Finite Group Databasebeta

← #10691227

Subgroups — #10691227

Order 768.

684 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82822C211yesmaximal, characteristic
2#74005C211yesmaximal, characteristic
3C3×#1520C211yesmaximal, characteristic, complemented
4#71690C211yesmaximal, characteristic, complemented
5C2×#4468C211yesmaximal, characteristic
6#67676C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#9199—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×(C2×D3)⋊C4C2211yescharacteristic
11C2×D3×C22⋊C4—21no—
12C2×C3×C22⋊C8C2211yescharacteristic
13C3×#380C2211yescharacteristic
14#5231C412yescomplemented
15#5212C2211yescharacteristic
16#5212C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#4994—22no—
19#4993C412yescomplemented
20#4989—22no—
21#4955—21no—
22#4946—21no—
23#4942—21no—
24#4668—22no—
25C3×#328—22no—
26#4538—22no—
27#4468—22no—
28#2626—31no—
29#2315—31no—
30#1520—31no—
31#1520D311yescharacteristic, complemented
32C2×#323—31no—
33#1289—31no—
34#1289—31no—
35C24×D3D411yescharacteristic
36C22×C3×Q8C2×C411yescharacteristic, complemented
37C23×C3×C4C2311yescharacteristic
38C23×Dic3D411yescharacteristic
39C22×C4×D3—22no—
40C22×C3×C8—21no—
41C3×C22⋊Q8—21no—
42C3×C22⋊Q8—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×(C2×D3)⋊C4—21no—
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—21no—
55C2×(C2×D3)⋊C4—21no—
56C2×(C2×D3)⋊C4—22no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4—22no—
59C2×(C2×D3)⋊C4C2×C412yes—
60C2×(C2×D3)⋊C4D412yes—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×Dic3⋊C4—22no—
64C2×Dic3⋊C4—22no—
65C22×C3⋊C8—21no—
66D3×C22⋊C4—42no—
67D3×C22⋊C4—42no—
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#95D412yes—
71C3⋊#95—22no—
72C3⋊#95—22no—
73C3⋊#95C2×C412yescomplemented
74C3⋊#95D412yes—
75#830—21no—
76#830—21no—
77C2×C22⋊Q8C2×D311yescharacteristic
78C22×C22⋊C4—31no—
79C22×C22⋊C4—61no—
80C2×C22⋊C8—31no—
81C2×C22⋊C8C2×D311yescharacteristic
82#399—32no—
83#396—62no—
84#393—32no—
85#385—62no—
86#385—61no—
87#385—62no—
88#380—31no—
89#380—31no—
90#380—61no—
91#380C2×D311yescharacteristic
92#379—61no—
93#328—22no—
94#328—62no—
95#323—62no—
96C24×C3C2×D411yescharacteristic
97C23×D3—21no—
98C23×D3—21no—
99C23×D3—21no—
100C23×D3—21no—
101C23×D3—81no—
102C23×D3—81no—
103C23×D3C22⋊C412yes—
104C23×D3SD1612yes—
105C23×D3Q1612yes—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—81no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—22no—
112C22×C3×C4—81no—
113C22×C3×C4C22×C411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C22×Dic3—21no—
120C22×Dic3—21no—
121C22×Dic3—22no—
122C22×Dic3—22no—
123C22×Dic3—81no—
124C22×Dic3C22⋊C412yes—
125C2×C4×D3—44no—
126C2×C4×D3—82no—
127C2×C4×D3—82no—
128C2×C3×C8—41no—
129C2×C3×C8—81no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—81no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139#217—41no—
140#217—41no—
141#217—41no—
142#217—41no—
143(C2×D3)⋊C4—22no—
144(C2×D3)⋊C4—22no—
145(C2×D3)⋊C4—22no—
146(C2×D3)⋊C4—22no—
147(C2×D3)⋊C4—44no—
148(C2×D3)⋊C4—82no—
149(C2×D3)⋊C4—82no—
150C4⋊Dic3—81no—
151C4⋊Dic3—81no—
152Dic3⋊C4—82no—
153Dic3⋊C4—82no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—81no—
156C25—31no—
157C22×Q8C4×D311yescharacteristic, complemented
158C23×C4—31no—
159C23×C4—62no—
160C23×C4C22×D311yescharacteristic
161C22×C8—21no—
162C22×C8—61no—
163C22⋊Q8—21no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166C22⋊Q8—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—62no—
170C2×C4⋊C4—62no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4C4×D311yescharacteristic, complemented
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—32no—
177C2×C22⋊C4—32no—
178C2×C22⋊C4—61no—
179C2×C22⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—62no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—62no—
186C2×C22⋊C4—62no—
187C2×C22⋊C4—122no—
188C2×C22⋊C4—122no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8—61no—
192C22⋊C8—61no—
193#95—32no—
194#95—32no—
195#95—62no—
196#95—62no—
197#95#6712yes—
198C23×C3—41no—
199C23×C3—41no—
200C23×C3C2×C22⋊C411yescharacteristic
201C23×C3#12011yescharacteristic
202C23×C3#12311yescharacteristic
203C23×C3C2×SD1611yescharacteristic
204C23×C3C2×Q1611yescharacteristic
205C23×C3AGL(1,ℤ/8)11yescharacteristic
206C23×C3#13711yescharacteristic
207C22×D3—22no—
208C22×D3—22no—
209C22×D3—22no—
210C22×D3—22no—
211C22×D3—22no—
212C22×D3—22no—
213C22×D3—42no—
