Finite Group Databasebeta

← #10692855

Subgroups — #10692855

Order 768.

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

normal subgroups only

This group has 490 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#73720C211yesmaximal, characteristic
3#70827C211yesmaximal, characteristic
4#70814C211yesmaximal, characteristic, complemented
5#70814C211yesmaximal, characteristic, complemented
6C3×#1285C211yesmaximal, characteristic, complemented
7#67621C211yesmaximal, characteristic, complemented
8#9449—31nomaximal, Sylow
9C2×C4×D12C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#4994—21no—
15#4994—21no—
16#4939C411yescharacteristic
17#4939C411yescharacteristic
18#4936C411yescharacteristic
19#4935C411yescharacteristic
20#4926C2211yescharacteristic
21#4926C2211yescharacteristic
22#4836—22no—
23#4726—21no—
24D3×C22⋊C8—21no—
25#4706—22no—
26#4668—21no—
27#4668—21no—
28#4468—21no—
29#4468—21no—
30#4465—22no—
31#4465—22no—
32#2973—31no—
33#2566—31no—
34#1534—31no—
35#1534—31no—
36#1466—31no—
37#1285—31no—
38#1285D311yescharacteristic, complemented
39C22×D12C2×C411yescharacteristic
40C22×C4×D3D411yescharacteristic
41C22×C4×D3D411yescharacteristic
42C2×C3×C4⋊C4—21no—
43C2×C3×C42C2×C411yescharacteristic
44C2×C3×C42C2×C411yescharacteristic
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)⋊C4D411yescharacteristic
51C2×(C2×D3)⋊C4D411yescharacteristic
52C2×C4⋊Dic3—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3C2×C411yescharacteristic
55C2×C4⋊Dic3C2×C411yescharacteristic
56C2×C4⋊Dic3C2×C411yescharacteristic
57C2×Dic3⋊C4—21no—
58C2×Dic3⋊C4—21no—
59C2×ΓL(1,𝔽25)—22no—
60C2×D3×C8—22no—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12—21no—
64C4×D12—21no—
65#870—21no—
66#870—21no—
67#870—21no—
68#870—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74C3×#97D412yes—
75C3×#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78#830—21no—
79#830—21no—
80(C2×D3)⋊C8—22no—
81(C2×D3)⋊C8—22no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84#803—21no—
85#803D412yes—
86C2×#125—31no—
87C2×C4×D4—31no—
88#435—62no—
89#435—62no—
90#432—31no—
91#432—31no—
92#432C2×D311yescharacteristic
93#432C2×D311yescharacteristic
94#407—61no—
95C2×C22⋊C8—61no—
96#396—61no—
97#396—61no—
98#390—31no—
99#390—31no—
100#385—61no—
101#385—61no—
102#384—31no—
103#382—31no—
104C4×C4⋊C4C2×D311yescharacteristic
105#330—62no—
106#330—62no—
107#323—61no—
108#323—61no—
109C23×D3C22⋊C411yescharacteristic
110C23×D3C22⋊C411yescharacteristic
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4C22×C411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×Dic3—21no—
117C22×Dic3—21no—
118C22×Dic3C22⋊C411yescharacteristic
119C22×Dic3C22⋊C411yescharacteristic
120C2×D12—22no—
121C2×D12—21no—
122C2×D12—21no—
123C2×D12—41no—
124C2×C4×D3—21no—
125C2×C4×D3—21no—
126C2×C4×D3—21no—
127C2×C4×D3—21no—
128C2×C4×D3—81no—
129C2×C4×D3—81no—
130C2×C3×C8—22no—
131C2×C3×C8—41no—
132C3×C4⋊C4—42no—
133C3×C42—41no—
134C3×C42—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—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—41no—
161ΓL(1,𝔽25)—82no—
162D3×C8—82no—
163C22×D4—31no—
164C23×C4—31no—
165C23×C4—31no—
166C2×M16—62no—
167C22×C8—62no—
168#125—61no—
169#125—61no—
170#125—61no—
171#125—61no—
172C4×D4—61no—
173C4×D4—61no—
174C4×D4—61no—
175C4×D4—61no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—31no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C22⋊C4—31no—
185C2×C22⋊C4—31no—
186C2×C22⋊C4—61no—
187C2×C22⋊C4—61no—
188C2×C22⋊C4—61no—
189C2×C22⋊C4—61no—
190C2×C42C4×D311yescharacteristic
191C2×C42C4×D311yescharacteristic
192C2×C42C22×D311yescharacteristic
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C22⋊C8—21no—
198C22⋊C8—21no—
