Finite Group Databasebeta

← #10690317

Subgroups — #10690317

Order 768.

750 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2329C211yesmaximal, characteristic
2C2×#5050C211yesmaximal, characteristic
3C2×#4538C211yesmaximal, characteristic
4#69525C211yesmaximal, characteristic
5#69424C211yesmaximal, characteristic
6#67624C211yesmaximal, characteristic
7#67613C211yesmaximal, characteristic
8#9040—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×C3×C22⋊Q8C2211yescharacteristic
11C22×#216—21no—
12C3×#391—21no—
13C3×#389—21no—
14C3×#387—21no—
15C3×#387—21no—
16C3×#385—21no—
17C3×#380C2211yescharacteristic
18C3×C4×C22⋊C4—21no—
19C2×C3×#95C2211yescharacteristic
20C2×C3⋊#95—21no—
21C2×C3⋊#95—21no—
22C2×C3⋊#95C2211yescharacteristic
23C2×#830C2211yescharacteristic
24C2×#830C2211yescharacteristic
25#5050—22no—
26#5050C414yescomplemented
27C2×#810—21no—
28C2×#810—21no—
29C2×#810—21no—
30#4538—22no—
31C2×#451—31no—
32#2329D311yescharacteristic
33#1787—31no—
34C2×#328—31no—
35#1446—31no—
36#1294—31no—
37#1280—31no—
38C22×C3×Q8D411yescharacteristic
39C23×C3×C4D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C23×Dic3—21no—
43C23×Dic3—21no—
44C22×Dic6—21no—
45C22×Dic6—21no—
46C22×Dic6—21no—
47C22×Dic6—21no—
48C22×Dic6—21no—
49C22×Dic6—42no—
50C22×Dic6—42no—
51C22×Dic6C2×C412yes—
52C3×C22⋊Q8—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8C2×C412yes—
55C3×C22⋊Q8D412yes—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C4⋊C4D411yescharacteristic
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C42—21no—
65C2×C3×C42—21no—
66C2×#216—24no—
67C2×#216—42no—
68C2×#216—42no—
69C22×C3⋊C8—21no—
70C22×C3⋊C8—21no—
71C3×#95—21no—
72C3×#95—21no—
73C3×#95—21no—
74C3×#95—21no—
75C3×#95—21no—
76C3×#95—21no—
77C3×#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—42no—
81C3⋊#95—42no—
82#830—21no—
83#830—21no—
84#830—21no—
85#830C2×C412yes—
86#810—24no—
87#810—42no—
88#810—42no—
89C23×Q8—31no—
90C22×Q16—61no—
91C2×C22⋊Q8C2×D311yescharacteristic
92#451—34no—
93#451—62no—
94C2×#103—61no—
95C2×#103—61no—
96C2×#103—61no—
97C2×C22⋊C8—31no—
98C2×C22⋊C8—31no—
99#391—21no—
100#389—21no—
101#387—21no—
102#387—21no—
103#385—21no—
104#380C2×D311yescharacteristic
105C4×C22⋊C4—21no—
106C2×#95—31no—
107C2×#95—61no—
108C2×#95—61no—
109C2×#95C2×D311yescharacteristic
110#328—62no—
111C24×C3C2×D411yescharacteristic
112C2×C3×Q8—21no—
113C2×C3×Q8—41no—
114C2×C3×Q8—41no—
115C2×C3×Q8C22⋊C412yes—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—41no—
126C22×C3×C4—41no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4Q1612yescomplemented
132C22×C3×C4C22×C411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×Dic3—22no—
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—22no—
142C22×Dic3—41no—
143C22×Dic3—41no—
144C22×Dic3—41no—
145C22×Dic3—41no—
146C22×Dic3—41no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3—41no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C2×Dic6—22no—
153C2×Dic6—22no—
154C2×Dic6—22no—
155C2×Dic6—22no—
156C2×Dic6—22no—
157C2×Dic6—22no—
158C2×Dic6—44no—
159C2×Dic6—44no—
160C2×Dic6—44no—
161C2×Dic6—44no—
162C2×Dic6—41no—
