Finite Group Databasebeta

← #10696698

Subgroups — #10696698

Order 768.

556 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 87 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#83166C211yesmaximal, characteristic
2#83159C211yesmaximal, characteristic, complemented
3#79065C211yesmaximal, characteristic, complemented
4C3×#1965C211yesmaximal, characteristic, complemented
5#69859C211yesmaximal, characteristic, complemented
6#69856C211yesmaximal, characteristic, complemented
7#67710C211yesmaximal, characteristic, complemented
8#10112—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10#5477C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×#100—22no—
14C3×C4×M16C2211yescharacteristic
15#5217—22no—
16#5215—22no—
17#5134—22no—
18#5073—21no—
19#5072—22no—
20#5071—22no—
21#5070—21no—
22#5070—21no—
23#5070—21no—
24#4925C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4850—21no—
27#4848—21no—
28#4846—21no—
29#4841—21no—
30#4464—22no—
31#4456—22no—
32#3254—31no—
33#3205—31no—
34#3138—31no—
35#1965D311yescharacteristic, complemented
36#1525—31no—
37#1485—31no—
38#1299—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4D412yes—
42C2×#235—21no—
43C2×C3×M16—21no—
44C2×C3×M16D412yes—
45C3×#127—21no—
46C3×#127—21no—
47C3×#127—21no—
48C3×#127—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#213—24no—
51#930—21no—
52#930—21no—
53#926—21no—
54#924—21no—
55#924—21no—
56C2×#208—21no—
57C2×#204—22no—
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×C4⋊C8—21no—
70C3×#100—44no—
71C3×#97—21no—
72C3×C4×C8—21no—
73#832—24no—
74#832—24no—
75#832—24no—
76#830—21no—
77#821—22no—
78#818—22no—
79#816—22no—
80C4⋊(C3⋊C8)D412yes—
81#803D412yes—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84#533—31no—
85#532—31no—
86C2×#127C2×D311yescharacteristic
87#493—61no—
88#490—62no—
89#486—61no—
90#486—61no—
91#486—61no—
92#477—61no—
93#474—61no—
94#468—62no—
95#461—61no—
96#460—61no—
97#459—62no—
98#447—62no—
99#431—31no—
100#423C2×D311yescharacteristic
101#418—62no—
102C2×#100—22no—
103C4×M16—31no—
104C4×M16C2×D311yescharacteristic
105#329—62no—
106Q8⋊C8—62no—
107C24×C3—22no—
108C2×C3×D4—22no—
109C2×C3×D4—22no—
110C2×C3×D4—44no—
111C2×C3×D4—42no—
112C2×C3×D4—42no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—82no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
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—
130C3×M16—22no—
131C3×M16—41no—
132C3×M16—42no—
133C2×C3×C8—22no—
134C2×C3×C8—41no—
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139#217—41no—
140#213—84no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149Dic3⋊C4—41no—
150Dic3⋊C4—41no—
151C4×Dic3—41no—
152#208—42no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155#204—82no—
156C2×#41—61no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4D1212yes—
160C2×SD16—62no—
161C2×D8—64no—
162C2×M16—21no—
163C2×M16—61no—
164C2×M16#6712yes—
165C4⋊Q8—31no—
166C4⋊Q8—31no—
167C4⋊Q8—31no—
168#127—21no—
169#127—21no—
170#127—21no—
171#127—21no—
172#127—31no—
173#125—61no—
174#124—61no—
175C22⋊Q8—61no—
176C22⋊Q8—61no—
177#121—61no—
178#121—61no—
179C4×Q8—61no—
180C4×Q8—61no—
181C4×D4—61no—
182C4×D4—61no—
183#117—61no—
184C2×C42C22×D311yescharacteristic
185#107—62no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—32no—
189#103—62no—
190#102—62no—
191#102—64no—
192#102—64no—
193#102—64no—
194#100—44no—
195C22⋊C8—61no—
196#97—21no—
197#97—32no—
198C4×C8—21no—
199C4×C8—61no—
200C4×C8—61no—
201C23×C3—44no—
202C23×C3—42no—
