Finite Group Databasebeta

← #10692448

Subgroups — #10692448

Order 768.

418 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#3238C211yesmaximal, characteristic
2#83130C211yesmaximal, characteristic
3#71780C211yesmaximal, characteristic, complemented
4#71777C211yesmaximal, characteristic, complemented
5#71755C211yesmaximal, characteristic
6#67755C211yesmaximal, characteristic, complemented
7#67749C211yesmaximal, characteristic, complemented
8#9375—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C3×#529C2211yescharacteristic
11C3×#501C412yescomplemented
12C3×#489C412yescomplemented
13C3×#483—21no—
14C3×#480—21no—
15C3×#477—21no—
16C3×#476—21no—
17C3×#432C2211yescharacteristic
18#5024—21no—
19#5011—21no—
20#5011—21no—
21D4×C3⋊C8—21no—
22#5000—21no—
23#4999—21no—
24C4×C4⋊Dic3C2211yescharacteristic
25#4926C2211yescharacteristic
26#4926C2211yescharacteristic
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28#4546—21no—
29#4546—21no—
30#4538—21no—
31#4537—21no—
32#4529—21no—
33#4529—21no—
34#3238D311yescharacteristic
35#2992—31no—
36#1988—31no—
37#1558—31no—
38#1509—31no—
39#1290—31no—
40#1285—31no—
41C22×C3×Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8C2×C412yes—
46C3×#126—21no—
47C3×#125C2×C411yescharacteristic
48C3×#124C2×C411yescharacteristic
49C3×#123—21no—
50C3×C22⋊Q8D411yescharacteristic
51C3×#121D411yescharacteristic
52C3×C4×Q8D411yescharacteristic
53C3×C4×D4—21no—
54C3×C4×D4D411yescharacteristic
55C3×#117—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×C4⋊Dic3—21no—
60C2×C4×Dic3—21no—
61C2×#208—21no—
62C22×C3⋊C8—21no—
63C22×C3⋊C8D411yescharacteristic
64C22×C3⋊C8D411yescharacteristic
65C3×#107—22no—
66C3×#106—22no—
67C3×C4⋊C8—21no—
68C3×#103—22no—
69C3×#102—22no—
70C3×C22⋊C8—21no—
71C3×#97C2×C412yes—
72C3⋊#95—21no—
73#830—21no—
74#830—21no—
75#830—21no—
76#830—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D411yescharacteristic
83C4⋊(C3⋊C8)D411yescharacteristic
84#803—21no—
85#803—21no—
86#803—21no—
87C4×C3⋊C8—21no—
88C2×C4⋊Q8C2×D311yescharacteristic
89#529C2×D311yescharacteristic
90#501Dic312yescomplemented
91#489Dic312yescomplemented
92#483—21no—
93#480—21no—
94#477—21no—
95#476—21no—
96#446—61no—
97#435—61no—
98#435—61no—
99C8×D4—61no—
100#433—61no—
101#432—31no—
102#432—31no—
103#432C2×D311yescharacteristic
104#424—61no—
105C2×C4⋊C8—31no—
106C4×C4⋊C4—31no—
107#340—61no—
108#340—61no—
109#333—61no—
110#333—61no—
111#328—61no—
112#327—61no—
113C2×C3×Q8—22no—
114C2×C3×Q8—41no—
115C2×C3×Q8—81no—
116C2×C3×Q8—81no—
117C2×C3×Q8C22⋊C411yescharacteristic
118C2×C3×D4—21no—
119C2×C3×D4C22⋊C411yescharacteristic
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×Dic3—41no—
126C2×C3×C8—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4C22⋊C411yescharacteristic
136C3×C4⋊C4C22⋊C411yescharacteristic
137C3×C22⋊C4—21no—
138C3×C22⋊C4—21no—
139C3×C22⋊C4—21no—
140C3×C22⋊C4—21no—
141C3×C22⋊C4—21no—
142C3×C42—21no—
143C3×C42C22×C411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147C4⋊Dic3—22no—
148C4⋊Dic3—41no—
149C4×Dic3—81no—
150#208—81no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—81no—
159C22×Q8—21no—
160C2×M16—61no—
161C22×C8—31no—
162C22×C8—31no—
163C22×C8—61no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8C2×Dic312yes—
168#126—21no—
169#125C2×Dic311yescharacteristic
170#124C2×Dic311yescharacteristic
171#123—21no—
172C22⋊Q8#6711yescharacteristic
173#121#6711yescharacteristic
174C4×Q8#6711yescharacteristic
175C4×D4—21no—
176C4×D4#6711yescharacteristic
177#117—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—61no—
181C2×C42—61no—
182C2×C42C22×D311yescharacteristic
183#107—22no—
184#106—22no—
185C4⋊C8—21no—
186C4⋊C8—31no—
