Finite Group Databasebeta

← #10696072

Subgroups — #10696072

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#83066C211yesmaximal, characteristic, complemented
2#83012C211yesmaximal, characteristic, complemented
3#79343C211yesmaximal, characteristic
4C3×#1534C211yesmaximal, characteristic, complemented
5#70835C211yesmaximal, characteristic, complemented
6#70799C211yesmaximal, characteristic, complemented
7#67752C211yesmaximal, characteristic, complemented
8#10019—31nomaximal, Sylow
9C2×C3×#125C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#432C2211yescharacteristic
14#5069—21no—
15#5068—22no—
16#5067—21no—
17#4993—21no—
18#4992—21no—
19#4991—21no—
20#4989—22no—
21#4981—21no—
22#4966—21no—
23#4966—21no—
24#4937C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4843—21no—
27#4842—21no—
28#4735—21no—
29#4732—21no—
30#4711—22no—
31C3×#330—22no—
32#4483—22no—
33#4470—22no—
34#3142—31no—
35#3109—31no—
36#2633—31no—
37#1534D311yescharacteristic, complemented
38#1514—31no—
39#1490—31no—
40#1288—31no—
41C2×#235—21no—
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C3×#125—21no—
45C3×#125—21no—
46C3×#125—21no—
47C3×#125—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C42C2311yescharacteristic
53#930—21no—
54#930—21no—
55C2×(C2×D3)⋊C4—22no—
56C2×(C2×D3)⋊C4—21no—
57C2×(C2×D3)⋊C4—22no—
58C2×C4⋊Dic3D411yescharacteristic
59C2×Dic3⋊C4—22no—
60C2×#208—21no—
61C2×D24—22no—
62#875—21no—
63#874D411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12—21no—
69C4×D12—21no—
70#869D411yescharacteristic
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×#97—21no—
76C3×#97D412yes—
77C3⋊#95—22no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95D412yes—
81#821—22no—
82#821—22no—
83#817—22no—
84#810—22no—
85#810—22no—
86#809—22no—
87#809—22no—
88#803—21no—
89C4×C3⋊C8D412yes—
90#532—31no—
91C2×#125C2×D311yescharacteristic
92C2×C4×D4—31no—
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#432C2×D311yescharacteristic
102#432C2×D311yescharacteristic
103#417—61no—
104#417—61no—
105C4×M16—31no—
106#399—61no—
107#398—61no—
108#397—61no—
109#394—61no—
110#385—62no—
111#383—31no—
112#330—22no—
113#327—62no—
114D4⋊C8—62no—
115C23×D3—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C2×#67—41no—
122C22×Dic3—22no—
123C22×Dic3—21no—
124#235—44no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—42no—
131C2×C4×D3—41no—
132C2×C4×D3—41no—
133C2×C4×D3—41no—
134C2×C4×D3—82no—
135C2×Dic6—21no—
136C2×C3×C8—22no—
137C2×C3×C8—41no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150(C2×D3)⋊C4—44no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—82no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160Dic3⋊C4—82no—
161#208—42no—
162C2×C3⋊C8—22no—
163D24—44no—
164C2×#41—61no—
165C22×D4—31no—
166C23×C4—61no—
167C2×D8—62no—
168C2×M16—61no—
169C4⋊Q8—31no—
170#127—31no—
171#125—21no—
172#125—21no—
173#125—21no—
174#125—21no—
175#124—61no—
176#121—61no—
177#121—61no—
178C4×D4—61no—
179C4×D4—61no—
180C4×D4—61no—
181C4×D4—61no—
182C4×D4—61no—
183C4×D4—61no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4D1211yescharacteristic
189C2×C4⋊C4D1211yescharacteristic
190C2×C22⋊C4—62no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—62no—
193C2×C42C22×D311yescharacteristic
194#106—62no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197#103—62no—
198#103—62no—
199#102—62no—
200#102—62no—
201#102—62no—
202#102—62no—
203C22⋊C8—21no—
204C22⋊C8—21no—
205#97—21no—
206#97—61no—
207#97#6712yes—
208C4×C8—32no—
209#95—32no—
210#95—62no—
211#95—61no—
212#95—61no—
213C23×C3#12011yescharacteristic
214C22×D3—22no—
215C22×D3—42no—
216C22×D3—41no—
217C22×D3—41no—
218C22×D3—42no—
219C22×D3—41no—
220C2×C3×C4—22no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12711yescharacteristic
234#67—82no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—42no—
241D12—22no—
242D12—22no—
243D12—41no—
244D12—41no—
245D12—42no—
246D12—42no—
247D12—81no—
248C4×D3—82no—
249C4×D3—82no—
250C4×D3—82no—
251Dic6—42no—
252C3×C8—42no—
253C3×C8—81no—
254C3⋊C8—42no—
255C24—61no—
256#41—124no—
257C2×Q8—61no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—61no—
263C2×D4—122no—
264C2×D4—121no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—62no—
268C22×C4—61no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4—121no—
272C22×C4—242no—
273C22×C4C2×D1211yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4D3×D411yescharacteristic
276D8—124no—
277M16—122no—
278C2×C8—22no—
279C2×C8—41no—
280C2×C8—62no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—242no—
294C22⋊C4—124no—
295C22⋊C4—121no—
296C22⋊C4—121no—
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#39211yescharacteristic
307C22×C3#45412yes—
308C2×D3—44no—
309C2×D3—42no—
310C2×D3—82no—
311C2×D3—81no—
312C2×D3—81no—
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—
323C3×C4—81no—
324Dic3—82no—
325Dic3—81no—
326Dic3—81no—
327C23—62no—
328C23—122no—
329C23—121no—
330C23—121no—
331C23—122no—
332C23—121no—
333C23#88211yescharacteristic
334Q8—122no—
335D4—62no—
336D4—62no—
337D4—121no—
338D4—121no—
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—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—121no—
355C2×C4—121no—
356C2×C4—122no—
357C2×C4—242no—
358C2×C4—242no—
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#200311yescharacteristic
371C2×C3#200811yescharacteristic
372C2×C3#201011yescharacteristic
373D3—82no—
374D3—161no—
375C22—21no—
376C22—21no—
377C22—124no—
378C22—122no—
379C22—242no—
380C22—241no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22—241no—
385C22#475312yes—
386C22#496011yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—81no—
394C4—81no—
395C4—242no—
396C4—241no—
397C4—241no—
398C3#1001911yescharacteristic, complemented, Sylow
399C2—22no—
400C2—242no—
401C2—481no—
402C2#7011611yescharacteristic
403C2#7040611yescharacteristic
404C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1534C2
socleC22×C3#392
prime radical#79343C2
hypercenter#1534D3
hypocenterC3#10019
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1534

Complements

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

NH ≅ G/Ntype
1. #83066C2semidirect
2. #83012C2semidirect
4. C3×#1534C2semidirect
5. #70835C2semidirect
6. #70799C2semidirect
7. #67752C2semidirect
37. #1534D3semidirect
398. C3#10019semidirect

Maximal subgroups

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

subgroupindex
#830662
#830122
#793432
C3×#15342
#708352
#707992
#677522
#100193