Finite Group Databasebeta

← #34023855

Subgroups — #34023855

Order 18688.

1130 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626407C211yesmaximal, characteristic
2#22626407C211yesmaximal, characteristic
3#22626122C212yesmaximal, complemented
4#22626122C212yesmaximal, complemented
5#22626121C212yesmaximal, complemented
6#22626120C212yesmaximal, complemented
7#22625827C212yesmaximal
8#22625826C212yesmaximal
9C2×#17989212C211yesmaximal, characteristic
10#17989537—24no—
11#17989537—24no—
12#17989529C2212yes—
13#17989529C2212yes—
14#17989528C2212yes—
15#17989528C2212yes—
16C4×C22⋊#64408C2212yes—
17C4×C22⋊#64408C2212yes—
18#17989465C2214yes—
19#17989465C2214yes—
20#17989465C2214yes—
21#17989465C2214yes—
22#17989212—24no—
23#17989212—24no—
24C2×#14556571C411yescharacteristic, complemented
25C2×#14556571C2211yescharacteristic
26C2×#14556570C411yescharacteristic, complemented
27C2×#14556569C411yescharacteristic, complemented
28C2×#14556569C411yescharacteristic, complemented
29C2×#14556569C2211yescharacteristic
30C2×#14556569C2211yescharacteristic
31C2×#14556568C411yescharacteristic, complemented
32C2×#14556568C411yescharacteristic, complemented
33C2×#14556567C411yescharacteristic, complemented
34C2×#14556567C2211yescharacteristic
35C22×#11924005C2211yescharacteristic
36C2×C22⋊C4×D73C2211yescharacteristic
37C2×C22⋊C4×D73C2211yescharacteristic
38C2×C22⋊Q8×C73C411yescharacteristic, complemented
39C2×C4⋊#64408—22no—
40C2×C4⋊#64408—22no—
41C2×C22⋊#64408—22no—
42C2×C22⋊#64408—22no—
43C2×C22⋊#64408—22no—
44C2×C22⋊#64408—22no—
45C2×C22⋊#64408C2314yes—
46C2×C22⋊#64408C2314yes—
47C2×C4×#64408—22no—
48C2×C4×#64408—22no—
49(C2×#64408)⋊C4—22no—
50(C2×#64408)⋊C4—22no—
51(C2×#64408)⋊C4—22no—
52(C2×#64408)⋊C4—22no—
53C24×D73C2311yescharacteristic
54C22×Dic146C2×C411yescharacteristic
55C22×Dic146C2×C411yescharacteristic
56C22×Dic146C2×C411yescharacteristic
57C23×Dic73C2×C411yescharacteristic
58C22×C4×D73C2×C411yescharacteristic
59C22×C4×D73C2×C411yescharacteristic
60C22×C4×D73C2311yescharacteristic
61C22×C4×D73C2311yescharacteristic
62#14556571—22no—
63#14556571—22no—
64#14556571—24no—
65#14556570—24no—
66#14556569—22no—
67#14556569—22no—
68#14556569—22no—
69#14556569—22no—
70#14556569—24no—
71#14556569—24no—
72#14556568—24no—
73#14556568—24no—
74#14556567—22no—
75#14556567—22no—
76#14556567—24no—
77C2×C4⋊Dic73C2×C411yescharacteristic
78C2×C4⋊Dic73C2×C411yescharacteristic
79C2×C4⋊Dic73C2×C411yescharacteristic
80C2×#11924007C2×C411yescharacteristic
81C2×#11924007C2×C411yescharacteristic
82C2×#11924007C2×C411yescharacteristic
83C2×#11924007C2×C411yescharacteristic
84C2×#11924007C2×C411yescharacteristic
85C2×#11924007C2×C411yescharacteristic
86C2×C22⋊Dic73C2×C411yescharacteristic
