Finite Group Databasebeta

← #10692531

Subgroups — #10692531

Order 768.

496 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 89 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#78518C211yesmaximal, characteristic, complemented
2#73426C211yesmaximal, characteristic
3#70826C211yesmaximal, characteristic, complemented
4#70804C211yesmaximal, characteristic, complemented
5#69860C211yesmaximal, characteristic, complemented
6C3×#1300C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#9389—31nomaximal, Sylow
9C2×#875C2211yescharacteristic
10C2×#872C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×#376C2211yescharacteristic
14#5126—21no—
15#5118—21no—
16#4940—21no—
17#4939—21no—
18#4926C2211yescharacteristic
19C4×#208C2211yescharacteristic
20#4709—22no—
21#4690—22no—
22#4673C411yescharacteristic
23#4672C411yescharacteristic
24#4671C411yescharacteristic
25#4670—21no—
26#4665—21no—
27#4656C411yescharacteristic
28#4469—21no—
29#4468—21no—
30#4464—22no—
31#4464—22no—
32#2921—31no—
33#2610—31no—
34#1527—31no—
35#1527—31no—
36#1464—31no—
37#1300—31no—
38#1300D311yescharacteristic, complemented
39C22×D12—21no—
40C22×D12C2×C411yescharacteristic
41C22×C4×D3D411yescharacteristic
42C22×Dic6C2×C411yescharacteristic
43C2×C3×M16—21no—
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×(C2×D3)⋊C4—21no—
52C2×(C2×D3)⋊C4C2×C411yescharacteristic
53C2×(C2×D3)⋊C4D411yescharacteristic
54C2×C4⋊Dic3—21no—
55C2×Dic3⋊C4C2×C411yescharacteristic
56C2×Dic3⋊C4D411yescharacteristic
57C2×C4×Dic3D411yescharacteristic
58C2×#208—21no—
59#875—21no—
60#875—21no—
61#875—21no—
62#875—21no—
63#872—21no—
64#872—21no—
65#872—21no—
66#872—21no—
67C3×C4⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#97—21no—
70C3×#97D412yescomplemented
71C3×C4×C8—21no—
72C3×#95—21no—
73C3×#95C2×C411yescharacteristic
74C3×#95C2×C411yescharacteristic
75C3⋊#95—21no—
76C3⋊#95—21no—
77#830—21no—
78(C2×D3)⋊C8—24no—
79Dic3⋊C8—24no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82#803D412yescomplemented
83C4×C3⋊C8—21no—
84C2×#124—31no—
85C2×#117—31no—
86#433—62no—
87#433—62no—
88#432—31no—
89#432C2×D311yescharacteristic
90#423—61no—
91#407—61no—
92C4×M16—31no—
93C4×M16C2×D311yescharacteristic
94#395—31no—
95#394—31no—
96#393—31no—
97#392—31no—
98#390—61no—
99#388—61no—
100#388—61no—
101#386—61no—
102#376C2×D311yescharacteristic
103#329—62no—
104#329—62no—
105#324—61no—
106#323—61no—
107C23×D3—21no—
108C23×D3C22⋊C411yescharacteristic
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4C22×C411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×Dic3—21no—
115C22×Dic3C22⋊C411yescharacteristic
116C22×Dic3C22⋊C411yescharacteristic
117C22×Dic3C22⋊C411yescharacteristic
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—42no—
121C2×D12—42no—
122C2×D12—81no—
123C2×D12—81no—
124C2×C4×D3—21no—
125C2×C4×D3—21no—
126C2×C4×D3—41no—
127C2×C4×D3—41no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—81no—
131C3×M16—41no—
132C3×M16—41no—
133C2×C3×C8—22no—
134C2×C3×C8—41no—
135C3×C42—42no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42#4111yescharacteristic
139C3×C42#4111yescharacteristic
140(C2×D3)⋊C4—21no—
141(C2×D3)⋊C4—21no—
142(C2×D3)⋊C4—21no—
143(C2×D3)⋊C4—21no—
144(C2×D3)⋊C4—42no—
145(C2×D3)⋊C4—42no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—81no—
151(C2×D3)⋊C4—81no—
152C4⋊Dic3—81no—
153Dic3⋊C4—21no—
154Dic3⋊C4—21no—
155Dic3⋊C4—21no—
156Dic3⋊C4—21no—
157C4×Dic3—21no—
158C4×Dic3—21no—
159#208—41no—
160#208—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C22×Q8—31no—
164C22×D4—31no—
165C22×D4—61no—
166C23×C4—31no—
167C2×M16—21no—
168C2×M16—61no—
169#124—61no—
170#124—61no—
171#124—61no—
172#124—61no—
173#117—61no—
174#117—61no—
175#117—61no—
176#117—61no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—61no—
180C2×C22⋊C4—31no—
181C2×C22⋊C4—31no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C42—21no—
189C2×C42—31no—
190C2×C42C22×D311yescharacteristic
191C4⋊C8—21no—
192C4⋊C8—64no—
193C4⋊C8—61no—
194C22⋊C8—21no—
195C22⋊C8—64no—
196C22⋊C8—61no—
197#97—21no—
198#97—32no—
199#97—61no—
200#97D1212yescomplemented
