Finite Group Databasebeta

← #10697380

Subgroups — #10697380

Order 768.

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

normal subgroups only

This group has 440 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#83372C211yesmaximal, characteristic, complemented
2#83358C211yesmaximal, characteristic
3#79319C211yesmaximal, characteristic
4C3×#1515C211yesmaximal, characteristic, complemented
5#69566C211yesmaximal, characteristic, complemented
6#69564C211yesmaximal, characteristic
7C3⋊#1308C211yesmaximal, characteristic, complemented
8#10210—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×#383C2211yescharacteristic
14#5094—21no—
15#5091—21no—
16#5089—21no—
17#5086—21no—
18#5069—21no—
19#5064—21no—
20#5047—22no—
21#5031—21no—
22#5027—21no—
23#4996—22no—
24#4926C2211yescharacteristic
25#4926C2211yescharacteristic
26#4846—21no—
27#4842—21no—
28#4737—22no—
29#4713—21no—
30#4702—21no—
31C3×#328—22no—
32#4452—22no—
33#4451—22no—
34#3237—31no—
35#3235—31no—
36#3216—31no—
37#1515—31no—
38#1515D311yescharacteristic, complemented
39#1514—31no—
40#1308—31no—
41C2×#235—21no—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×#117—21no—
50C2×C3×C42C2311yescharacteristic
51C2×#216—22no—
52C2×#215—22no—
53#930—21no—
54#930—21no—
55#926—21no—
56#924—21no—
57#924—21no—
58#875—21no—
59#874D411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65#869D411yescharacteristic
66C4×Dic6—21no—
67C4×Dic6—21no—
68C3×C4⋊C8—21no—
69C3×C22⋊C8D412yes—
70C3×#97—21no—
71C3×#95—21no—
72C3×#95—21no—
73C3×#95—21no—
74#830D412yes—
75#830D412yes—
76#821—22no—
77#821—22no—
78#818—22no—
79#817—22no—
80#810—22no—
81#810—22no—
82#810—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#533—31no—
91#533C2×D311yescharacteristic
92#532—31no—
93#501—61no—
94#501—61no—
95#497—61no—
96#491—61no—
97#490—61no—
98#489—61no—
99#485—62no—
100#482—62no—
101#477—61no—
102#476—61no—
103#475—61no—
104#474—61no—
105#462—61no—
106#460—61no—
107#452—62no—
108#432—31no—
109#432—31no—
110#432C2×D311yescharacteristic
111#383C2×D311yescharacteristic
112#328—22no—
113#328—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—
122#235—82no—
123C2×D12—21no—
124C2×D12—21no—
125C2×D12—21no—
126C2×C4×D3—41no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×C3×C8—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—41no—
136C3×C4⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143#217—41no—
144#216—82no—
145#215—82no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156C4×Dic3—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×#41—61no—
160C2×Q16—62no—
161C2×SD16—62no—
162C4⋊Q8—31no—
163C4⋊Q8—31no—
164C4⋊Q8—31no—
165C4⋊Q8D1211yescharacteristic
166C4⋊Q8D1211yescharacteristic
167#127—31no—
168#125—21no—
169#125—61no—
170#124—61no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8—61no—
174C22⋊Q8—61no—
175#121—61no—
176#121—61no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—61no—
180C4×Q8—61no—
181C4×D4—61no—
182C4×D4—61no—
183#117—21no—
184#117—61no—
185C2×C42C22×D311yescharacteristic
186#107—62no—
187#107—62no—
188#106—62no—
189#106—62no—
190C4⋊C8—21no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#103—62no—
194#103—62no—
195#103—62no—
196#102—62no—
197#102—62no—
198#102—62no—
199C22⋊C8—32no—
200C22⋊C8—32no—
201C22⋊C8#6712yes—
202#97—21no—
203#97—61no—
204#97—61no—
205#95—21no—
206#95—21no—
207#95—21no—
208C23×C3#12711yescharacteristic
209C22×D3—41no—
210C3×Q8—41no—
211C3×Q8—41no—
212C3×Q8—81no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—81no—
219C2×C3×C4—81no—
220C2×C3×C4—81no—
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227#67—82no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—41no—
231D12—42no—
232D12—42no—
233D12—81no—
234C4×D3—82no—
235Dic6—42no—
236Dic6—42no—
237Dic6—81no—
238C3×C8—42no—
239C3⋊C8—42no—
240C3⋊C8—42no—
241#41—242no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—61no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—121no—
251C22×C4—41no—
252C22×C4—121no—
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255C22×C4C2×#6711yescharacteristic
256Q16—242no—
257SD16—242no—
258C2×C8—22no—
259C2×C8—62no—
260C2×C8—62no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C22⋊C4—41no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C42—41no—
280C42—121no—
281C42C2×D1211yescharacteristic
282C42#23511yescharacteristic
283C42D3×D411yescharacteristic
284C42D3×D411yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3#39211yescharacteristic
289C2×D3—81no—
290C2×D3—81no—
291C2×D3—81no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—81no—
299C3×C4—81no—
300C3×C4—161no—
301Dic3—81no—
302Dic3—81no—
303Dic3—81no—
304C23—121no—
305C23#94011yescharacteristic
306Q8—41no—
307Q8—41no—
308Q8—81no—
309Q8—122no—
310Q8—122no—
311Q8—241no—
312D4—122no—
313D4—122no—
314D4—241no—
315D4—242no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4—242no—
328C2×C4#88211yescharacteristic
329C2×C4#88211yescharacteristic
330C2×C4#88411yescharacteristic
331C2×C4#88411yescharacteristic
332C2×C4#93011yescharacteristic
333C2×C4#93711yescharacteristic
334C8—42no—
335C8—122no—
336C8—122no—
337C2×C3—41no—
338C2×C3#200811yescharacteristic
339C2×C3#200911yescharacteristic
340C2×C3#200911yescharacteristic
341D3—161no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—241no—
346C22—241no—
347C22—241no—
348C22#496011yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—81no—
356C4—81no—
357C4—161no—
358C4—241no—
359C4—241no—
360C4—241no—
361C3#1021011yescharacteristic, complemented, Sylow
362C2—41no—
363C2—481no—
364C2#7005311yescharacteristic
365C2#7035111yescharacteristic
366C2#7040911yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1515C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1515D3
hypocenterC3#10210
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. #83372C2semidirect
4. C3×#1515C2semidirect
5. #69566C2semidirect
7. C3⋊#1308C2semidirect
38. #1515D3semidirect
361. C3#10210semidirect

Maximal subgroups

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

subgroupindex
#833722
#833582
#793192
C3×#15152
#695662
#695642
C3⋊#13082
#102103