87C2×C22⋊Dic73C2×C411yescharacteristic
88C2×#11924005—22no—
89C2×#11924005—22no—
90C2×#11924005C2×C411yescharacteristic
91C2×#11924005C2×C411yescharacteristic
92C2×#11924005C2×C411yescharacteristic
93C2×#11924005C2×C411yescharacteristic
94C2×#11924005C2311yescharacteristic
95C2×#11924005C2311yescharacteristic
96C2×#11924005C2311yescharacteristic
97C2×#11924005C2311yescharacteristic
98C22⋊C4×D73—22no—
99C22⋊C4×D73—22no—
100C22⋊C4×D73D414yes—
101C22⋊C4×D73D414yes—
102C22×Q8×C73C2×C411yescharacteristic
103C23×C4×C73C2×C411yescharacteristic
104C22⋊Q8×C73—24no—
105C2×C4⋊C4×C73C2×C411yescharacteristic
106C2×C4⋊C4×C73C2×C411yescharacteristic
107C2×C4⋊C4×C73C2×C411yescharacteristic
108C2×C22⋊C4×C73C2×C411yescharacteristic
109C2×C22⋊C4×C73C2×C411yescharacteristic
110C22×#64408—24no—
111C22×#64408—24no—
112C4⋊#64408—82no—
113C4⋊#64408—82no—
114C22⋊#64408—28no—
115C22⋊#64408—28no—
116C22⋊#64408—82no—
117C22⋊#64408—82no—
118C22⋊#64408—82no—
119C22⋊#64408—82no—
120C4×#64408—82no—
121C4×#64408—82no—
122C23×D73—24no—
123C23×D73—81no—
124C23×D73—81no—
125C23×D73C22×C411yescharacteristic
126C23×D73C2×D412yes—
127C23×D73C2×D412yes—
128C23×D73C2411yescharacteristic
129C2×Dic146—22no—
130C2×Dic146—22no—
131C2×Dic146—22no—
132C2×Dic146—81no—
133C2×Dic146—81no—
134C2×Dic146—81no—
135C22×Dic73—22no—
136C22×Dic73—81no—
137C22×Dic73C22×C411yescharacteristic
138C22×Dic73C22×C411yescharacteristic
139C22×Dic73C22×C411yescharacteristic
140C22×Dic73C22×C411yescharacteristic
141C22×Dic73C22×C411yescharacteristic
142C22×Dic73C22×C411yescharacteristic
143C2×C4×D73—22no—
144C2×C4×D73—22no—
145C2×C4×D73—81no—
146C2×C4×D73—81no—
147C2×C4×D73—81no—
148C2×C4×D73—81no—
149C2×C4×D73C2×D412yes—
150C2×C4×D73C2×D412yes—
151C2×C4×D73C2×D412yes—
152C2×C4×D73C2×D412yes—
153C4⋊Dic73—22no—
154C4⋊Dic73—22no—
155C4⋊Dic73—22no—
156#11924007—22no—
157#11924007—22no—
158#11924007—22no—
159#11924007—22no—
160#11924007—22no—
161#11924007—22no—
162C22⋊Dic73—22no—
163C22⋊Dic73C22⋊C414yes—
164#11924005—22no—
165#11924005—22no—
166#11924005—21no—
167#11924005—21no—
168#11924005—21no—
169#11924005—21no—
170#11924005—21no—
171#11924005—21no—
172#11924005—21no—
173#11924005—21no—
174#11924005C22⋊C414yes—
175#11924005C22⋊C414yes—
176C24×C73C22×C411yescharacteristic
177C2×Q8×C73—22no—
178C2×Q8×C73—81no—
179C22×C4×C73—22no—
180C22×C4×C73—81no—
181C22×C4×C73C22×C411yescharacteristic
182C22×C4×C73C22×C411yescharacteristic
183C22×C4×C73C22×C411yescharacteristic
184C22×C4×C73C22×C411yescharacteristic
185C22×C4×C73C22×C411yescharacteristic
186C22×C4×C73C22×C411yescharacteristic
187C4⋊C4×C73—22no—
188C4⋊C4×C73—22no—
189C4⋊C4×C73—22no—
190C22⋊C4×C73—22no—
191C22⋊C4×C73C22⋊C414yes—
192C23×C73—22no—
193C23×C73—41no—
194C23×C73—41no—