163C2×Dic6—41no—
164C2×Dic6—41no—
165C2×Dic6—41no—
166C2×Dic6—41no—
167C2×Dic6—41no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—41no—
172C3×C4⋊C4—41no—
173C3×C4⋊C4—81no—
174C3×C4⋊C4—81no—
175C3×C4⋊C4C22⋊C412yes—
176C3×C22⋊C4—22no—
177C3×C22⋊C4—22no—
178C3×C22⋊C4—41no—
179C3×C22⋊C4—41no—
180C3×C42—81no—
181C3×C42—81no—
182#216—48no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8—41no—
185C2×C3⋊C8—41no—
186C2×C3⋊C8—41no—
187C2×C3⋊C8—41no—
188C2×C3⋊C8—41no—
189C22×Q8—32no—
190C22×Q8—61no—
191C22×Q8—61no—
192C22×Q8—61no—
193C22×Q8—61no—
194C22×Q8—61no—
195C22×Q8—122no—
196C22×Q8—122no—
197C22×Q8#6711yescharacteristic
198C23×C4—61no—
199C23×C4—61no—
200C23×C4#6711yescharacteristic
201C23×C4#6711yescharacteristic
202C23×C4C22×D311yescharacteristic
203C2×Q16—64no—
204C2×Q16—122no—
205C2×Q16—122no—
206C22×C8—61no—
207C22×C8—61no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8C4×D312yes—
211C22⋊Q8D1212yes—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4#6711yescharacteristic
216C2×C4⋊C4#6711yescharacteristic
217C2×C4⋊C4#6711yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C42—21no—
221C2×C42—21no—
222#103—64no—
223#103—122no—
224#103—122no—
225C22⋊C8—32no—
226C22⋊C8—61no—
227C22⋊C8—61no—
228C22⋊C8—61no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—61no—
237#95—61no—
238#95—122no—
239#95—122no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3C4×D411yescharacteristic
245C23×C3#12011yescharacteristic
246C23×C3#12111yescharacteristic
247C23×C3C2×Q1611yescharacteristic
248C23×C3#13511yescharacteristic
249C23×C3#13711yescharacteristic
250C23×C3#13711yescharacteristic
251C3×Q8—42no—
252C3×Q8—81no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—81no—
282C2×C3×C4—81no—
283C2×C3×C4C2×C22⋊C411yescharacteristic
284C2×C3×C4C4×D411yescharacteristic
285C2×C3×C4#12111yescharacteristic
286C2×C3×C4#12311yescharacteristic
287C2×Dic3—24no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—81no—
298C2×Dic3—81no—
299C2×Dic3—81no—
300C2×Dic3—81no—
301Dic6—24no—
302Dic6—24no—
303Dic6—42no—
304Dic6—42no—
305Dic6—42no—
306Dic6—42no—
307Dic6—42no—
308Dic6—42no—
309C3⋊C8—42no—
310C3⋊C8—81no—
311C24C2×#6711yescharacteristic
312C2×Q8—21no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—62no—
316C2×Q8—62no—
317C2×Q8—62no—
318C2×Q8—62no—
319C2×Q8—62no—
320C2×Q8—62no—
321C2×Q8—121no—
322C2×Q8—121no—
323C2×Q8—121no—
324C2×Q8—121no—
325C2×Q8—124no—
326C2×Q8—124no—
327C2×Q8—124no—
328C2×Q8—124no—
329C2×Q8—121no—
330C2×Q8—121no—
331C2×Q8(C2×D3)⋊C412yes—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—121no—
352C22×C4—121no—
353C22×C4—121no—
354C22×C4—121no—
355C22×C4—121no—
356C22×C4—121no—
357C22×C4—121no—
358C22×C4—121no—
359C22×C4—121no—
360C22×C4—121no—
361C22×C4#21612yescomplemented
362C22×C4C2×C4×D311yescharacteristic
363C22×C4C2×D1211yescharacteristic
364C22×C4#23511yescharacteristic
365C22×C4#23511yescharacteristic
366C22×C4C2×#6711yescharacteristic
367C22×C4C2×#6711yescharacteristic
368Q16—128no—
369C2×C8—62no—
370C2×C8—121no—
371C2×C8—121no—
372C2×C8—121no—
373C2×C8—121no—
374C2×C8—121no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—81no—