199C22⋊C8—62no—
200C22⋊C8—62no—
201C22⋊C8—61no—
202C22⋊C8—61no—
203#97—21no—
204#97—32no—
205#97—61no—
206#97D1212yes—
207#95—21no—
208#95—61no—
209#95—61no—
210C23×C3C2×C22⋊C411yescharacteristic
211C22×D3—21no—
212C22×D3—21no—
213C22×D3—21no—
214C22×D3—21no—
215C22×D3—21no—
216C22×D3—41no—
217C22×D3—41no—
218C22×D3—41no—
219C22×D3—41no—
220C22×D3#10112yes—
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—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247D12—42no—
248D12—42no—
249C4×D3—42no—
250C4×D3—42no—
251C4×D3—81no—
252C4×D3—81no—
253C3×C8—42no—
254C3×C8—81no—
255C3⋊C8—42no—
256C3⋊C8—81no—
257C24—31no—
258C24—31no—
259C2×D4—62no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—121no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—241no—
274C22×C4—241no—
275C22×C4C2×C4×D311yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4D3×D411yescharacteristic
278M16—242no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—62no—
282C2×C8—121no—
283C2×C8—242no—
284C4⋊C4—42no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—121no—
292C4⋊C4—121no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C4⋊C4—241no—
296C4⋊C4—241no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C22⋊C4—61no—
301C22⋊C4—122no—
302C22⋊C4—122no—
303C22⋊C4—241no—
304C22⋊C4—241no—
305C42—41no—
306C42—41no—
307C42C2×D1211yescharacteristic
308C42#23511yescharacteristic
309C42D3×D411yescharacteristic
310C42#23911yescharacteristic
311C22×C3—21no—
312C22×C3—21no—
313C22×C3#38611yescharacteristic
314C22×C3C2×#10111yescharacteristic
315C22×C3#41311yescharacteristic
316C2×D3—22no—
317C2×D3—41no—
318C2×D3—41no—
319C2×D3—42no—
320C2×D3—42no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—81no—
325C2×D3—81no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—42no—
333C3×C4—81no—
334Dic3—41no—
335Dic3—41no—
336Dic3—81no—
337Dic3—81no—
338C23—32no—
339C23—61no—
340C23—61no—
341C23—61no—
342C23—61no—
343C23—61no—
344C23—121no—
345C23—121no—
346C23—121no—
347C23—121no—
348C23D3×C22⋊C411yescharacteristic
349D4—122no—
350D4—122no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
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—122no—
364C2×C4—122no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—241no—
374C2×C4—241no—
375C2×C4C4×D1211yescharacteristic
376C2×C4#88211yescharacteristic
377C2×C4#88411yescharacteristic
378C2×C4#89311yescharacteristic
379C2×C4#89411yescharacteristic
380C2×C4#89511yescharacteristic
381C8—42no—
382C8—81no—
383C8—122no—
384C8—241no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3#188411yescharacteristic
388C2×C3#190111yescharacteristic
389C2×C3#191111yescharacteristic
390D3—42no—
391D3—81no—
392C22—21no—
393C22—21no—
394C22—62no—
395C22—122no—
396C22—122no—
397C22—121no—
398C22—121no—
399C22—121no—
400C22—121no—
401C22—121no—
402C22—241no—
403C22—241no—
404C22#466511yescharacteristic
405C22#474511yescharacteristic
406C22D3×#10111yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—42no—
414C4—81no—
415C4—121no—
416C4—121no—
417C4—241no—
418C4—241no—
419C3#944911yescharacteristic, complemented, Sylow
420C2—21no—
421C2—21no—
422C2—122no—
423C2—241no—
424C2#6870511yescharacteristic
425C2#6877711yescharacteristic
426C2#6899311yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1285C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1285D3
hypocenterC3#9449
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
4. #70814C2semidirect
5. #70814C2semidirect
6. C3×#1285C2semidirect
7. #67621C2semidirect
38. #1285D3semidirect
419. C3#9449semidirect

Maximal subgroups

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

subgroupindex
#793352
#737202
#708272
#708142
C3×#12852
#676212
#94493