195C23×C73C2×C22⋊C412yes—
196C23×C73C23×C411yescharacteristic
197C22×D73—22no—
198C22×D73—24no—
199C22×D73—24no—
200C22×D73—41no—
201C22×D73—41no—
202C22×D73—41no—
203C22×D73—41no—
204C22×D73—42no—
205C22×D73—42no—
206C22×D73—42no—
207C22×D73—42no—
208C22×D73—88no—
209C22×D73C2×C22⋊C412yes—
210C22×D73C22×D411yescharacteristic
211C22×D73C22×D411yescharacteristic
212C22×D732−1+414yes—
213C2×#64408—44no—
214C2×#64408—44no—
215C2×#64408—44no—
216C2×#64408—44no—
217C2×#64408—44no—
218C2×#64408—44no—
219Q8×C73—82no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—41no—
225C2×C4×C73—41no—
226C2×C4×C73—41no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—41no—
230C2×C4×C73—82no—
231C2×C4×C73C2×C22⋊C412yes—
232C2×C4×C73C2×C22⋊C412yes—
233C2×Dic73—21no—
234C2×Dic73—21no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—41no—
240C2×Dic73—41no—
241C2×Dic73—41no—
242C2×Dic73—41no—
243C2×Dic73—82no—
244C2×Dic73C2×C22⋊C412yes—
245C2×Dic73C2×C22⋊C412yes—
246C4×D73—42no—
247C4×D73—42no—
248C4×D73—42no—
249C4×D73—42no—
250C4×D73—82no—
251C4×D73—82no—
252Dic146—82no—
253Dic146—82no—
254Dic146—82no—
255C22×C73—21no—
256C22×C73—21no—
257C22×C73—22no—
258C22×C73—22no—
259C22×C73—22no—
260C22×C73—44no—
261C22×C73C22×C22⋊C411yescharacteristic
262C22×C73#51912yes—
263C2×D73—21no—
264C2×D73—21no—
265C2×D73—21no—
266C2×D73—21no—
267C2×D73—22no—
268C2×D73—22no—
269C2×D73—22no—
270C2×D73—44no—
271C2×D73—44no—
272C2×D73—84no—
273C2×D73—84no—
274C2×D73#53612yes—
275C2×D73#53612yes—
276#64408—84no—
277#64408—84no—
278C4×C73—22no—
279C4×C73—22no—
280C4×C73—41no—
281C4×C73—41no—
282C4×C73—41no—
283C4×C73—41no—
284Dic73—22no—
285Dic73—22no—
286Dic73—41no—
287Dic73—41no—
288Dic73—41no—
289Dic73—41no—
290#21827—731nomaximal, Sylow
291C2×C73—22no—
292C2×C73—42no—
293C2×C73#231011yescharacteristic
294C2×C73#288311yescharacteristic
295C2×C73#288311yescharacteristic
296D73—22no—
297D73—22no—
298D73—42no—
299C2×#543—731no—
300#2312—731no—
301#2312—731no—
302#1996—732no—
303#1996—732no—
304#1996—732no—
305#1996—732no—
306#1819—732no—
307#1819—732no—
308C73#2182711yescharacteristic, complemented, Sylow
309#543—1464no—
310#543—1464no—
311C2×C22⋊Q8—731no—
312C2×C22⋊Q8—731no—
313C2×C22⋊Q8—731no—
314C2×C22⋊Q8—731no—
315C2×C22⋊Q8—731no—
316C2×C22⋊Q8—731no—
317C2×C22⋊Q8—731no—
318C2×C22⋊Q8—731no—
319C2×C22⋊Q8—731no—
320C2×C22⋊Q8—731no—
321C2×C22⋊Q8—731no—
322C2×C22⋊Q8#6440811yescharacteristic, complemented
323C22×C22⋊C4—731no—
324C22×C22⋊C4—731no—
325C22×C22⋊C4—731no—
326#388—1464no—
327#388—1464no—
328#380—732no—
329#380—732no—
330#379—732no—
331#379—732no—
332C4×C22⋊C4—732no—
333C4×C22⋊C4—732no—
334#342—734no—
335#342—734no—
336#342—734no—
337#342—734no—
338C25—731no—
339C22×Q8—731no—
340C22×Q8—731no—
341C22×Q8—731no—
