Finite Group Databasebeta

← #10697634

Subgroups — #10697634

Order 768.

604 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 103 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#79305C211yesmaximal, characteristic
2#79068C211yesmaximal, characteristic, complemented
3#73712C211yesmaximal, characteristic
4#70766C211yesmaximal, characteristic, complemented
5#70502C211yesmaximal, characteristic
6#69471C211yesmaximal, characteristic, complemented
7C3×#1303C211yesmaximal, characteristic, complemented
8#10250—31nomaximal, Sylow
9C2×#875—21no—
10C2×#874C2211yescharacteristic
11#5470C2211yescharacteristic
12C2×C4×Dic6C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C3×C4×C4⋊C4C2211yescharacteristic
16#5121—21no—
17#5120—21no—
18#4988—22no—
19#4939—21no—
20#4925C2211yescharacteristic
21#4851—21no—
22#4850—21no—
23#4848—22no—
24#4829—22no—
25#4731—22no—
26#4694—21no—
27#4693—21no—
28#4672—22no—
29#4485—22no—
30#4484—22no—
31#4452—22no—
32#3208—31no—
33#3078—31no—
34#2548—31no—
35#1545—31no—
36#1503—31no—
37#1479—31no—
38#1303D311yescharacteristic, complemented
39C22×D12—21no—
40C22×D12—21no—
41C22×D12D412yes—
42C22×Dic6D411yescharacteristic
43C2×C3×M16—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C42—21no—
46C2×C3×C42C2311yescharacteristic
47C2×(C2×D3)⋊C4—24no—
48C2×(C2×D3)⋊C4—24no—
49#926—21no—
50C2×C4⋊Dic3D411yescharacteristic
51#924—21no—
52#924—21no—
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C2×#204—24no—
56#875—42no—
57#875—42no—
58#874—21no—
59#874—21no—
60#874—21no—
61#874—21no—
62#870—21no—
63#869D411yescharacteristic
64#869D411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C4×Dic6—21no—
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×C4⋊C8D412yes—
74C3×C22⋊C8—21no—
75C3×C4×C8—21no—
76C3×#95D412yes—
77#830—21no—
78#821—24no—
79#818—22no—
80#817—22no—
81#816—22no—
82#816—22no—
83C4⋊(C3⋊C8)—21no—
84C4×C3⋊C8—21no—
85#533—31no—
86C2×#127—31no—
87C2×#124—61no—
88C2×C4×Q8—31no—
89#492—61no—
90#486—61no—
91#479—62no—
92#477—61no—
93#475—61no—
94#460—62no—
95#450—62no—
96#431—31no—
97#431C2×D311yescharacteristic
98#428—61no—
99#423C2×D311yescharacteristic
100#419—61no—
101#394—62no—
102#390—61no—
103#390—62no—
104C4×C4⋊C4C2×D311yescharacteristic
105#331—62no—
106#328—62no—
107Q8⋊C8—62no—
108C23×D3—22no—
109C22×C3×C4—22no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—21no—
114C22×Dic3—21no—
115C2×D12—22no—
116C2×D12—22no—
117C2×D12—44no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—82no—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—21no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6SD1612yescomplemented
135C3×M16—41no—
136C3×M16—41no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C3×C4⋊C4—42no—
140C3×C42—41no—
141C3×C42—41no—
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42#4111yescharacteristic
146#217—41no—
147(C2×D3)⋊C4—48no—
148(C2×D3)⋊C4—84no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155Dic3⋊C4—41no—
156Dic3⋊C4—41no—
157Dic3⋊C4—41no—
158Dic3⋊C4—41no—
159C4×Dic3—41no—
160C4×Dic3—41no—
161C4×Dic3—41no—
162C2×C3⋊C8—41no—
163#204—84no—
164C22×Q8—31no—
165C22×D4—32no—
166C22×D4—61no—
167C22×D4—61no—
168C2×SD16—64no—
169C2×M16—21no—
170C4⋊Q8—31no—
171C4⋊Q8—31no—
172#127—61no—
173#127—61no—
174#127—61no—
175#127—61no—
176#125—61no—
177#124—122no—
178#124—122no—
179C22⋊Q8—61no—
180C22⋊Q8—61no—
181C4×Q8—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×Q8—61no—
186C4×Q8—61no—
187#117—61no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—61no—
191C2×C22⋊C4—64no—
192C2×C22⋊C4—64no—
193C2×C42—21no—
194C2×C42—61no—
195C2×C42C22×D311yescharacteristic
196#107—62no—
197#106—62no—
198C4⋊C8—21no—
199C4⋊C8—61no—
200C4⋊C8D1212yes—
201C4⋊C8D1212yes—
202#103—62no—
203#103—62no—
204#102—64no—
205C22⋊C8—21no—
206C22⋊C8—61no—
207C4×C8—21no—
208C4×C8—61no—
209#95D1212yes—
210C23×C3#12011yescharacteristic
211C22×D3—44no—
212C22×D3—44no—
213C22×D3—42no—
214C22×D3—42no—
215C22×D3—42no—
