Finite Group Databasebeta

← #10691174

Subgroups — #10691174

Order 768.

478 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 91 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#79314C211yesmaximal, characteristic, complemented
2#74310C211yesmaximal, characteristic
3#70804C211yesmaximal, characteristic
4C4×#824C211yesmaximal, characteristic, complemented
5#69860C211yesmaximal, characteristic
6C3×#1288C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#9188—31nomaximal, Sylow
9C2×#875C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×#383C2211yescharacteristic
14C2×#824—21no—
15#4926C2211yescharacteristic
16C4×#208C2211yescharacteristic
17#4828—22no—
18#4719—21no—
19#4718—21no—
20#4701—22no—
21#4677—21no—
22#4673—21no—
23#4672C411yescharacteristic
24#4672C411yescharacteristic
25#4670—21no—
26#4670—21no—
27#4656C411yescharacteristic
28#4656C411yescharacteristic
29#4554—21no—
30#4464—22no—
31#4464—22no—
32#2983—31no—
33#2632—31no—
34C4×#101—31no—
35#1527—31no—
36#1527—31no—
37#1288—31no—
38#1288D311yescharacteristic, complemented
39C22×D12C2×C411yescharacteristic
40C22×Dic6C2×C411yescharacteristic
41C22×Dic6D411yescharacteristic
42C2×C3×M16—21no—
43C2×C3×C42C2311yescharacteristic
44C2×(C2×D3)⋊C4—21no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3D411yescharacteristic
53C2×Dic3⋊C4C2×C411yescharacteristic
54C2×Dic3⋊C4C2×C411yescharacteristic
55C2×C4×Dic3D411yescharacteristic
56C2×C4×Dic3D411yescharacteristic
57C2×#208—21no—
58#875—21no—
59#875—21no—
60#875—21no—
61#875—21no—
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6—21no—
65C4×Dic6—21no—
66C3×C4⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×#97—21no—
69C3×#97D412yes—
70C3×C4×C8—21no—
71C3×#95—21no—
72C3×#95—21no—
73C3×#95C2×C411yescharacteristic
74C3×#95C2×C411yescharacteristic
75C3⋊#95—21no—
76C3⋊#95—21no—
77#830—21no—
78#824—24no—
79#815—22no—
80Dic3⋊C8—22no—
81Dic3⋊C8—22no—
82C8×Dic3—22no—
83C4⋊(C3⋊C8)—21no—
84#803—21no—
85#803D412yes—
86C4×C3⋊C8—21no—
87C2×#124—31no—
88C2×C4×Q8—31no—
89#446—62no—
90#446—62no—
91#432—31no—
92#432—61no—
93#432C2×D311yescharacteristic
94#431—61no—
95C2×#101—61no—
96C4×M16—31no—
97C4×M16C2×D311yescharacteristic
98#399—61no—
99#395—31no—
100#395—31no—
101#394—31no—
102#394—31no—
103#393—61no—
104#388—61no—
105#388—61no—
106#383C2×D311yescharacteristic
107#343—61no—
108#329—62no—
109#329—62no—
110C23×D3—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4C22×C411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×Dic3—21no—
117C22×Dic3C22⋊C411yescharacteristic
118C22×Dic3C22⋊C411yescharacteristic
119C22×Dic3C22⋊C411yescharacteristic
120C22×Dic3C22⋊C411yescharacteristic
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—81no—
124C2×Dic6—21no—
125C2×Dic6—21no—
126C2×Dic6—22no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—41no—
130C2×Dic6—81no—
131C3×M16—22no—
132C3×M16—41no—
133C2×C3×C8—22no—
134C2×C3×C8—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138C3×C42#4111yescharacteristic
139(C2×D3)⋊C4—42no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—42no—
145(C2×D3)⋊C4—81no—
146(C2×D3)⋊C4—81no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—81no—
150C4⋊Dic3—81no—
151Dic3⋊C4—21no—
152Dic3⋊C4—21no—
153Dic3⋊C4—21no—
154Dic3⋊C4—21no—
155C4×Dic3—21no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158C4×Dic3—21no—
159#208—22no—
160#208—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C22×Q8—31no—
164C22×Q8—31no—
165C22×D4—31no—
166C2×M16—21no—
167C2×M16—61no—
168#124—61no—
169#124—61no—
170#124—61no—
171#124—61no—
172C4×Q8—61no—
173C4×Q8—61no—
174C4×Q8—61no—
175C4×Q8—61no—
176C2×C4⋊C4—31no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C22⋊C4—61no—
182C2×C22⋊C4—61no—
183C2×C22⋊C4—61no—
184C2×C22⋊C4—61no—
185C2×C22⋊C4—61no—
186C2×C22⋊C4—61no—
187C2×C42—31no—
188C2×C42—31no—
189C2×C42C22×D311yescharacteristic