342C22×Q8C2×#6440811yescharacteristic
343C23×C4—731no—
344C23×C4—731no—
345C23×C4—731no—
346C23×C4—731no—
347C23×C4—731no—
348C23×C4C2×#6440811yescharacteristic
349C22⋊Q8—24no—
350C22⋊Q8—1464no—
351C22⋊Q8—1464no—
352C22⋊Q8—1464no—
353C22⋊Q8—1464no—
354C22⋊Q8—1464no—
355C22⋊Q8—1464no—
356C22⋊Q8—1462no—
357C22⋊Q8—1462no—
358C22⋊Q8—1462no—
359C22⋊Q8—1462no—
360C22⋊Q8—1462no—
361C22⋊Q8—1462no—
362C22⋊Q8—1462no—
363C22⋊Q8—1462no—
364C22⋊Q8—1464no—
365C2×C4⋊C4—731no—
366C2×C4⋊C4—731no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—731no—
369C2×C4⋊C4—731no—
370C2×C4⋊C4—731no—
371C2×C4⋊C4—731no—
372C2×C4⋊C4—731no—
373C2×C4⋊C4—731no—
374C2×C4⋊C4—1462no—
375C2×C4⋊C4—1462no—
376C2×C4⋊C4C2×#6440811yescharacteristic
377C2×C4⋊C4C2×#6440811yescharacteristic
378C2×C4⋊C4C2×#6440811yescharacteristic
379C2×C22⋊C4—731no—
380C2×C22⋊C4—731no—
381C2×C22⋊C4—731no—
382C2×C22⋊C4—734no—
383C2×C22⋊C4—734no—
384C2×C22⋊C4—731no—
385C2×C22⋊C4—731no—
386C2×C22⋊C4—731no—
387C2×C22⋊C4—731no—
388C2×C22⋊C4—731no—
389C2×C22⋊C4—731no—
390C2×C22⋊C4—731no—
391C2×C22⋊C4—734no—
392C2×C22⋊C4—734no—
393C2×C22⋊C4—1462no—
394C2×C22⋊C4—1462no—
395C2×C22⋊C4—1462no—
396C2×C22⋊C4—1462no—
397C2×C22⋊C4—1462no—
398C2×C22⋊C4—1462no—
399C2×C22⋊C4—1462no—
400C2×C22⋊C4—1462no—
401C2×C22⋊C4C2×#6440811yescharacteristic
402C2×C22⋊C4C2×#6440811yescharacteristic
403C2×C42—1462no—
404C2×C42—1462no—
405#95—1462no—
406#95—1462no—
407#95—1462no—
408#95—1462no—
409C24—5841no—
410C24—5841no—
411C24—731no—
412C24—731no—
413C24—732no—
414C24—732no—
415C24—1464no—
416C24C22×#6440811yescharacteristic
417C2×Q8—22no—
418C2×Q8—81no—
419C2×Q8—5841no—
420C2×Q8—5841no—
421C2×Q8—5841no—
422C2×Q8—1462no—
423C2×Q8—1462no—
424C2×Q8—1462no—
425C22×C4—22no—
426C22×C4—81no—
427C22×C4—5841no—
428C22×C4—5841no—
429C22×C4—5841no—
430C22×C4—5841no—
431C22×C4—5841no—
432C22×C4—731no—
433C22×C4—731no—
434C22×C4—732no—
435C22×C4—732no—
436C22×C4—732no—
437C22×C4—732no—
438C22×C4—731no—
439C22×C4—731no—
440C22×C4—731no—
441C22×C4—731no—
442C22×C4—1462no—
443C22×C4—1462no—
444C22×C4—1464no—
445C22×C4—1464no—
446C22×C4—1462no—
447C22×C4C22×#6440811yescharacteristic
448C22×C4C22×#6440811yescharacteristic
449C22×C4C22×#6440811yescharacteristic
450C22×C4C22×#6440811yescharacteristic
451C22×C4C22×#6440811yescharacteristic
452C22×C4C22×#6440811yescharacteristic
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—5842no—
457C4⋊C4—5842no—
458C4⋊C4—1462no—
459C4⋊C4—1462no—
460C4⋊C4—1462no—
461C4⋊C4—1462no—
462C4⋊C4—1462no—
463C4⋊C4—1462no—
464C4⋊C4—1462no—
465C4⋊C4—1462no—
466C4⋊C4—1462no—
467C22⋊C4—22no—
468C22⋊C4—5842no—
469C22⋊C4—5842no—
470C22⋊C4—5842no—
471C22⋊C4—5842no—
472C22⋊C4—734no—
473C22⋊C4—734no—
474C22⋊C4—734no—
475C22⋊C4—1461no—
476C22⋊C4—1461no—
