Finite Group Databasebeta

← #10692486

Subgroups — #10692486

Order 768.

530 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 95 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#82954C211yesmaximal, characteristic, complemented
2#82947C211yesmaximal, characteristic
3C3×#1527C211yesmaximal, characteristic, complemented
4#70835C211yesmaximal, characteristic, complemented
5#69867C211yesmaximal, characteristic, complemented
6#67750C211yesmaximal, characteristic, complemented
7#67666C211yesmaximal, characteristic, complemented
8#9381—31nomaximal, Sylow
9C2×C3×#124C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13#5229—21no—
14#5227—21no—
15#5220—21no—
16#5217—21no—
17C2×#825—21no—
18C2×#825—21no—
19#5058C412yescomplemented
20#5057—22no—
21#5055C412yescomplemented
22#4951—21no—
23#4951—21no—
24#4950—21no—
25Dic3×C22⋊C4—21no—
26#4937C2211yescharacteristic
27C4×C4⋊Dic3C2211yescharacteristic
28#4926C2211yescharacteristic
29C3×#329—22no—
30#4470—21no—
31#4470—21no—
32#3213—31no—
33#2337—31no—
34#1980—31no—
35#1527D311yescharacteristic, complemented
36#1514—31no—
37#1288—31no—
38#1285—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4D411yescharacteristic
41C23×Dic3—21no—
42C2×#235—21no—
43C2×C3×M16—21no—
44C3×#124—21no—
45C3×#124—21no—
46C3×#124C2×C412yes—
47C3×#124D412yes—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—21no—
54C2×#215—22no—
55C2×#213—22no—
56#930—21no—
57#930—21no—
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3—21no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3D411yescharacteristic
62C2×C4×Dic3D411yescharacteristic
63#875—21no—
64#874C2×C411yescharacteristic, complemented
65C4×D12—21no—
66C4×D12—21no—
67#869C2×C411yescharacteristic, complemented
68C3×C4⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×#97—21no—
72C3×C4×C8—21no—
73#835—22no—
74#832—22no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95D411yescharacteristic
81C3⋊#95D411yescharacteristic
82#830—21no—
83#825—42no—
84#825—42no—
85C4⋊(C3⋊C8)—21no—
86#803C2×C412yescomplemented
87#532—31no—
88C2×#124C2×D311yescharacteristic
89#496—32no—
90#489—32no—
91#463—62no—
92#448—61no—
93#447—61no—
94#432—31no—
95#432C2×D311yescharacteristic
96C2×#104—61no—
97C2×#104—61no—
98C4×M16C2×D311yescharacteristic
99#391—61no—
100#388—61no—
101#388—61no—
102#386—61no—
103#385—61no—
104#383—31no—
105C4×C4⋊C4—31no—
106C4×C22⋊C4—61no—
107#329—22no—
108#327—61no—
109#327—61no—
110C24×C3—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—81no—
113C2×C3×Q8C22⋊C412yes—
114C2×C3×D4—21no—
115C2×C3×D4—81no—
116C2×C3×D4C22⋊C412yes—
117C22×C3×C4—21no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C2×#67—41no—
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C22×Dic3—21no—
126C22×Dic3—42no—
127C22×Dic3—81no—
128C22×Dic3—81no—
129#235—41no—
130#235—41no—
131#235—41no—
132#235—41no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—21no—
136C2×C4×D3—41no—
137C2×Dic6—21no—
138C3×M16—22no—
139C3×M16—41no—
140C2×C3×C8—41no—
141C2×C3×C8—41no—
142C3×C22⋊C4—22no—
143C3×C22⋊C4—22no—
144C3×C22⋊C4—22no—
145C3×C22⋊C4—22no—
146C3×C42C22×C411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149C3×C42#4111yescharacteristic
150#217—42no—
151#217—81no—
152#217—81no—
153#215—82no—
154#213—82no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—41no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—42no—
160C4⋊Dic3—81no—
161C4×Dic3—41no—
162C4×Dic3—41no—
163C4×Dic3—81no—
164C2×C3⋊C8—22no—
165C2×#41—61no—
166C22×Q8D1211yescharacteristic
167C22×D4D1211yescharacteristic
168C23×C4—61no—
169C2×SD16—62no—
170C2×D8—62no—
171C2×M16—21no—
172C4⋊Q8—31no—
173#127—31no—
174#124—21no—
175#124—21no—
176#124—61no—
177#124C4×D312yes—
178#124#6712yes—
179#121—61no—
180#121—61no—
181C4×D4—61no—
182C4×D4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C42—31no—
191C2×C42—31no—
192C2×C42—61no—
193C2×C42C22×D311yescharacteristic
194C4⋊C8—21no—
195C4⋊C8—61no—
196#104—122no—
197#104—122no—
198#103—62no—
199#102—62no—
200C22⋊C8—21no—
201C22⋊C8—61no—
202#97—21no—
203#97—21no—
204#97—32no—
205C4×C8—21no—
206#95—31no—
207#95—31no—
208#95—61no—
