Finite Group Databasebeta

← #10690314

Subgroups — #10690314

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
1C2×#5228C211yesmaximal, characteristic
2#73874C211yesmaximal, characteristic
3C3×#1787C211yesmaximal, characteristic
4#72134C211yesmaximal, characteristic
5C2×#4467C211yesmaximal, characteristic
6#67674C211yesmaximal, characteristic
7#67624C211yesmaximal, characteristic
8#9040—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×#216—21no—
11C2×#924C2211yescharacteristic
12C2×C3×#103—21no—
13C2×C3×#103—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C3×#380C2211yescharacteristic
16#5228—22no—
17#5228C414yescomplemented
18C2×#835—21no—
19C2×C3⋊#95—21no—
20C2×C3⋊#95—21no—
21C2×C3⋊#95C2211yescharacteristic
22C2×C3⋊#95C2211yescharacteristic
23C2×#830C2211yescharacteristic
24#4978—21no—
25#4976—21no—
26#4946—21no—
27Dic3×C22⋊C4—21no—
28#4653—21no—
29#4646—21no—
30#4467—22no—
31C2×#451—31no—
32#2329—31no—
33#1787D311yescharacteristic
34C2×#328—31no—
35#1446—31no—
36#1294—31no—
37#1280—31no—
38C22×C3×Q8—21no—
39C22×C3×Q8—21no—
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—42no—
44C22×C3×Q8—42no—
45C22×C3×Q8C2×C412yes—
46C23×C3×C4—21no—
47C23×C3×C4C2311yescharacteristic
48C23×Dic3—21no—
49C23×Dic3D411yescharacteristic
50C23×Dic3D411yescharacteristic
51C22×Dic6D411yescharacteristic
52C22×C3×C8—21no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C22⋊C4—21no—
56C2×#217—21no—
57C2×#216—24no—
58C2×#216—42no—
59C2×#216—42no—
60C2×C4⋊Dic3D411yescharacteristic
61#924—21no—
62#924—21no—
63#924C2×C412yes—
64#924D412yes—
65C2×Dic3⋊C4—21no—
66C2×Dic3⋊C4—21no—
67C2×Dic3⋊C4—21no—
68C2×C4×Dic3—21no—
69C2×C4×Dic3—21no—
70C22×C3⋊C8—21no—
71C3×#103—42no—
72C3×#103—42no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×#95—21no—
76#835—24no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84C3⋊#95—21no—
85C3⋊#95—42no—
86C3⋊#95—42no—
87#830—21no—
88#830C2×C412yes—
89C23×Q8C2×D311yescharacteristic
90C22×Q16—61no—
91C2×C22⋊Q8—31no—
92#451—34no—
93#451—62no—
94C2×#103—21no—
95C2×#103—21no—
96C2×#103—61no—
97C2×C22⋊C8—31no—
98C2×C22⋊C8C2×D311yescharacteristic
99#391—61no—
100#389—61no—
101#387—61no—
102#387—61no—
103#385—61no—
104#380C2×D311yescharacteristic
105C4×C22⋊C4—61no—
106C2×#95—31no—
107C2×#95—31no—
108C2×#95—61no—
109C2×#95—61no—
110#328—62no—
111C24×C3C2×D411yescharacteristic
112C2×C3×Q8—22no—
113C2×C3×Q8—22no—
114C2×C3×Q8—22no—
115C2×C3×Q8—22no—
116C2×C3×Q8—22no—
117C2×C3×Q8—22no—
118C2×C3×Q8—44no—
119C2×C3×Q8—41no—
120C2×C3×Q8—41no—
121C2×C3×Q8—41no—
122C2×C3×Q8—41no—
123C2×C3×Q8—44no—
124C2×C3×Q8—44no—
125C2×C3×Q8—44no—
126C2×C3×Q8—41no—
127C2×C3×Q8—41no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—41no—
136C22×C3×C4—41no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4C22×C411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—21no—
148C22×Dic3—21no—
149C22×Dic3—22no—
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—41no—
154C22×Dic3—41no—
155C22×Dic3—41no—
156C22×Dic3—41no—
157C22×Dic3—41no—
158C22×Dic3—41no—
159C22×Dic3—41no—
160C22×Dic3—41no—
161C22×Dic3—41no—
162C22×Dic3—41no—
163C22×Dic3Q1612yescomplemented
164C2×Dic6—21no—
165C2×Dic6—41no—
166C2×Dic6—41no—
167C2×Dic6C22⋊C412yes—
168C2×C3×C8—41no—
169C2×C3×C8—41no—
170C2×C3×C8—41no—
171C3×C4⋊C4—41no—
172C3×C4⋊C4—41no—
173C3×C22⋊C4—41no—
174C3×C22⋊C4—41no—
175#217—22no—
176#217—22no—
177#216—48no—
178C4⋊Dic3—21no—
179C4⋊Dic3C22⋊C412yes—
180Dic3⋊C4—22no—
181Dic3⋊C4—22no—
182Dic3⋊C4—81no—
183Dic3⋊C4—81no—
184C4×Dic3—81no—
185C4×Dic3—81no—
186C2×C3⋊C8—22no—
187C2×C3⋊C8—41no—
188C2×C3⋊C8—41no—
189C22×Q8—21no—
190C22×Q8—21no—
191C22×Q8—21no—
192C22×Q8—21no—
193C22×Q8—21no—
194C22×Q8—31no—
195C22×Q8—42no—
196C22×Q8—42no—
197C22×Q8C4×D312yes—
198C23×C4—21no—
199C23×C4—31no—
200C23×C4—31no—
201C23×C4—61no—
202C23×C4C22×D311yescharacteristic
203C2×Q16—64no—
204C2×Q16—122no—
205C2×Q16—122no—
206C22×C8—21no—
207C22×C8—61no—
208C22⋊Q8—32no—
209C22⋊Q8—32no—
210C22⋊Q8—61no—
211C22⋊Q8—61no—
212C2×C4⋊C4—31no—
213C2×C4⋊C4—61no—
214C2×C4⋊C4—61no—
215C2×C4⋊C4—61no—
216C2×C4⋊C4#6711yescharacteristic
217C2×C4⋊C4#6711yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—61no—