190C4⋊C8—21no—
191C4⋊C8—62no—
192C4⋊C8—62no—
193C4⋊C8—61no—
194#101—64no—
195C22⋊C8—21no—
196C22⋊C8—61no—
197#97—21no—
198#97—32no—
199#97—62no—
200#97—61no—
201#97D1212yes—
202C4×C8—21no—
203C4×C8—62no—
204C4×C8—61no—
205#95—21no—
206#95—21no—
207#95—61no—
208#95—61no—
209#95C4×D311yescharacteristic
210#95C4×D311yescharacteristic
211C23×C3C2×C22⋊C411yescharacteristic
212C22×D3—41no—
213C22×D3—41no—
214C22×D3—41no—
215C22×D3—41no—
216C22×D3—41no—
217C22×D3—41no—
218C22×D3—41no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12311yescharacteristic
235C2×Dic3—21no—
236C2×Dic3—21no—
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—
247C2×Dic3#10412yes—
248D12—81no—
249D12—81no—
250Dic6—42no—
251Dic6—42no—
252Dic6—81no—
253Dic6—81no—
254C3×C8—42no—
255C3×C8—42no—
256C3⋊C8—42no—
257C3⋊C8—42no—
258C24—61no—
259C2×Q8—62no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×Q8—121no—
265C2×Q8—241no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—241no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4—61no—
276C22×C4C2×C4×D311yescharacteristic
277C22×C4D3×D411yescharacteristic
278C22×C4D3×D411yescharacteristic
279M16—22no—
280M16—41no—
281M16—62no—
282M16—121no—
283C2×C8—22no—
284C2×C8—41no—
285C2×C8—62no—
286C2×C8—121no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—241no—
294C4⋊C4—241no—
295C22⋊C4—122no—
296C22⋊C4—122no—
297C22⋊C4—121no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C22⋊C4—121no—
301C22⋊C4—241no—
302C22⋊C4—241no—
303C42—61no—
304C42—61no—
305C42—61no—
306C42—61no—
307C42C2×D1211yescharacteristic
308C42#23511yescharacteristic
309C42D3×D411yescharacteristic
310C42#23911yescharacteristic
311C22×C3—21no—
312C22×C3—21no—
313C22×C3#38611yescharacteristic
314C22×C3C2×#10411yescharacteristic
315C22×C3#42111yescharacteristic
316C2×D3—81no—
317C2×D3—81no—
318C2×D3—81no—
319C2×D3—81no—
320C2×D3—81no—
321C2×D3—81no—
322C2×D3—81no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—81no—
330C3×C4—81no—
331Dic3—22no—
332Dic3—41no—
333Dic3—41no—
334Dic3—41no—
335Dic3—81no—
336C23—121no—
337C23—121no—
338C23—121no—
339C23—121no—
340C23—121no—
341C23—121no—
342C23—121no—
343C23D3×C22⋊C411yescharacteristic
344Q8—122no—
345Q8—122no—
346Q8—241no—
347Q8—241no—
348D4—241no—
349D4—241no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—32no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4C4×D1211yescharacteristic
374C2×C4#88211yescharacteristic
375C2×C4#88411yescharacteristic
376C2×C4#89311yescharacteristic
377C2×C4#89411yescharacteristic
378C2×C4#89511yescharacteristic
379C8—42no—
380C8—42no—
381C8—122no—
382C8—122no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3#187111yescharacteristic
386C2×C3#191211yescharacteristic
387C2×C3#191411yescharacteristic
388D3—161no—
389C22—21no—
390C22—21no—
391C22—241no—
392C22—241no—
393C22—241no—
394C22—241no—
395C22—241no—
396C22—241no—
397C22—241no—
398C22#466511yescharacteristic
399C22D3×#10411yescharacteristic
400C22#482111yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—62no—
408C4—81no—
409C4—81no—
410C4—121no—
411C4—121no—
412C4—121no—
413C4—241no—
414C3#918811yescharacteristic, complemented, Sylow
415C2—21no—
416C2—21no—
417C2—481no—
418C2#6870811yescharacteristic
419C2#6875511yescharacteristic
420C2#6904011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1288C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1288D3
hypocenterC3#9188
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1288

Complements

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

NH ≅ G/Ntype
1. #79314C2semidirect
4. C4×#824C2semidirect
6. C3×#1288C2semidirect
7. #67633C2semidirect
38. #1288D3semidirect
414. C3#9188semidirect

Maximal subgroups

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

subgroupindex
#793142
#743102
#708042
C4×#8242
#698602
C3×#12882
#676332
#91883