Finite Group Databasebeta

← #10691454

Subgroups — #10691454

Order 768.

724 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3100C211yesmaximal, characteristic
2C4×D4×Dic3C211yesmaximal, characteristic, complemented
3#71815C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic, complemented
5#71598C211yesmaximal, characteristic, complemented
6#67746C211yesmaximal, characteristic, complemented
7#67736C211yesmaximal, characteristic, complemented
8#9233—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C2×D4×Dic3—21no—
12C22×C4×Dic3—21no—
13C3×#480—21no—
14C3×#474C412yescomplemented
15C3×#459C412yescomplemented
16C3×#447—21no—
17C3×C4×D8—22no—
18C3×#431C2211yescharacteristic
19C2×#837—21no—
20C2×#837—21no—
21C4×#217—21no—
22Dic3×C4⋊C4—21no—
23Dic3×C22⋊C4—22no—
24C4×C4⋊Dic3C2211yescharacteristic
25C42×Dic3C2211yescharacteristic
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4537—22no—
29D4⋊(C3⋊C8)—22no—
30#3100D311yescharacteristic
31C42×D4—31no—
32#1962—31no—
33#1502—31no—
34#1486—31no—
35#1303—31no—
36#1276—31no—
37C2×C3×#41—21no—
38C22×C3×D4D411yescharacteristic
39C23×C3×C4—21no—
40C23×Dic3—22no—
41C2×C3×D8—22no—
42C3×C4⋊Q8C2×C411yescharacteristic, complemented
43C3×#127C2×C411yescharacteristic, complemented
44C3×#124—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4C2×C412yes—
52C3×C4×D4D412yes—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C22⋊C4—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#217—22no—
57D4×Dic3—28no—
58C2×C4⋊Dic3—21no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3—22no—
61C2×C4×Dic3—21no—
62C2×C4×Dic3—21no—
63C2×C4×Dic3—42no—
64C2×C4×Dic3—42no—
65C2×C4×Dic3D411yescharacteristic
66C2×C4×Dic3D411yescharacteristic
67C2×#208—21no—
68C3×#107—22no—
69C3×C4⋊C8C2×C412yescomplemented
70C3×#102—22no—
71C3×#102—22no—
72C3×C22⋊C8—21no—
73C3×C4×C8—21no—
74#837—42no—
75#837—42no—
76C3⋊#95—22no—
77C3⋊#95D412yes—
78#830—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4×C3⋊C8—21no—
83#532C2×D311yescharacteristic
84C2×C4×D4—61no—
85C2×C4×D4C2×D311yescharacteristic
86C22×C42—61no—
87#480—21no—
88#474Dic312yescomplemented
89#459Dic312yescomplemented
90#447—21no—
91C4×D8—22no—
92#431—31no—
93#431C2×D311yescharacteristic
94#423—31no—
95C2×#104—61no—
96C2×#104—61no—
97C4×C4⋊C4—31no—
98C4×C4⋊C4—61no—
99C4×C22⋊C4—62no—
100C4×C22⋊C4—61no—
101C43—31no—
102#327—62no—
103D4⋊C8—62no—
104C24×C3—21no—
105C3×#41—41no—
106C3×#41—41no—
107C3×#41—41no—
108C3×#41—41no—
109C2×C3×Q8—21no—
110C2×C3×D4—24no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—41no—
116C2×C3×D4C22⋊C412yes—
117C22×C3×C4—21no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×Dic3—21no—
129C22×Dic3—28no—
130C22×Dic3—21no—
131C22×Dic3—22no—
132C22×Dic3—44no—
133C22×Dic3—44no—
134C3×D8—44no—
135C2×C3×C8—22no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4C22⋊C412yes—
140C3×C22⋊C4—22no—
141C3×C22⋊C4—22no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42C22×C411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147C3×C42#4111yescharacteristic
148#217—28no—
149C4⋊Dic3—24no—
150C4×Dic3—24no—
151C4×Dic3—21no—
152C4×Dic3—21no—
153C4×Dic3—21no—
154C4×Dic3—42no—
155C4×Dic3—44no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158C4×Dic3D812yescomplemented
159#208—41no—
160#208—41no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×#41—21no—
164C22×D4#6711yescharacteristic
165C23×C4—21no—
166C23×C4—62no—
167C2×D8—22no—
168C2×M16—61no—
169C4⋊Q8C2×Dic311yescharacteristic, complemented
170#127C2×Dic311yescharacteristic, complemented
171#124—21no—
172#121—21no—
173#121—21no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—68no—
179C4×D4C2×Dic312yes—
180C4×D4#6712yes—
181C2×C4⋊C4—61no—
182C2×C4⋊C4#6711yescharacteristic
183C2×C22⋊C4—21no—
184C2×C22⋊C4—62no—
185C2×C42—31no—
186C2×C42—31no—
187C2×C42—61no—
188C2×C42—62no—
189C2×C42—61no—
190C2×C42—61no—
191C2×C42—122no—
192C2×C42—122no—
193C2×C42C22×D311yescharacteristic
194#107—22no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198C4⋊C8C2×Dic312yescomplemented
199#104—122no—
200#104—122no—
201#102—22no—
202#102—22no—
203C22⋊C8—21no—
204C22⋊C8—61no—
205C4×C8—21no—
206C4×C8—61no—
207#95—32no—
208#95—62no—
209C23×C3—24no—
210C23×C3—22no—
211C23×C3—42no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3#12011yescharacteristic
216C3×Q8—41no—
217C3×Q8—41no—
218C3×D4—24no—
219C3×D4—22no—
220C3×D4—42no—
221C3×D4—41no—
222C3×D4—41no—
223C3×D4—81no—
224C3×D4—81no—
225C3×D4—81no—
226C3×D4#10414yescomplemented
227C2×C3×C4—22no—
228C2×C3×C4—22no—
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—81no—