220C2×C42—61no—
221C2×C42—61no—
222#103—42no—
223#103—42no—
224#103—64no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—32no—
228C22⋊C8—61no—
229#95—21no—
230#95—61no—
231#95—61no—
232#95—61no—
233#95—61no—
234#95—61no—
235#95—61no—
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—24no—
252C3×Q8—24no—
253C3×Q8—42no—
254C3×Q8—42no—
255C3×Q8—42no—
256C3×Q8—42no—
257C3×Q8—42no—
258C3×Q8—42no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—24no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
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×C4C2×C22⋊C411yescharacteristic
281C2×C3×C4C4×D411yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×C3×C4#12311yescharacteristic
284C2×Dic3—22no—
285C2×Dic3—22no—
286C2×Dic3—22no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—81no—
302C2×Dic3—81no—
303C2×Dic3—81no—
304C2×Dic3—81no—
305C2×Dic3—81no—
306C2×Dic3—81no—
307Dic6—42no—
308Dic6—81no—
309C3×C8—81no—
310C3⋊C8—42no—
311C24D3×D411yescharacteristic
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—32no—
319C2×Q8—44no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—61no—
330C2×Q8—121no—
331C2×Q8—121no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—32no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4—62no—
349C22×C4—61no—
350C22×C4—61no—
351C22×C4—61no—
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—121no—
362C22×C4C2×C4×D311yescharacteristic
363C22×C4#23511yescharacteristic
364C22×C4D3×D411yescharacteristic
365C22×C4D3×D411yescharacteristic
366C22×C4#23711yescharacteristic
367C22×C4C2×#6711yescharacteristic
368Q16—128no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—62no—
373C2×C8—121no—
374C2×C8—121no—
375C4⋊C4—32no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—61no—
379C4⋊C4—62no—
380C4⋊C4—62no—
381C4⋊C4—241no—
382C4⋊C4—241no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—62no—
386C22⋊C4—62no—
387C42—241no—
388C42—241no—
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—44no—
417C3×C4—81no—
418C3×C4—81no—
419Dic3—42no—
420Dic3—42no—
421Dic3—42no—
422Dic3—82no—
423Dic3—81no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23#67⋊C411yescharacteristic
429C23#88411yescharacteristic
430C23#91511yescharacteristic
431C23#91611yescharacteristic
432C23D3×Q1611yescharacteristic
433C23#91811yescharacteristic
434C23#93711yescharacteristic
435Q8—24no—
436Q8—24no—
437Q8—42no—
438Q8—42no—
439Q8—42no—
440Q8—42no—
441Q8—42no—
442Q8—42no—
443Q8—122no—
444Q8—241no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—121no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—241no—
484C2×C4—241no—
485C2×C4—241no—
486C2×C4—241no—
487C2×C4—241no—
488C2×C4—241no—
489C2×C4D3×C22⋊C411yescharacteristic
490C2×C4#88311yescharacteristic
491C2×C4C4×#6711yescharacteristic
492C2×C4#93811yescharacteristic
493C8—81no—
494C8—122no—
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#478911yescharacteristic
519C22#479011yescharacteristic
520C22#479511yescharacteristic
521C22#479811yescharacteristic
522C22#495611yescharacteristic
523C22#503211yescharacteristic
524C22#503211yescharacteristic
525C22#517311yescharacteristic
526C22#518611yescharacteristic
527C22#520211yescharacteristic
528C22#520511yescharacteristic
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—44no—
534C4—81no—
535C4—81no—
536C4—122no—
537C4—122no—
538C4—122no—
539C4—242no—
540C4—241no—
541C3#904011yescharacteristic, complemented, Sylow
542C2—21no—
543C2—21no—
544C2—21no—
545C2—21no—
546C2#6949911yescharacteristic
547C2#7028111yescharacteristic
548C2#7034811yescharacteristic
549C2#7097311yescharacteristic
550C2#7097411yescharacteristic
551C2#7098011yescharacteristic
552C2#7098311yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1787C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1787D3
hypocenterC3#9040
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
17. #5228C4semidirect
163. C22×Dic3Q16semidirect
541. C3#9040semidirect

Maximal subgroups

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

subgroupindex
C2×#52282
#738742
C3×#17872
#721342
C2×#44672
#676742
#676242
#90403