381C4⋊C4—81no—
382C4⋊C4(C2×D3)⋊C412yes—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C42—81no—
388C42—81no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3#38611yescharacteristic
402C22×C3C4×Q1611yescharacteristic
403C22×C3#44111yescharacteristic
404C22×C3#45111yescharacteristic
405C22×C3#45211yescharacteristic
406C22×C3#45511yescharacteristic
407C22×C3#45611yescharacteristic
408C22×C3B(4,𝔽2)11yescharacteristic
409C22×C3B(4,𝔽2)11yescharacteristic
410C22×C3#46211yescharacteristic
411C22×C3#46411yescharacteristic
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—42no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4—81no—
420C3×C4—81no—
421C3×C4—81no—
422Dic3—44no—
423Dic3—82no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23C4×#6711yescharacteristic
429C23#93011yescharacteristic
430C23#94211yescharacteristic
431C23C2×#21611yescharacteristic
432C23#95011yescharacteristic
433C23#95111yescharacteristic
434C23#95311yescharacteristic
435Q8—42no—
436Q8—64no—
437Q8—64no—
438Q8—81no—
439Q8—122no—
440Q8—122no—
441Q8—122no—
442Q8—122no—
443Q8—122no—
444Q8—122no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—64no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—241no—
486C2×C4—241no—
487C2×C4—241no—
488C2×C4—241no—
489C2×C4C2×(C2×D3)⋊C411yescharacteristic
490C2×C4C4×#6711yescharacteristic
491C2×C4#92911yescharacteristic
492C2×C4#93011yescharacteristic
493C8—122no—
494C8—241no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3#160211yescharacteristic
500C2×C3#160311yescharacteristic
501C2×C3#160511yescharacteristic
502C2×C3#160611yescharacteristic
503C2×C3#187811yescharacteristic
504C2×C3#188211yescharacteristic
505C2×C3#188911yescharacteristic
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22#502811yescharacteristic
519C22C4×#21611yescharacteristic
520C22#503011yescharacteristic
521C22#503111yescharacteristic
522C22#521311yescharacteristic
523C22#522811yescharacteristic
524C22#523911yescharacteristic
525C22#524211yescharacteristic
526C22#524411yescharacteristic
527C22#524411yescharacteristic
528C22#524611yescharacteristic
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—81no—
537C4—81no—
538C4—81no—
539C4—124no—
540C4—242no—
541C3#904011yescharacteristic, complemented, Sylow
542C2—21no—
543C2—21no—
544C2—21no—
545C2—21no—
546C2#7163211yescharacteristic
547C2#7165411yescharacteristic
548C2#7166611yescharacteristic
549C2#7188511yescharacteristic
550C2#7188911yescharacteristic
551C2#7189311yescharacteristic
552C2#7190411yescharacteristic
subgroupG/H
centerC23#953
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2329C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2329D3
hypocenterC3#9040
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2329

Complements

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

NH ≅ G/Ntype
26. #5050C4semidirect
131. C22×C3×C4Q16semidirect
361. C22×C4#216semidirect
541. C3#9040semidirect

Maximal subgroups

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

subgroupindex
C3×#23292
C2×#50502
C2×#45382
#695252
#694242
#676242
#676132
#90403