Finite Group Databasebeta

← #10691179

Subgroups — #10691179

Order 768.

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

normal subgroups only

This group has 502 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#79335C211yesmaximal, characteristic
2#74277C211yesmaximal, characteristic
3#70812C211yesmaximal, characteristic, complemented
4C4×#823C211yesmaximal, characteristic
5#69863C211yesmaximal, characteristic, complemented
6C3×#1288C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#9189—31nomaximal, Sylow
9C2×C4×D12C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×#383C2211yescharacteristic
14C2×#823—21no—
15#4926C2211yescharacteristic
16C4×#208C2211yescharacteristic
17#4836—22no—
18#4726—21no—
19D3×C22⋊C8—21no—
20#4706—22no—
21#4677C411yescharacteristic
22#4677C411yescharacteristic
23#4674C411yescharacteristic
24#4674C411yescharacteristic
25#4672—21no—
26#4668—21no—
27#4668—21no—
28#4661—21no—
29#4554—21no—
30#4465—22no—
31#4465—22no—
32#2973—31no—
33#2633—31no—
34C4×#100—31no—
35#1528—31no—
36#1528—31no—
37#1288—31no—
38#1288D311yescharacteristic, complemented
39C22×D12D411yescharacteristic
40C22×C4×D3D411yescharacteristic
41C22×C4×D3D411yescharacteristic
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)⋊C4C2×C411yescharacteristic
49C2×(C2×D3)⋊C4C2×C411yescharacteristic
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3C2×C411yescharacteristic
53C2×C4⋊Dic3C2×C411yescharacteristic
54C2×C4⋊Dic3D411yescharacteristic
55C2×Dic3⋊C4—21no—
56C2×Dic3⋊C4—21no—
57C2×#208—21no—
58C2×ΓL(1,𝔽25)—22no—
59C2×D3×C8—22no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12—21no—
64#870—21no—
65#870—21no—
66#870—21no—
67#870—21no—
68C3×C4⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×#97D412yes—
72C3×C4×C8—21no—
73C3×#95—21no—
74C3×#95—21no—
75C3×#95C2×C411yescharacteristic
76C3×#95C2×C411yescharacteristic
77C3⋊#95—21no—
78C3⋊#95—21no—
79#830—21no—
80#823—24no—
81(C2×D3)⋊C8—22no—
82(C2×D3)⋊C8—22no—
83C4⋊(C3⋊C8)—21no—
84#803—21no—
85#803D412yes—
86C4×C3⋊C8—21no—
87C2×#125—31no—
88C2×C4×D4—31no—
89#435—62no—
90#435—62no—
91#432—31no—
92#432C2×D311yescharacteristic
93C2×#100—61no—
94#407—61no—
95C2×C22⋊C8—61no—
96C4×M16—31no—
97C4×M16C2×D311yescharacteristic
98#399—31no—
99#399—31no—
100#398—61no—
101#397—31no—
102#397—31no—
103#394—61no—
104#385—61no—
105#385—61no—
106#383C2×D311yescharacteristic
107#343—61no—
108#330—62no—
109#330—62no—
110C23×D3C22⋊C411yescharacteristic
111C23×D3C22⋊C411yescharacteristic
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4C22×C411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×Dic3—21no—
118C22×Dic3—21no—
119C22×Dic3C22⋊C411yescharacteristic
120C22×Dic3C22⋊C411yescharacteristic
121C2×D12—22no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—41no—
125C2×C4×D3—21no—
126C2×C4×D3—21no—
127C2×C4×D3—21no—
128C2×C4×D3—21no—
129C2×C4×D3—81no—
130C2×C4×D3—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—21no—
140(C2×D3)⋊C4—21no—
141(C2×D3)⋊C4—21no—
142(C2×D3)⋊C4—21no—
143(C2×D3)⋊C4—42no—
144(C2×D3)⋊C4—42no—
145(C2×D3)⋊C4—81no—
146(C2×D3)⋊C4—81no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—81no—
154C4⋊Dic3—81no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159#208—22no—
160#208—41no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163ΓL(1,𝔽25)—82no—
164D3×C8—82no—
165C22×D4—31no—
166C23×C4—31no—
167C23×C4—31no—
168C2×M16—21no—
169C2×M16—62no—
170C2×M16—61no—
171C22×C8—62no—
172#125—61no—
173#125—61no—
174#125—61no—
175#125—61no—
176C4×D4—61no—
177C4×D4—61no—
178C4×D4—61no—
179C4×D4—61no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C22⋊C4—31no—
188C2×C22⋊C4—31no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C42C22×D311yescharacteristic
194C4⋊C8—21no—
195C4⋊C8—61no—
196#100—64no—
197C22⋊C8—21no—
198C22⋊C8—62no—
199C22⋊C8—62no—
200C22⋊C8—61no—
201#97—21no—
202#97—32no—
203#97—61no—
