Finite Group Databasebeta

← #10692447

Subgroups — #10692447

Order 768.

418 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 93 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#79341C211yesmaximal, characteristic
2#79322C211yesmaximal, characteristic
3#70824C211yesmaximal, characteristic, complemented
4#69490C211yesmaximal, characteristic, complemented
5#69470C211yesmaximal, characteristic
6C3×#1285C211yesmaximal, characteristic, complemented
7#67755C211yesmaximal, characteristic, complemented
8#9375—31nomaximal, Sylow
9#5479C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#4926C2211yescharacteristic
15C2×C4⋊(C3⋊C8)C2211yescharacteristic
16#4850—21no—
17#4842—21no—
18#4840—21no—
19#4839—21no—
20#4736—21no—
21#4730—21no—
22#67⋊C8—21no—
23#4721—21no—
24#4714C412yescomplemented
25#4706—21no—
26#4702C412yescomplemented
27#4701—21no—
28#4529—21no—
29#4529—21no—
30#4481—21no—
31#4481—21no—
32#4452—21no—
33#4451—21no—
34#3238—31no—
35#2992—31no—
36#1988—31no—
37#1558—31no—
38#1509—31no—
39#1290—31no—
40#1285D311yescharacteristic, complemented
41C22×Dic6—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C42—21no—
44C2×C3×C42C2311yescharacteristic
45#930D411yescharacteristic
46#929—21no—
47C4×#67—21no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50#924D411yescharacteristic
51C22×C3⋊C8D411yescharacteristic
52C22×C3⋊C8D411yescharacteristic
53C2×ΓL(1,𝔽25)—21no—
54C2×D3×C8—21no—
55#876—21no—
56#875C2×C411yescharacteristic
57C4×D12D411yescharacteristic
58#872—21no—
59#870C2×C411yescharacteristic
60#869—21no—
61#869—21no—
62#869—21no—
63#869C2×C412yes—
64C4×Dic6D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C4⋊C8D411yescharacteristic
67C3×C4⋊C8D411yescharacteristic
68C3×C22⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×#97—21no—
71C3×#97C2×C412yes—
72C3×#95—21no—
73#830—21no—
74#821—22no—
75(C2×D3)⋊C8—21no—
76(C2×D3)⋊C8—21no—
77#818—22no—
78#817—22no—
79#816—22no—
80#815—21no—
81Dic3⋊C8—21no—
82Dic3⋊C8—21no—
83C8×Dic3—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87#803—21no—
88C2×C4⋊Q8—31no—
89#529—31no—
90#501—32no—
91#489—32no—
92#483—61no—
93#480—61no—
94#477—61no—
95#476—61no—
96#446—61no—
97#435—61no—
98#435—61no—
99C8×D4—61no—
100#433—61no—
101#432—31no—
102#432C2×D311yescharacteristic
103#432C2×D311yescharacteristic
104#424—61no—
105C2×C4⋊C8—31no—
106C4×C4⋊C4C2×D311yescharacteristic
107#340—61no—
108#340—61no—
109#333—61no—
110#333—61no—
111#328—61no—
112#327—61no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4111yescharacteristic
117C2×#67—21no—
118C22×Dic3—41no—
119C2×D12C22⋊C411yescharacteristic
120C2×C4×D3—21no—
121C2×Dic6—22no—
122C2×Dic6—41no—
123C2×Dic6—81no—
124C2×Dic6—81no—
125C2×Dic6C22⋊C411yescharacteristic
126C2×C3×C8—22no—
127C2×C3×C8—21no—
128C2×C3×C8—21no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—41no—
131C3×C42—81no—
132C3×C42C22×C411yescharacteristic
133C3×C42C2×D411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136#217—21no—
137(C2×D3)⋊C4—21no—
138(C2×D3)⋊C4—21no—
139(C2×D3)⋊C4—21no—
140(C2×D3)⋊C4—21no—
141C4⋊Dic3—22no—
142C4⋊Dic3—22no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3C22⋊C411yescharacteristic
146C4⋊Dic3C22⋊C411yescharacteristic
147Dic3⋊C4—21no—
148Dic3⋊C4—21no—
149Dic3⋊C4—21no—
150Dic3⋊C4—21no—
151C4×Dic3—21no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157ΓL(1,𝔽25)—81no—
158D3×C8—81no—
159C22×Q8—61no—
160C2×M16—61no—
161C22×C8—31no—
162C22×C8—31no—
163C22×C8—61no—
164C4⋊Q8—32no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168#126—61no—
169#125—31no—
170#124—31no—
171#123—61no—
172C22⋊Q8—31no—
173#121—31no—
174C4×Q8—31no—
175C4×D4—31no—
176C4×D4—61no—
177#117—61no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C42—21no—
182C2×C42C22×D311yescharacteristic
183#107—62no—