238C2×C3×C4—81no—
239C2×C3×C4C2×C22⋊C411yescharacteristic
240C2×C3×C4C4×D411yescharacteristic
241C2×C3×C4C4×D411yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12311yescharacteristic
245C2×C3×C4C2×D811yescharacteristic
246C2×C3×C4#13511yescharacteristic
247C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
248C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
249C2×Dic3—22no—
250C2×Dic3—22no—
251C2×Dic3—22no—
252C2×Dic3—22no—
253C2×Dic3—24no—
254C2×Dic3—416no—
255C2×Dic3—41no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—41no—
259C3×C8—42no—
260C3⋊C8—81no—
261C3⋊C8—81no—
262C24—21no—
263#41—41no—
264#41—41no—
265#41—41no—
266#41—41no—
267C2×Q8—21no—
268C2×D4—24no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—41no—
274C2×D4#21712yes—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—61no—
284C22×C4—68no—
285C22×C4—62no—
286C22×C4—61no—
287C22×C4—124no—
288C22×C4—124no—
289C22×C4D3×D411yescharacteristic
290C22×C4D3×D411yescharacteristic
291C22×C4C2×#6711yescharacteristic
292D8—44no—
293M16—121no—
294M16—121no—
295C2×C8—22no—
296C2×C8—121no—
297C2×C8—121no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—64no—
302C4⋊C4#21712yes—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—68no—
308C42—32no—
309C42—64no—
310C42—61no—
311C42—61no—
312C42—61no—
313C42—124no—
314C42—122no—
315C42—121no—
316C42—121no—
317C42#23711yescharacteristic
318C42#23711yescharacteristic
319C42C22×Dic311yescharacteristic
320C42C2×#6711yescharacteristic
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—24no—
324C22×C3—44no—
325C22×C3—44no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3#38611yescharacteristic
332C22×C3#44711yescharacteristic
333C22×C3#44911yescharacteristic
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—42no—
338C3×C4—81no—
339C3×C4C2×#10412yes—
340C3×C4C4×D811yescharacteristic
341C3×C4#44211yescharacteristic
342C3×C4#45911yescharacteristic
343C3×C4#46311yescharacteristic
344Dic3—24no—
345Dic3—42no—
346Dic3—44no—
347C23—24no—
348C23—22no—
349C23—42no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23#93711yescharacteristic
354Q8—41no—
355Q8—41no—
356D4—24no—
357D4—22no—
358D4—41no—
359D4—41no—
360D4—42no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4#83714yescomplemented
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—64no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—1216no—
383C2×C4—121no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—121no—
387C2×C4D3×D811yescharacteristic
388C2×C4#91111yescharacteristic
389C2×C4#91111yescharacteristic
390C2×C4#91211yescharacteristic
391C2×C4D4×Dic311yescharacteristic
392C2×C4D4×Dic311yescharacteristic
393C2×C4#93511yescharacteristic
394C2×C4#93911yescharacteristic
395C2×C4#93911yescharacteristic
396C2×C4C2×#21711yescharacteristic
397C8—42no—
398C8—241no—
399C8—241no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—44no—
403C2×C3—161no—
404C2×C3#187911yescharacteristic
405C2×C3#189611yescharacteristic
406C2×C3#190611yescharacteristic
407C22—21no—
408C22—21no—
409C22—24no—
410C22—44no—
411C22—44no—
412C22—42no—
413C22—42no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22#515611yescharacteristic
418C22#515811yescharacteristic
419C22#522011yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—42no—
424C4—64no—
425C4—81no—
426C4—124no—
427C4—122no—
428C4Dic3×D811yescharacteristic
429C4#515011yescharacteristic
430C4#515211yescharacteristic
431C4#515311yescharacteristic
432C4C2×#83712yes—
433C3#923311yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—44no—
437C2—161no—
438C2#7171111yescharacteristic
439C2#7203711yescharacteristic
440C2#7211411yescharacteristic
subgroupG/H
centerC2×C4#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3100C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3100D3
hypocenterC3#9233
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
2. C4×D4×Dic3C2semidirect
3. #71815C2semidirect
4. #71811C2semidirect
5. #71598C2semidirect
6. #67746C2semidirect
7. #67736C2semidirect
14. C3×#474C4semidirect
15. C3×#459C4semidirect
42. C3×C4⋊Q8C2×C4semidirect
43. C3×#127C2×C4semidirect
69. C3×C4⋊C8C2×C4semidirect
88. #474Dic3semidirect
89. #459Dic3semidirect
158. C4×Dic3D8semidirect
169. C4⋊Q8C2×Dic3semidirect
170. #127C2×Dic3semidirect
198. C4⋊C8C2×Dic3semidirect
226. C3×D4#104semidirect
364. D4#837semidirect
433. C3#9233semidirect

Maximal subgroups

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

subgroupindex
C3×#31002
C4×D4×Dic32
#718152
#718112
#715982
#677462
#677362
#92333