204#97D1212yes—
205C4×C8—21no—
206C4×C8—61no—
207#95—21no—
208#95—21no—
209#95—61no—
210#95—61no—
211#95C4×D311yescharacteristic
212#95C4×D311yescharacteristic
213C23×C3C2×C22⋊C411yescharacteristic
214C22×D3—21no—
215C22×D3—21no—
216C22×D3—21no—
217C22×D3—21no—
218C22×D3—21no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—41no—
222C22×D3—41no—
223C22×D3#10412yes—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
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—41no—
233C2×C3×C4—41no—
234C2×C3×C4C4×D411yescharacteristic
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12311yescharacteristic
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250D12—42no—
251D12—42no—
252C4×D3—42no—
253C4×D3—42no—
254C4×D3—81no—
255C4×D3—81no—
256C3×C8—42no—
257C3×C8—42no—
258C3⋊C8—42no—
259C3⋊C8—42no—
260C24—31no—
261C24—31no—
262C2×D4—62no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—121no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—241no—
277C22×C4—241no—
278C22×C4C2×C4×D311yescharacteristic
279C22×C4D3×D411yescharacteristic
280C22×C4D3×D411yescharacteristic
281M16—22no—
282M16—41no—
283M16—62no—
284M16—121no—
285M16—242no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—62no—
289C2×C8—121no—
290C2×C8—242no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C4⋊C4—121no—
300C4⋊C4—121no—
301C4⋊C4—241no—
302C4⋊C4—241no—
303C22⋊C4—61no—
304C22⋊C4—61no—
305C22⋊C4—61no—
306C22⋊C4—61no—
307C22⋊C4—122no—
308C22⋊C4—122no—
309C22⋊C4—241no—
310C22⋊C4—241no—
311C42C2×D1211yescharacteristic
312C42#23511yescharacteristic
313C42D3×D411yescharacteristic
314C42#23911yescharacteristic
315C22×C3—21no—
316C22×C3—21no—
317C22×C3#38611yescharacteristic
318C22×C3C2×#10411yescharacteristic
319C22×C3#42111yescharacteristic
320C2×D3—22no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—42no—
324C2×D3—42no—
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—81no—
329C2×D3—81no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—81no—
337C3×C4—81no—
338Dic3—41no—
339Dic3—41no—
340Dic3—81no—
341Dic3—81no—
342C23—32no—
343C23—61no—
344C23—61no—
345C23—61no—
346C23—61no—
347C23—61no—
348C23—121no—
349C23—121no—
350C23—121no—
351C23—121no—
352C23D3×C22⋊C411yescharacteristic
353D4—122no—
354D4—122no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—122no—
368C2×C4—122no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—241no—
378C2×C4—241no—
379C2×C4C4×D1211yescharacteristic
380C2×C4#88211yescharacteristic
381C2×C4#88411yescharacteristic
382C2×C4#89311yescharacteristic
383C2×C4#89411yescharacteristic
384C2×C4#89511yescharacteristic
385C8—42no—
386C8—42no—
387C8—122no—
388C8—122no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3#187111yescharacteristic
392C2×C3#191111yescharacteristic
393C2×C3#191511yescharacteristic
394D3—42no—
395D3—81no—
396C22—21no—
397C22—21no—
398C22—62no—
399C22—122no—
400C22—122no—
401C22—121no—
402C22—121no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—241no—
407C22—241no—
408C22#466511yescharacteristic
409C22D3×#10411yescharacteristic
410C22#482111yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—81no—
418C4—81no—
419C4—121no—
420C4—121no—
421C4—241no—
422C4—241no—
423C3#918911yescharacteristic, complemented, Sylow
424C2—21no—
425C2—21no—
426C2—122no—
427C2—241no—
428C2#6870511yescharacteristic
429C2#6875511yescharacteristic
430C2#6904111yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1288C2
socleC22×C3#386
prime radicalC22×D12D4
cosocleC2×C3×C42C23
hypercenter#1288D3
hypocenterC3#9189
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
3. #70812C2semidirect
5. #69863C2semidirect
6. C3×#1288C2semidirect
7. #67633C2semidirect
38. #1288D3semidirect
423. C3#9189semidirect

Maximal subgroups

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

subgroupindex
#793352
#742772
#708122
C4×#8232
#698632
C3×#12882
#676332
#91893