184#106—62no—
185C4⋊C8—21no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8D1211yescharacteristic
192C4⋊C8D1211yescharacteristic
193#103—62no—
194#102—62no—
195C22⋊C8—21no—
196C22⋊C8—21no—
197C22⋊C8—61no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200#97—21no—
201#97—61no—
202#97—61no—
203#97C4×D312yes—
204C4×C8—61no—
205#95—21no—
206C23×C3#12111yescharacteristic
207C22×D3—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—81no—
215C2×C3×C4—81no—
216C2×C3×C4C2×C22⋊C411yescharacteristic
217C2×C3×C4C4×D411yescharacteristic
218C2×C3×C4C4×D411yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12311yescharacteristic
222#67—41no—
223#67—41no—
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—42no—
228C2×Dic3—81no—
229C2×Dic3—81no—
230D12—41no—
231C4×D3—41no—
232C4×D3—41no—
233Dic6—41no—
234Dic6—82no—
235Dic6—82no—
236C3×C8—42no—
237C3×C8—41no—
238C3×C8—41no—
239C3⋊C8—41no—
240C3⋊C8—41no—
241C3⋊C8—81no—
242C2×Q8—31no—
243C2×Q8—62no—
244C2×Q8—121no—
245C2×Q8—241no—
246C2×Q8—241no—
247C2×D4—31no—
248C2×D4—61no—
249C22×C4—41no—
250C22×C4—61no—
251C22×C4—121no—
252C22×C4#23511yescharacteristic
253C22×C4D3×D411yescharacteristic
254C22×C4D3×D411yescharacteristic
255M16—241no—
256C2×C8—22no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—121no—
262C2×C8—121no—
263C2×C8—121no—
264C2×C8—241no—
265C4⋊C4—22no—
266C4⋊C4—31no—
267C4⋊C4—31no—
268C4⋊C4—41no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—62no—
274C4⋊C4—62no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C22⋊C4—61no—
278C22⋊C4—61no—
279C22⋊C4—61no—
280C22⋊C4—61no—
281C22⋊C4—61no—
282C42—61no—
283C42—81no—
284C42C2×C4×D311yescharacteristic
285C42C2×D1211yescharacteristic
286C42D3×D411yescharacteristic
287C42#23911yescharacteristic
288C22×C3—21no—
289C22×C3—21no—
290C22×C3#38611yescharacteristic
291C22×C3#47111yescharacteristic
292C22×C3#47211yescharacteristic
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—41no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—82no—
302C3×C4#45511yescharacteristic
303C3×C4#45611yescharacteristic
304Dic3—41no—
305Dic3—41no—
306Dic3—41no—
307Dic3—82no—
308C23—61no—
309C23#88411yescharacteristic
310Q8—121no—
311Q8—242no—
312Q8—242no—
313D4—121no—
314D4—121no—
315D4—121no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—122no—
330C2×C4—241no—
331C2×C4—241no—
332C2×C4C4×D1211yescharacteristic
333C2×C4D3×C22⋊C411yescharacteristic
334C2×C4#88211yescharacteristic
335C2×C4#89311yescharacteristic
336C2×C4#89411yescharacteristic
337C2×C4#89511yescharacteristic
338C8—42no—
339C8—41no—
340C8—41no—
341C8—121no—
342C8—121no—
343C8—241no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3#188211yescharacteristic
347C2×C3#190711yescharacteristic
348C2×C3#190911yescharacteristic
349D3—81no—
350C22—21no—
351C22—21no—
352C22—121no—
353C22—121no—
354C22—121no—
355C22#466511yescharacteristic
356C22#474911yescharacteristic
357C22#475411yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—82no—
364C4—121no—
365C4—121no—
366C4—121no—
367C4—242no—
368C4#482511yescharacteristic
369C4#482611yescharacteristic
370C3#937511yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—241no—
374C2#6871211yescharacteristic
375C2#6877511yescharacteristic
376C2#6900611yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1285C2
socleC22×C3#386
prime radical#930D4
cosocleC2×C3×C42C23
hypercenter#1285D3
hypocenterC3#9375
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
3. #70824C2semidirect
4. #69490C2semidirect
6. C3×#1285C2semidirect
7. #67755C2semidirect
24. #4714C4semidirect
26. #4702C4semidirect
40. #1285D3semidirect
370. C3#9375semidirect

Maximal subgroups

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

subgroupindex
#793412
#793222
#708242
#694902
#694702
C3×#12852
#677552
#93753