Finite Group Databasebeta

← #10696071

Subgroups — #10696071

Order 768.

540 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 79 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#82391C211yesmaximal, characteristic, complemented
2#82310C211yesmaximal, characteristic
3#78529C211yesmaximal, characteristic, complemented
4C3×#1490C211yesmaximal, characteristic, complemented
5#70835C211yesmaximal, characteristic, complemented
6#70814C211yesmaximal, characteristic, complemented
7#67666C211yesmaximal, characteristic, complemented
8#10019—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5222—21no—
15#5217—21no—
16#5210—22no—
17#5133—21no—
18#5133—21no—
19#5015—21no—
20#5014—22no—
21#5008—21no—
22#4983—21no—
23#4968—21no—
24#4953—21no—
25#4949—21no—
26#4937C2211yescharacteristic
27#4926C2211yescharacteristic
28#4713—21no—
29#4703—21no—
30#4697—22no—
31C3×D4⋊C8—22no—
32#4470—22no—
33#4465—22no—
34#3142—31no—
35#3109—31no—
36#2633—31no—
37#1534—31no—
38#1514—31no—
39#1490D311yescharacteristic, complemented
40#1288—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×#235—21no—
44C2×C3×M16—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C22⋊C4—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#217—22no—
53C2×#217—22no—
54C2×#213—22no—
55#930—21no—
56#930—21no—
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4⋊Dic3D411yescharacteristic
59C2×Dic3⋊C4—21no—
60C2×Dic3⋊C4—21no—
61C2×Dic3⋊C4—22no—
62#875—21no—
63#874D411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66#870—21no—
67#870—21no—
68#870—21no—
69#870—21no—
70#869D411yescharacteristic
71C3×C4⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×#97—21no—
74C3×#97—21no—
75C3×C4×C8D412yes—
76#832—22no—
77C3⋊#95—22no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95D412yes—
81#830—21no—
82#821—22no—
83#821—22no—
84#816—22no—
85#816—22no—
86#809—22no—
87#808—22no—
88C4⋊(C3⋊C8)—21no—
89#803D412yes—
90#532—31no—
91C2×#125—31no—
92C2×C4×D4C2×D311yescharacteristic
93#491—61no—
94#490—61no—
95#488—62no—
96#481—61no—
97#476—61no—
98#459—61no—
99#447—61no—
100#442—62no—
101#432—31no—
102#432C2×D311yescharacteristic
103#417—61no—
104#417—61no—
105C4×M16C2×D311yescharacteristic
106#399—61no—
107#398—61no—
108#397—61no—
109#394—61no—
110#385—62no—
111#383—31no—
112#330—62no—
113#327—62no—
114D4⋊C8—22no—
115C24×C3—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—42no—
119C22×C3×C4—21no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—82no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C2×#67—41no—
127C22×Dic3—22no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130#235—44no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×C4×D3—41no—
135C2×Dic6—21no—
136C3×M16—42no—
137C2×C3×C8—22no—
138C2×C3×C8—41no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147#217—44no—
148#217—82no—
149#213—44no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158Dic3⋊C4—41no—
159Dic3⋊C4—41no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162Dic3⋊C4—82no—
163C2×C3⋊C8—22no—
164C2×#41—61no—
165C22×D4D1211yescharacteristic
166C23×C4—21no—
167C2×D8—62no—
168C2×M16—21no—
169C4⋊Q8—31no—
170#127—31no—
171#125—61no—
172#125—61no—
173#125—61no—
174#125—61no—
175#124—61no—
176#121—61no—
177#121—61no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—61no—
183C4×D4—61no—
184C2×C4⋊C4—31no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4D1211yescharacteristic
190C2×C22⋊C4—21no—
191C2×C22⋊C4—62no—
192C2×C22⋊C4—62no—
193C2×C42C22×D311yescharacteristic
194#106—62no—
195C4⋊C8—21no—
196C4⋊C8—61no—
197#103—62no—
198#103—62no—
199#102—62no—
200#102—62no—
201#102—62no—
202#102—62no—
203C22⋊C8—21no—