216C2×C3×C4—22no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12711yescharacteristic
227C2×C3×C4C2×SD1611yescharacteristic
228C2×C3×C4#13511yescharacteristic
229C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
230C2×C3×C4#13711yescharacteristic
231C2×Dic3—22no—
232C2×Dic3—41no—
233C2×Dic3—41no—
234C2×Dic3—42no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237C2×Dic3—41no—
238D12—44no—
239D12—82no—
240D12—82no—
241D12—82no—
242D12—81no—
243D12—81no—
244D12—81no—
245D12—81no—
246Dic6—22no—
247Dic6—22no—
248Dic6—41no—
249Dic6—41no—
250Dic6—42no—
251Dic6—81no—
252C3×C8—42no—
253C3×C8—42no—
254C3⋊C8—81no—
255C24—62no—
256C2×Q8—32no—
257C2×Q8—61no—
258C2×Q8—61no—
259C2×Q8—61no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×D4—62no—
265C2×D4—62no—
266C2×D4—124no—
267C2×D4—121no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—242no—
276C22×C4—22no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4C2×D1211yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4D3×D411yescharacteristic
282SD16—244no—
283M16—41no—
284M16—41no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—121no—
288C4⋊C4—42no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C4⋊C4—121no—
298C4⋊C4—121no—
299C22⋊C4—128no—
300C22⋊C4—121no—
301C22⋊C4—244no—
302C42—41no—
303C42—41no—
304C42—121no—
305C42—121no—
306C42—121no—
307C42C2×D1211yescharacteristic
308C42C2×D1211yescharacteristic
309C42D3×D411yescharacteristic
310C42#23911yescharacteristic
311C22×C3—21no—
312C22×C3—21no—
313C22×C3#39211yescharacteristic
314C22×C3#45011yescharacteristic
315C22×C3#45211yescharacteristic
316C2×D3—84no—
317C2×D3—84no—
318C2×D3—82no—
319C2×D3—82no—
320C2×D3—82no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—82no—
325C3×C4#44811yescharacteristic
326C3×C4#44911yescharacteristic
327C3×C4#45512yes—
328C3×C4#46511yescharacteristic
329C3×C4#46811yescharacteristic
330Dic3—42no—
331Dic3—81no—
332Dic3—81no—
333Dic3—81no—
334C23—124no—
335C23—124no—
336C23—122no—
337C23—122no—
338C23—122no—
339C23#88211yescharacteristic
340Q8—62no—
341Q8—62no—
342Q8—121no—
343Q8—121no—
344Q8—122no—
345Q8—241no—
346D4—124no—
347D4—242no—
348D4—242no—
349D4—242no—
350D4—241no—
351D4—241no—
352D4—241no—
353D4—241no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—62no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—122no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4#87411yescharacteristic
367C2×C4#88211yescharacteristic
368C2×C4#88211yescharacteristic
369C2×C4#89511yescharacteristic
370C2×C4#89511yescharacteristic
371C2×C4#89511yescharacteristic
372C2×C4#91111yescharacteristic
373C2×C4D3×SD1611yescharacteristic
374C2×C4#91611yescharacteristic
375C2×C4#91811yescharacteristic
376C8—42no—
377C8—42no—
378C8—241no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3#200011yescharacteristic
382C2×C3#200411yescharacteristic
383C2×C3#200811yescharacteristic
384D3—162no—
385C22—21no—
386C22—21no—
387C22—244no—
388C22—244no—
389C22—242no—
390C22—242no—
391C22—242no—
392C22#467111yescharacteristic
393C22#477611yescharacteristic
394C22#480811yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—82no—
399C4—122no—
400C4—241no—
401C4—241no—
402C4—241no—
403C4#478111yescharacteristic
404C4#480611yescharacteristic
405C4#482512yes—
406C4#486411yescharacteristic
407C4#488611yescharacteristic
408C3#1025011yescharacteristic, complemented, Sylow
409C2—21no—
410C2—21no—
411C2—482no—
412C2#6876211yescharacteristic
413C2#6888711yescharacteristic
414C2#6902811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#392
prime radicalC2×#874C22
hypercenter#1303D3
hypocenterC3#10250
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
2. #79068C2semidirect
4. #70766C2semidirect
6. #69471C2semidirect
7. C3×#1303C2semidirect
38. #1303D3semidirect
134. C2×Dic6SD16semidirect
408. C3#10250semidirect

Maximal subgroups

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

subgroupindex
#793052
#790682
#737122
#707662
#705022
#694712
C3×#13032
#102503