187C4⋊C8—31no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190C4⋊C8—61no—
191C4⋊C8—61no—
192C4⋊C8—61no—
193#103—22no—
194#102—22no—
195C22⋊C8—21no—
196C22⋊C8—61no—
197C22⋊C8—61no—
198C22⋊C8—61no—
199C22⋊C8—61no—
200#97—61no—
201#97—61no—
202#97—61no—
203#97C2×Dic312yes—
204C4×C8—61no—
205#95—61no—
206C23×C3—21no—
207C23×C3#12111yescharacteristic
208C3×Q8—41no—
209C3×Q8—82no—
210C3×Q8—82no—
211C3×D4—41no—
212C3×D4—41no—
213C3×D4—41no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—42no—
225C2×C3×C4—81no—
226C2×C3×C4—81no—
227C2×C3×C4C2×C22⋊C411yescharacteristic
228C2×C3×C4C4×D411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12311yescharacteristic
233C2×Dic3—42no—
234C2×Dic3—81no—
235C2×Dic3—81no—
236C3×C8—42no—
237C3⋊C8—41no—
238C3⋊C8—41no—
239C3⋊C8—41no—
240C3⋊C8—41no—
241C3⋊C8—81no—
242C2×Q8—22no—
243C2×Q8—41no—
244C2×Q8—81no—
245C2×Q8—81no—
246C2×Q8#21711yescharacteristic
247C2×D4—21no—
248C2×D4#21711yescharacteristic
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—121no—
252C22×C4D3×D411yescharacteristic
253C22×C4#23711yescharacteristic
254C22×C4C2×#6711yescharacteristic
255M16—241no—
256C2×C8—22no—
257C2×C8—61no—
258C2×C8—61no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—121no—
262C2×C8—121no—
263C2×C8—121no—
264C2×C8—241no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—62no—
274C4⋊C4—121no—
275C4⋊C4#21711yescharacteristic
276C4⋊C4#21711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C42—21no—
283C42—241no—
284C42D3×D411yescharacteristic
285C42#23711yescharacteristic
286C42C22×Dic311yescharacteristic
287C42C2×#6711yescharacteristic
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3#38611yescharacteristic
294C22×C3#47111yescharacteristic
295C22×C3#47211yescharacteristic
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—41no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—82no—
305C3×C4#45511yescharacteristic
306C3×C4#45611yescharacteristic
307Dic3—82no—
308C23—21no—
309C23#93911yescharacteristic
310Q8—41no—
311Q8—82no—
312Q8—82no—
313D4—41no—
314D4—41no—
315D4—41no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—122no—
330C2×C4—241no—
331C2×C4—241no—
332C2×C4D4×Dic311yescharacteristic
333C2×C4D4×Dic311yescharacteristic
334C2×C4#93511yescharacteristic
335C2×C4#93711yescharacteristic
336C2×C4#93911yescharacteristic
337C2×C4C2×#21711yescharacteristic
338C8—42no—
339C8—121no—
340C8—121no—
341C8—121no—
342C8—121no—
343C8—241no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—81no—
347C2×C3#188211yescharacteristic
348C2×C3#190711yescharacteristic
349C2×C3#190911yescharacteristic
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22#519911yescharacteristic
356C22#520011yescharacteristic
357C22#522011yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—82no—
367C4—242no—
368C4#520211yescharacteristic
369C4#520511yescharacteristic
370C3#937511yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—81no—
374C2#7186511yescharacteristic
375C2#7209111yescharacteristic
376C2#7210011yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3238C2
socleC22×C3#386
prime radicalC3×#121D4
cosocleC2×C3×C42C23
hypercenter#3238D3
hypocenterC3#9375
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3238

Complements

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

NH ≅ G/Ntype
3. #71780C2semidirect
4. #71777C2semidirect
6. #67755C2semidirect
7. #67749C2semidirect
11. C3×#501C4semidirect
12. C3×#489C4semidirect
90. #501Dic3semidirect
91. #489Dic3semidirect
370. C3#9375semidirect

Maximal subgroups

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

subgroupindex
C3×#32382
#831302
#717802
#717772
#717552
#677552
#677492
#93753