204C22⋊C8—61no—
205#97—21no—
206#97—21no—
207#97—32no—
208C4×C8#6712yes—
209#95—32no—
210#95—62no—
211#95—61no—
212#95—61no—
213C23×C3—22no—
214C23×C3—42no—
215C23×C3—41no—
216C23×C3—41no—
217C23×C3—42no—
218C23×C3#12011yescharacteristic
219C22×D3—41no—
220C3×D4—22no—
221C3×D4—22no—
222C3×D4—41no—
223C3×D4—41no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—41no—
229C2×C3×C4—82no—
230C2×C3×C4—82no—
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12711yescharacteristic
237#67—82no—
238C2×Dic3—41no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—42no—
247D12—42no—
248D12—42no—
249D12—81no—
250C4×D3—82no—
251Dic6—42no—
252C3×C8—42no—
253C3×C8—81no—
254C3⋊C8—42no—
255C24—21no—
256#41—124no—
257C2×Q8—61no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—42no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—121no—
265C22×C4—21no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—62no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—82no—
272C22×C4—121no—
273C22×C4C2×D1211yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4D3×D411yescharacteristic
276D8—124no—
277M16—42no—
278C2×C8—22no—
279C2×C8—41no—
280C2×C8—62no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C4⋊C4—121no—
292C4⋊C4—121no—
293C4⋊C4—242no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—124no—
297C22⋊C4—121no—
298C22⋊C4—121no—
299C22⋊C4—242no—
300C42#23511yescharacteristic
301C42D3×D411yescharacteristic
302C42D3×D411yescharacteristic
303C42C2×#6711yescharacteristic
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—44no—
307C22×C3—42no—
308C22×C3—82no—
309C22×C3—81no—
310C22×C3—81no—
311C22×C3#39211yescharacteristic
312C22×C3#45412yes—
313C2×D3—81no—
314C2×D3—81no—
315C2×D3—81no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—81no—
323Dic3—82no—
324Dic3—81no—
325Dic3—81no—
326Dic3—81no—
327C23—22no—
328C23—42no—
329C23—41no—
330C23—41no—
331C23—42no—
332C23—121no—
333C23#88211yescharacteristic
334Q8—122no—
335D4—22no—
336D4—22no—
337D4—41no—
338D4—41no—
339D4—122no—
340D4—122no—
341D4—241no—
342D4—242no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—121no—
353C2×C4—122no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—122no—
359C2×C4—242no—
360C2×C4#88211yescharacteristic
361C2×C4#88411yescharacteristic
362C2×C4#88411yescharacteristic
363C2×C4#93011yescharacteristic
364C2×C4#93711yescharacteristic
365C2×C4#94011yescharacteristic
366C8—42no—
367C8—81no—
368C8—122no—
369C2×C3—22no—
370C2×C3—82no—
371C2×C3#200311yescharacteristic
372C2×C3#200811yescharacteristic
373C2×C3#201011yescharacteristic
374D3—161no—
375C22—21no—
376C22—21no—
377C22—44no—
378C22—42no—
379C22—82no—
380C22—81no—
381C22—81no—
382C22—241no—
383C22—241no—
384C22—241no—
385C22#482712yes—
386C22#496011yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—81no—
394C4—242no—
395C4—241no—
396C4—241no—
397C4—241no—
398C3#1001911yescharacteristic, complemented, Sylow
399C2—22no—
400C2—82no—
401C2—481no—
402C2#6999311yescharacteristic
403C2#7033111yescharacteristic
404C2#7038811yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1490C2
socleC22×C3#392
prime radical#82310C2
hypercenter#1490D3
hypocenterC3#10019
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1490

Complements

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

NH ≅ G/Ntype
1. #82391C2semidirect
3. #78529C2semidirect
4. C3×#1490C2semidirect
5. #70835C2semidirect
6. #70814C2semidirect
7. #67666C2semidirect
39. #1490D3semidirect
398. C3#10019semidirect

Maximal subgroups

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

subgroupindex
#823912
#823102
#785292
C3×#14902
#708352
#708142
#676662
#100193