477C22⋊C4—1461no—
478C22⋊C4—1461no—
479C22⋊C4—1461no—
480C22⋊C4—1461no—
481C22⋊C4—1462no—
482C22⋊C4—1461no—
483C22⋊C4—1461no—
484C22⋊C4—1462no—
485C22⋊C4—1468no—
486C22⋊C4—1468no—
487C22⋊C4—1462no—
488C22⋊C4C22⋊#6440814yes—
489C42—5842no—
490C42—5842no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—2921no—
495C23—2921no—
496C23—2921no—
497C23—2921no—
498C23—2922no—
499C23—2922no—
500C23—2922no—
501C23—2922no—
502C23—5848no—
503C23—734no—
504C23—731no—
505C23—731no—
506C23—732no—
507C23—1464no—
508C23—1464no—
509C23—1462no—
510C23C2×C22⋊#6440812yes—
511C23C23×#6440811yescharacteristic
512Q8—82no—
513Q8—5842no—
514Q8—5842no—
515Q8—5842no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—82no—
527C2×C4—2922no—
528C2×C4—2922no—
529C2×C4—2922no—
530C2×C4—2922no—
531C2×C4—2921no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—2921no—
535C2×C4—2924no—
536C2×C4—2924no—
537C2×C4—2924no—
538C2×C4—2924no—
539C2×C4—2924no—
540C2×C4—2924no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—5842no—
544C2×C4—5842no—
545C2×C4—5842no—
546C2×C4—732no—
547C2×C4—732no—
548C2×C4—1461no—
549C2×C4—1461no—
550C2×C4—1461no—
551C2×C4—1461no—
552C2×C4C2×C22⋊#6440812yes—
553C2×C4C2×C22⋊#6440812yes—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—44no—
560C22—2924no—
561C22—2924no—
562C22—5844no—
563C22—5844no—
564C22—732no—
565C22—732no—
566C22—1462no—
567C22—1462no—
568C22—1462no—
569C22—1461no—
570C22—1461no—
571C22—1461no—
572C22—1461no—
573C22C22×C22⋊#6440811yescharacteristic
574C22#1798965512yes—
575C4—22no—
576C4—22no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—2921no—
582C4—2921no—
583C4—2921no—
584C4—2921no—
585C4—5844no—
586C4—5844no—
587C4—1462no—
588C4—1462no—
589C2—22no—
590C2—42no—
591C2—2922no—
592C2—1462no—
593C2—1462no—
594C2#2262640511yescharacteristic
595C2#2262653011yescharacteristic
596C2#2262653111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#64408
commutatorC23×C73C23×C4
FrattiniC23C23×#64408
FittingC2×C22⋊Q8×C73C4
socleC22×C73C22×C22⋊C4
prime radicalC24×D73C23
cosocleC23×D73C24
hypercenterC2×C22⋊Q8#64408
hypocenterC73#21827
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22626122C2semidirect
4. #22626122C2semidirect
5. #22626121C2semidirect
6. #22626120C2semidirect
24. C2×#14556571C4semidirect
26. C2×#14556570C4semidirect
27. C2×#14556569C4semidirect
28. C2×#14556569C4semidirect
31. C2×#14556568C4semidirect
32. C2×#14556568C4semidirect
33. C2×#14556567C4semidirect
38. C2×C22⋊Q8×C73C4semidirect
308. C73#21827semidirect
322. C2×C22⋊Q8#64408semidirect

Maximal subgroups

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

subgroupindex
#226264072
#226261222
#226261212
#226261202
#226258272
#226258262
C2×#179892122
#2182773