Finite Group Databasebeta

← #10697373

Subgroups — #10697373

Order 768.

474 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 75 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#83361C211yesmaximal, characteristic, complemented
2#83358C211yesmaximal, characteristic, complemented
3#79350C211yesmaximal, characteristic
4C3×#1515C211yesmaximal, characteristic, complemented
5#69564C211yesmaximal, characteristic, complemented
6#69564C211yesmaximal, characteristic, complemented
7C3⋊#1308C211yesmaximal, characteristic, complemented
8#10209—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#383C2211yescharacteristic
14#5089—21no—
15#5087—21no—
16#5086—21no—
17#5084—21no—
18#5067—21no—
19#5064—21no—
20#5047—22no—
21#5031—21no—
22#5027—21no—
23#5003—22no—
24#4926C2211yescharacteristic
25#4926C2211yescharacteristic
26#4843—21no—
27#4841—21no—
28#4734—22no—
29#4714—21no—
30#4712—21no—
31C3×#328—22no—
32#4451—22no—
33#4451—22no—
34#3235—31no—
35#3216—31no—
36#3214—31no—
37#1515D311yescharacteristic, complemented
38#1514—31no—
39#1514—31no—
40#1308—31no—
41C2×#235—21no—
42C2×#235—21no—
43C3×C4⋊Q8D411yescharacteristic
44C3×C4⋊Q8D411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×#117—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#216—22no—
53C2×#215—22no—
54#930—21no—
55#930—21no—
56#930—21no—
57#930—21no—
58C2×D24—22no—
59C2×#204—22no—
60#875—21no—
61#875—21no—
62#874D411yescharacteristic
63#874D411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68#869D411yescharacteristic
69#869D411yescharacteristic
70C3×C4⋊C8—21no—
71C3×C22⋊C8D412yes—
72C3×#97—21no—
73C3×#95—21no—
74C3×#95—21no—
75C3×#95—21no—
76#830D412yes—
77#830D412yes—
78#821—22no—
79#816—22no—
80#810—22no—
81#809—22no—
82#809—22no—
83#809—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88#803—21no—
89#803—21no—
90#533C2×D311yescharacteristic
91#532—31no—
92#532—31no—
93#501—61no—
94#497—61no—
95#496—61no—
96#490—61no—
97#489—61no—
98#489—61no—
99#482—62no—
100#476—61no—
101#474—61no—
102#462—61no—
103#461—61no—
104#460—61no—
105#459—61no—
106#452—62no—
107#449—62no—
108#432—31no—
109#432—31no—
110#432C2×D311yescharacteristic
111#383C2×D311yescharacteristic
112#328—22no—
113#327—62no—
114#327—62no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C2×#67—41no—
122C2×#67—41no—
123#235—82no—
124#235—82no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×C4×D3—41no—
132C2×C4×D3—41no—
133C2×Dic6—21no—
134C2×Dic6—21no—
135C2×C3×C8—22no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42—41no—
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42#4111yescharacteristic
148#216—82no—
149#215—82no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—22no—
160D24—82no—
161#204—82no—
162C2×#41—61no—
163C2×#41—61no—
164C2×Q16—62no—
165C2×SD16—62no—
166C2×SD16—62no—
167C2×D8—62no—
168C4⋊Q8—31no—
169C4⋊Q8—31no—
170C4⋊Q8D1211yescharacteristic
171C4⋊Q8D1211yescharacteristic
172#127—31no—
173#127—31no—
174#125—21no—
175#124—61no—
176#124—61no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179#121—61no—
180#121—61no—
181#121—61no—
182#121—61no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×D4—61no—
186C4×D4—61no—
187C4×D4—61no—
188C4×D4—61no—
189#117—21no—
190C2×C42C22×D311yescharacteristic
191#107—62no—
192#106—62no—
193C4⋊C8—21no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196#103—62no—
197#103—62no—
198#102—62no—
199#102—62no—
200#102—62no—
201#102—62no—
202C22⋊C8—32no—
203C22⋊C8—32no—
204C22⋊C8#6712yes—
205#97—21no—
206#97—61no—
207#97—61no—
208#95—21no—
209#95—21no—
210#95—21no—
211C23×C3#12711yescharacteristic
212C22×D3—41no—
213C22×D3—41no—
214C3×Q8—41no—
215C3×Q8—41no—
216C3×Q8—81no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—81no—
223C2×C3×C4—81no—
224C2×C3×C4—81no—
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231#67—82no—
232#67—82no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235D12—41no—
236D12—41no—
237D12—42no—
238D12—42no—
239D12—42no—
240D12—81no—
241D12—81no—
242C4×D3—82no—
243C4×D3—82no—
244Dic6—42no—
245Dic6—42no—
246C3×C8—42no—
247C3⋊C8—42no—
248C3⋊C8—42no—
249#41—242no—
250#41—242no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—61no—
254C2×Q8—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—121no—
262C2×D4—121no—
263C22×C4—41no—
264C22×C4—121no—
265C22×C4—121no—
266C22×C4D3×D411yescharacteristic
267C22×C4D3×D411yescharacteristic
268C22×C4C2×#6711yescharacteristic
269Q16—242no—
270SD16—242no—
271SD16—242no—
272D8—242no—
273C2×C8—22no—
274C2×C8—62no—
275C2×C8—62no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C22⋊C4—41no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C42—41no—
292C42C2×D1211yescharacteristic
293C42#23511yescharacteristic
294C42D3×D411yescharacteristic
295C42D3×D411yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3#39211yescharacteristic
300C2×D3—81no—
301C2×D3—81no—
302C2×D3—81no—
303C2×D3—81no—
304C2×D3—81no—
305C2×D3—81no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—81no—
313C3×C4—81no—
314C3×C4—161no—
315Dic3—81no—
316Dic3—81no—
317C23—121no—
318C23—121no—
319C23#94011yescharacteristic
320Q8—41no—
321Q8—41no—
322Q8—81no—
323Q8—122no—
324Q8—122no—
325D4—121no—
326D4—122no—
327D4—121no—
328D4—122no—
329D4—122no—
330D4—241no—
331D4—241no—
332D4—242no—
333D4—242no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—242no—
345C2×C4—242no—
346C2×C4#88211yescharacteristic
347C2×C4#88211yescharacteristic
348C2×C4#88411yescharacteristic
349C2×C4#88411yescharacteristic
350C2×C4#93011yescharacteristic
351C2×C4#93711yescharacteristic
352C8—42no—
353C8—122no—
354C8—122no—
355C2×C3—41no—
356C2×C3#200811yescharacteristic
357C2×C3#200811yescharacteristic
358C2×C3#200911yescharacteristic
359D3—161no—
360D3—161no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—241no—
365C22—241no—
366C22—241no—
367C22—241no—
368C22—241no—
369C22—241no—
370C22#496011yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—81no—
378C4—81no—
379C4—161no—
380C4—241no—
381C4—241no—
382C3#1020911yescharacteristic, complemented, Sylow
383C2—41no—
384C2—481no—
385C2—481no—
386C2#7005011yescharacteristic
387C2#7035111yescharacteristic
388C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1515C2
socleC22×C3#392
prime radical#79350C2
hypercenter#1515D3
hypocenterC3#10209
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
1. #83361C2semidirect
2. #83358C2semidirect
4. C3×#1515C2semidirect
5. #69564C2semidirect
6. #69564C2semidirect
7. C3⋊#1308C2semidirect
37. #1515D3semidirect
382. C3#10209semidirect

Maximal subgroups

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

subgroupindex
#833612
#833582
#793502
C3×#15152
#695642
C3⋊#13082
#102093