214C22×D3—42no—
215C22×D3—42no—
216C22×D3—42no—
217C22×D3—82no—
218C22×D3—82no—
219C22×D3—81no—
220C22×D3—81no—
221C22×D3—82no—
222C22×D3—82no—
223C22×D3#10314yes—
224C3×Q8—81no—
225C3×Q8—81no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4C4×D411yescharacteristic
240C2×C3×C4C4×D411yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×Dic3—22no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—81no—
253C4×D3—88no—
254C3×C8—81no—
255C3⋊C8—81no—
256C24—32no—
257C24—32no—
258C24—32no—
259C24—61no—
260C24—61no—
261C24—61no—
262C24—61no—
263C24—241no—
264C24—241no—
265C24D3×D411yescharacteristic
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—81no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—32no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—62no—
276C22×C4—62no—
277C22×C4—81no—
278C22×C4—124no—
279C22×C4—242no—
280C22×C4—242no—
281C22×C4—241no—
282C22×C4C2×C4×D311yescharacteristic
283C22×C4#23511yescharacteristic
284C22×C4D3×D411yescharacteristic
285C22×C4D3×D411yescharacteristic
286C22×C4#23711yescharacteristic
287C22×C4C2×#6711yescharacteristic
288C2×C8—41no—
289C2×C8—81no—
290C2×C8—121no—
291C2×C8—241no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—81no—
297C4⋊C4—242no—
298C4⋊C4—242no—
299C4⋊C4—241no—
300C4⋊C4—241no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—62no—
306C22⋊C4—62no—
307C22⋊C4—62no—
308C22⋊C4—62no—
309C22⋊C4—124no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C22⋊C4—242no—
315C22⋊C4—242no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3#38611yescharacteristic
327C22×C3C2×#10311yescharacteristic
328C22×C3#41711yescharacteristic
329C22×C3#45011yescharacteristic
330C22×C3#45111yescharacteristic
331C22×C3#48311yescharacteristic
332C22×C3#48411yescharacteristic
333C2×D3—24no—
334C2×D3—44no—
335C2×D3—44no—
336C2×D3—42no—
337C2×D3—42no—
338C2×D3—44no—
339C2×D3—82no—
340C2×D3—82no—
341C2×D3—82no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—82no—
347C3×C4—81no—
348C3×C4—81no—
349Dic3—42no—
350Dic3—82no—
351Dic3—82no—
352C23—34no—
353C23—41no—
354C23—41no—
355C23—62no—
356C23—62no—
357C23—62no—
358C23—62no—
359C23—62no—
360C23—62no—
361C23—122no—
362C23—122no—
363C23—122no—
364C23—122no—
365C23—241no—
366C23—241no—
367C23—242no—
368C23—242no—
369C23—242no—
370C23—242no—
371C23D3×C22⋊C411yescharacteristic
372C23#88311yescharacteristic
373C23D3×SD1611yescharacteristic
374C23#91411yescharacteristic
375C23D3×Q1611yescharacteristic
376C23#91811yescharacteristic
377C23#93711yescharacteristic
378Q8—81no—
379Q8—81no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—62no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4—122no—
401C2×C4—122no—
402C2×C4—248no—
403C2×C4—241no—
404C2×C4#67⋊C411yescharacteristic
405C2×C4#88411yescharacteristic
406C2×C4C4×#6711yescharacteristic
407C2×C4#93811yescharacteristic
408C8—81no—
409C8—241no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3#187211yescharacteristic
417C2×C3#191111yescharacteristic
418C2×C3#191311yescharacteristic
419D3—44no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—64no—
431C22—124no—
432C22—124no—
433C22—124no—
434C22—122no—
435C22—122no—
436C22—242no—
437C22—242no—
438C22—242no—
439C22D3×#10311yescharacteristic
440C22#480311yescharacteristic
441C22#480511yescharacteristic
442C22#481111yescharacteristic
443C22#495611yescharacteristic
444C22#516911yescharacteristic
445C22#518611yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—82no—
451C4—81no—
452C4—81no—
453C4—122no—
454C4—242no—
455C4—242no—
456C3#919911yescharacteristic, complemented, Sylow
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—21no—
463C2—124no—
464C2#7003411yescharacteristic
465C2#7026811yescharacteristic
466C2#7040811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1520C2
socleC22×C3#386
prime radicalC24×D3D4
cosocleC23×C3×C4C23
hypercenter#1520D3
hypocenterC3#9199
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
3. C3×#1520C2semidirect
4. #71690C2semidirect
6. #67676C2semidirect
7. #67625C2semidirect
14. #5231C4semidirect
19. #4993C4semidirect
31. #1520D3semidirect
36. C22×C3×Q8C2×C4semidirect
47. C2×C3×C4⋊C4C2×C4semidirect
73. C3⋊#95C2×C4semidirect
157. C22×Q8C4×D3semidirect
173. C2×C4⋊C4C4×D3semidirect
456. C3#9199semidirect

Maximal subgroups

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

subgroupindex
#828222
#740052
C3×#15202
#716902
C2×#44682
#676762
#676252
#91993