no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—42no—
556C2×C4—48no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—48no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—2922no—
568C2×C4—2928no—
569C2×C4—2928no—
570C2×C4—2922no—
571C2×C4—2922no—
572C2×C4—2922no—
573C2×C4—1464no—
574C2×C4—1464no—
575C2×C4#1455655912yes—
576C2×C4#1455655914yes—
577C2×C4#1455656012yes—
578C2×C4#1455656214yes—
579C2×C4#1455656312yes—
580C2×C4#1455656314yes—
581C2×C4#1455658714yes—
582C2×C4C2×D4×D7311yescharacteristic
583C2×C4C2×D4×D7311yescharacteristic
584C2×C4C2×D4×D7311yescharacteristic
585C2×C4C22×C22⋊D7311yescharacteristic
586C2×C4C22×C22⋊D7311yescharacteristic
587C2×C4C2×#1192402711yescharacteristic
588C8—2922no—
589C8—1464no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—416no—
598C22#1798849514yes—
599C22#1798874214yes—
600C22#1798874314yes—
601C22#1798874714yes—
602C22#1798874914yes—
603C22#1798882614yes—
604C22#1798884314yes—
605C22C2×#1455655911yescharacteristic
606C22C2×#1455656011yescharacteristic
607C22C2×#1455656311yescharacteristic
608C22C22×#1192401811yescharacteristic
609C22C2×#1455661111yescharacteristic
610C22C2×#1455662111yescharacteristic
611C22C2×#1455662211yescharacteristic
612C4—44no—
613C4—42no—
614C4—44no—
615C4—2924no—
616C4—2924no—
617C4#1798849514yes—
618C4C2×#1455655911yescharacteristic
619C4C2×#1455656211yescharacteristic
620C4C2×#1455656311yescharacteristic
621C4C2×#1455658711yescharacteristic
622C2—44no—
623C2#2261093918yesdirect product factor
624C2C2×#1798849511yescharacteristic
625C2C2×#1798874211yescharacteristic
626C2C2×#1798874311yescharacteristic
627C2C2×#1798874711yescharacteristic
628C2C2×#1798874911yescharacteristic
629C2C2×#1798882611yescharacteristic
630C2C2×#1798884311yescharacteristic
subgroupG/H
centerC24C2×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC22×C22⋊Q8×C73C2
socleC24×C73C2×D4
prime radicalC25×C73D4
hypercenterC22×C22⋊Q8D73
hypocenterC73C2×#2001
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C23×#11924018C2semidirect
3. C22×#14556515C2semidirect
4. C22×#14556514C2semidirect
6. #22610939C2direct (N × H)
7. C2×#17988349C2semidirect
13. C22×#11924018C22semidirect
24. C2×#14556515C22semidirect
26. C2×#14556514C22semidirect
29. #17988349C22semidirect
148. C24×C73Q16semidirect
219. C23×C73C2×Q16semidirect
308. C2×C73#2001semidirect
324. C73C2×#2001semidirect
429. C24#11924018semidirect
535. C23C2×#11924018semidirect
623. C2#22610939direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#119240182
C22×#145565702
C22×#145565152
C22×#145565142
C22×C22⋊#105968312
#226109392
C2×#179883492
C22×C22⋊Q8×C732
C2×#200173