209#95—61no—
210#95—61no—
211#95—61no—
212#95—61no—
213C23×C3—22no—
214C23×C3—41no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3#12011yescharacteristic
221C22×D3—41no—
222C3×Q8—42no—
223C3×Q8—81no—
224C3×D4—42no—
225C3×D4—81no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—22no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4C2×C22⋊C411yescharacteristic
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12311yescharacteristic
239#67—81no—
240#67—81no—
241C2×Dic3—22no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—84no—
255D12—42no—
256D12—41no—
257D12—41no—
258D12—81no—
259C4×D3—81no—
260C4×D3—81no—
261Dic6—41no—
262Dic6—41no—
263C3×C8—42no—
264C3×C8—81no—
265C3⋊C8—42no—
266C24—21no—
267#41—121no—
268#41—121no—
269#41—121no—
270#41—121no—
271C2×Q8—21no—
272C2×Q8—61no—
273C2×Q8—81no—
274C2×Q8(C2×D3)⋊C412yes—
275C2×D4—21no—
276C2×D4—61no—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—81no—
280C2×D4—121no—
281C2×D4(C2×D3)⋊C412yes—
282C22×C4—21no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—122no—
288C22×C4—121no—
289C22×C4—241no—
290C22×C4—241no—
291C22×C4C2×D1211yescharacteristic
292C22×C4D3×D411yescharacteristic
293C22×C4D3×D411yescharacteristic
294SD16—242no—
295D8—242no—
296M16—22no—
297M16—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—62no—
301C4⋊C4—61no—
302C4⋊C4—61no—
303C4⋊C4—122no—
304C4⋊C4—241no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—122no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C22⋊C4—241no—
313C22⋊C4—241no—
314C42—121no—
315C42—121no—
316C42—241no—
317C42C2×C4×D311yescharacteristic
318C42#23711yescharacteristic
319C42#23711yescharacteristic
320C42C2×#6711yescharacteristic
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—42no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3#38611yescharacteristic
331C22×C3#45311yescharacteristic
332C22×C3#45411yescharacteristic
333C2×D3—81no—
334C2×D3—81no—
335C2×D3—81no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—42no—
343Dic3—42no—
344Dic3—81no—
345Dic3—81no—
346Dic3—81no—
347Dic3—81no—
348C23—22no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—121no—
356C23#88211yescharacteristic
357Q8—42no—
358Q8—81no—
359Q8—121no—
360Q8—121no—
361D4—42no—
362D4—81no—
363D4—122no—
364D4—121no—
365D4—121no—
366D4—241no—
367D4—241no—
368D4—241no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—62no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—241no—
390C2×C4—241no—
391C2×C4—244no—
392C2×C4#67⋊C411yescharacteristic
393C2×C4#67⋊C411yescharacteristic
394C2×C4#88611yescharacteristic
395C2×C4C2×(C2×D3)⋊C411yescharacteristic
396C2×C4#93911yescharacteristic
397C2×C4#93911yescharacteristic
398C8—42no—
399C8—81no—
400C8—122no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—82no—
404C2×C3#188311yescharacteristic
405C2×C3#190611yescharacteristic
406C2×C3#191111yescharacteristic
407D3—161no—
408C22—21no—
409C22—21no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—241no—
418C22—241no—
419C22—241no—
420C22#482211yescharacteristic
421C22#482711yescharacteristic
422C22#495811yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—122no—
431C4—241no—
432C4—241no—
433C4—241no—
434C4—241no—
435C3#938111yescharacteristic, complemented, Sylow
436C2—21no—
437C2—21no—
438C2—82no—
439C2—481no—
440C2#7007111yescharacteristic
441C2#7036611yescharacteristic
442C2#7039511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1527C2
socleC22×C3#386
prime radical#82947C2
cosocleC2×C3×C42C23
hypercenter#1527D3
hypocenterC3#9381
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1527

Complements

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

NH ≅ G/Ntype
1. #82954C2semidirect
3. C3×#1527C2semidirect
4. #70835C2semidirect
5. #69867C2semidirect
6. #67750C2semidirect
7. #67666C2semidirect
19. #5058C4semidirect
21. #5055C4semidirect
35. #1527D3semidirect
64. #874C2×C4semidirect
67. #869C2×C4semidirect
86. #803C2×C4semidirect
435. C3#9381semidirect

Maximal subgroups

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

subgroupindex
#829542
#829472
C3×#15272
#708352
#698672
#677502
#676662
#93813