203C23×C3—82no—
204C23×C3—82no—
205C23×C3#12111yescharacteristic
206C22×D3—41no—
207C3×D4—44no—
208C3×D4—44no—
209C3×D4—82no—
210C3×D4—82no—
211C3×D4—81no—
212C3×D4—81no—
213C3×D4—82no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12711yescharacteristic
224#67—82no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228D12—41no—
229D12—41no—
230D12—42no—
231D12—81no—
232C4×D3—82no—
233Dic6—41no—
234Dic6—41no—
235Dic6—42no—
236Dic6—81no—
237C3×C8—42no—
238C3×C8—42no—
239C3⋊C8—42no—
240C3⋊C8—42no—
241C24—22no—
242#41—242no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—61no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—44no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—82no—
259C2×D4—121no—
260C22×C4—121no—
261C22×C4C2×D1211yescharacteristic
262C22×C4#23511yescharacteristic
263C22×C4C2×#6711yescharacteristic
264SD16—242no—
265D8—244no—
266M16—22no—
267M16—41no—
268M16—42no—
269M16—122no—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—62no—
273C2×C8—62no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—121no—
281C4⋊C4—121no—
282C22⋊C4—121no—
283C22⋊C4—121no—
284C22⋊C4—121no—
285C42—121no—
286C42D3×D411yescharacteristic
287C42D3×D411yescharacteristic
288C42D3×D411yescharacteristic
289C42D3×D411yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—82no—
294C22×C3—82no—
295C22×C3—82no—
296C22×C3—164no—
297C22×C3#39211yescharacteristic
298C2×D3—81no—
299C2×D3—81no—
300C2×D3—81no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4#45311yescharacteristic
305C3×C4#45411yescharacteristic
306C3×C4#45512yes—
307C3×C4#47212yes—
308Dic3—81no—
309Dic3—81no—
310Dic3—81no—
311C23—44no—
312C23—42no—
313C23—82no—
314C23—82no—
315C23—121no—
316C23#93011yescharacteristic
317Q8—122no—
318Q8—121no—
319Q8—121no—
320Q8—241no—
321D4—44no—
322D4—44no—
323D4—82no—
324D4—82no—
325D4—82no—
326D4—81no—
327D4—81no—
328D4—122no—
329D4—121no—
330D4—121no—
331D4—241no—
332D4—242no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—242no—
341C2×C4#88211yescharacteristic
342C2×C4#88211yescharacteristic
343C2×C4#88411yescharacteristic
344C2×C4#88411yescharacteristic
345C2×C4#93711yescharacteristic
346C2×C4#94011yescharacteristic
347C8—42no—
348C8—42no—
349C8—122no—
350C8—122no—
351C2×C3—41no—
352C2×C3—162no—
353C2×C3#200411yescharacteristic
354C2×C3#200611yescharacteristic
355C2×C3#201811yescharacteristic
356D3—161no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—82no—
361C22—82no—
362C22—82no—
363C22—164no—
364C22—241no—
365C22—241no—
366C22—241no—
367C22#496011yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—241no—
372C4—241no—
373C4—241no—
374C4#474612yes—
375C4#474912yes—
376C4#506211yescharacteristic
377C4#507711yescharacteristic
378C3#1011211yescharacteristic, complemented, Sylow
379C2—41no—
380C2—162no—
381C2—481no—
382C2#6966411yescharacteristic
383C2#7008411yescharacteristic
384C2#7009011yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1965C2
socleC22×C3#392
prime radical#83166C2
hypercenter#1965D3
hypocenterC3#10112
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1965

Complements

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

NH ≅ G/Ntype
2. #83159C2semidirect
3. #79065C2semidirect
4. C3×#1965C2semidirect
5. #69859C2semidirect
6. #69856C2semidirect
7. #67710C2semidirect
35. #1965D3semidirect
378. C3#10112semidirect

Maximal subgroups

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

subgroupindex
#831662
#831592
#790652
C3×#19652
#698592
#698562
#677102
#101123