201C4×C8—21no—
202C4×C8—61no—
203#95—21no—
204#95—61no—
205#95—61no—
206#95C4×D311yescharacteristic
207#95C4×D311yescharacteristic
208C23×C3C2×C22⋊C411yescharacteristic
209C22×D3—21no—
210C22×D3—21no—
211C22×D3—21no—
212C22×D3—41no—
213C22×D3—41no—
214C22×D3—41no—
215C22×D3—41no—
216C22×D3—41no—
217C22×D3—41no—
218C22×D3—41no—
219C22×D3—41no—
220C22×D3—41no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4C4×D411yescharacteristic
232C2×C3×C4C4×D411yescharacteristic
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12311yescharacteristic
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247D12—81no—
248D12—81no—
249D12—84no—
250C4×D3—42no—
251C4×D3—42no—
252Dic6—81no—
253Dic6—81no—
254C3×C8—42no—
255C3×C8—81no—
256C3⋊C8—42no—
257C3⋊C8—81no—
258C24—31no—
259C24—61no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×Q8—241no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—122no—
266C2×D4—122no—
267C2×D4—241no—
268C2×D4—241no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—121no—
278C22×C4—121no—
279C22×C4C2×C4×D311yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4D3×D411yescharacteristic
282M16—41no—
283M16—41no—
284M16—121no—
285M16—121no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—62no—
289C2×C8—121no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—241no—
295C22⋊C4—61no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—122no—
300C22⋊C4—122no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C22⋊C4—241no—
306C22⋊C4—241no—
307C42—42no—
308C42—61no—
309C42—61no—
310C42C2×D1211yescharacteristic
311C42#23511yescharacteristic
312C42D3×D411yescharacteristic
313C42#23911yescharacteristic
314C22×C3—21no—
315C22×C3—21no—
316C22×C3#38611yescharacteristic
317C22×C3#42111yescharacteristic
318C22×C3#42111yescharacteristic
319C2×D3—42no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—81no—
324C2×D3—81no—
325C2×D3—81no—
326C2×D3—81no—
327C2×D3—81no—
328C2×D3—81no—
329C2×D3—81no—
330C2×D3—81no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—42no—
338C3×C4—81no—
339Dic3—41no—
340Dic3—41no—
341Dic3—41no—
342Dic3—81no—
343C23—61no—
344C23—61no—
345C23—61no—
346C23—121no—
347C23—121no—
348C23—121no—
349C23—121no—
350C23—121no—
351C23—121no—
352C23—121no—
353C23—121no—
354C23—121no—
355C23D3×C22⋊C411yescharacteristic
356Q8—241no—
357Q8—241no—
358D4—241no—
359D4—241no—
360D4—244no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4C4×D1211yescharacteristic
384C2×C4#88211yescharacteristic
385C2×C4#88411yescharacteristic
386C2×C4#89311yescharacteristic
387C2×C4#89411yescharacteristic
388C2×C4#89511yescharacteristic
389C8—42no—
390C8—81no—
391C8—122no—
392C8—241no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3#188411yescharacteristic
396C2×C3#189711yescharacteristic
397C2×C3#191411yescharacteristic
398D3—81no—
399D3—161no—
400C22—21no—
401C22—21no—
402C22—122no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—241no—
407C22—241no—
408C22—241no—
409C22—241no—
410C22—241no—
411C22—241no—
412C22—241no—
413C22—241no—
414C22#466511yescharacteristic
415C22#482111yescharacteristic
416C22#482111yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—42no—
424C4—81no—
425C4—121no—
426C4—121no—
427C4—121no—
428C4—241no—
429C3#938911yescharacteristic, complemented, Sylow
430C2—21no—
431C2—21no—
432C2—241no—
433C2—481no—
434C2#6875711yescharacteristic
435C2#6877711yescharacteristic
436C2#6904011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1300C2
socleC22×C3#386
prime radical#4671C4
cosocleC2×C3×C42C23
hypercenter#1300D3
hypocenterC3#9389
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1300

Complements

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

NH ≅ G/Ntype
1. #78518C2semidirect
3. #70826C2semidirect
4. #70804C2semidirect
5. #69860C2semidirect
6. C3×#1300C2semidirect
7. #67615C2semidirect
38. #1300D3semidirect
70. C3×#97D4semidirect
82. #803D4semidirect
200. #97D12semidirect
429. C3#9389semidirect

Maximal subgroups

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

subgroupindex
#785182
#734262
#708262
#708042
#698602
C3×#13002
#676152
#93893