Finite Group Databasebeta

← #10692462

Subgroups — #10692462

Order 768.

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

normal subgroups only

This group has 434 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×#3239C211yesmaximal, characteristic
2#83130C211yesmaximal, characteristic
3#71778C211yesmaximal, characteristic, complemented
4#71776C211yesmaximal, characteristic, complemented
5#71755C211yesmaximal, characteristic
6#67755C211yesmaximal, characteristic, complemented
7#67749C211yesmaximal, characteristic, complemented
8#9377—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C3×#529C2211yescharacteristic
11C3×#501C412yescomplemented
12C3×#489C412yescomplemented
13C3×#484—21no—
14C3×#481—21no—
15C3×#475—21no—
16C3×#474—21no—
17C3×#432C2211yescharacteristic
18C2×#835—21no—
19C2×#835—21no—
20#5024—21no—
21#5011—21no—
22#5011—21no—
23D4×C3⋊C8—21no—
24#5000—21no—
25#4999—21no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29C2×C4⋊(C3⋊C8)C2211yescharacteristic
30#4547—21no—
31#4547—21no—
32#4538—21no—
33#4537—21no—
34#3239D311yescharacteristic
35#2992—31no—
36#1981—31no—
37#1556—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—
72#835—42no—
73#835—42no—
74C3⋊#95—21no—
75#830—21no—
76#830—21no—
77#830—21no—
78#830—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)D411yescharacteristic
85C4⋊(C3⋊C8)D411yescharacteristic
86#803—21no—
87#803—21no—
88#803—21no—
89C4×C3⋊C8—21no—
90C2×C4⋊Q8C2×D311yescharacteristic
91#529C2×D311yescharacteristic
92#501Dic312yescomplemented
93#489Dic312yescomplemented
94#484—21no—
95#481—21no—
96#475—21no—
97#474—21no—
98#446—61no—
99#435—61no—
100#435—61no—
101C8×D4—61no—
102#433—61no—
103#432—31no—
104#432—31no—
105#432C2×D311yescharacteristic
106#424—61no—
107C2×C4⋊C8—31no—
108C2×#103—61no—
109C2×#103—61no—
110C4×C4⋊C4—31no—
111#332—61no—
112#332—61no—
113#328—61no—
114#327—61no—
115C2×C3×Q8—22no—
116C2×C3×Q8—41no—
117C2×C3×Q8—41no—
118C2×C3×Q8—41no—
119C2×C3×Q8—41no—
120C2×C3×Q8—41no—
121C2×C3×Q8C22⋊C411yescharacteristic
122C2×C3×D4—21no—
123C2×C3×D4C22⋊C411yescharacteristic
124C22×C3×C4—21no—
125C22×C3×C4—41no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4111yescharacteristic
129C22×Dic3—41no—
130C2×C3×C8—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4C22⋊C411yescharacteristic
140C3×C4⋊C4C22⋊C411yescharacteristic
141C3×C22⋊C4—21no—
142C3×C22⋊C4—21no—
143C3×C22⋊C4—21no—
144C3×C22⋊C4—21no—
145C3×C22⋊C4—21no—
146C3×C42—21no—
147C3×C42C22×C411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151C4⋊Dic3—22no—
152C4⋊Dic3—41no—
153C4×Dic3—81no—
154#208—81no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—21no—
157C2×C3⋊C8—21no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—81no—
163C22×Q8—21no—
164C2×M16—61no—
165C22×C8—31no—
166C22×C8—31no—
167C22×C8—61no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8C2×Dic312yes—
172#126—21no—
173#125C2×Dic311yescharacteristic
174#124C2×Dic311yescharacteristic
175#123—21no—
176C22⋊Q8#6711yescharacteristic
177#121#6711yescharacteristic
178C4×Q8#6711yescharacteristic
179C4×D4—21no—
180C4×D4#6711yescharacteristic
181#117—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—61no—
185C2×C42—61no—
186C2×C42C22×D311yescharacteristic
187#107—22no—
188#106—22no—
189C4⋊C8—21no—
190C4⋊C8—31no—
191C4⋊C8—31no—
192C4⋊C8—61no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197#103—22no—
198#103—122no—
199#103—122no—
200#102—22no—
201C22⋊C8—21no—
202C22⋊C8—61no—
203C22⋊C8—61no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206#97—61no—
207#97—61no—
208#97—61no—
209#97C2×Dic312yes—
210C4×C8—61no—
211#95—61no—
212C23×C3—21no—
213C23×C3#12111yescharacteristic
214C3×Q8—41no—
215C3×Q8—42no—
216C3×Q8—42no—
217C3×Q8—81no—
218C3×Q8—81no—
219C3×D4—41no—
220C3×D4—41no—
221C3×D4—41no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4C2×C22⋊C411yescharacteristic
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4C4×D411yescharacteristic
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12311yescharacteristic
241C2×Dic3—42no—
242C2×Dic3—81no—
243C2×Dic3—81no—
244C3×C8—42no—
245C3⋊C8—41no—
246C3⋊C8—41no—
247C3⋊C8—41no—
248C3⋊C8—41no—
249C3⋊C8—81no—
250C2×Q8—22no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8#21711yescharacteristic
257C2×D4—21no—
258C2×D4#21711yescharacteristic
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—121no—
262C22×C4D3×D411yescharacteristic
263C22×C4#23711yescharacteristic
264C22×C4C2×#6711yescharacteristic
265M16—241no—
266C2×C8—22no—
267C2×C8—61no—
268C2×C8—61no—
269C2×C8—61no—
270C2×C8—61no—
271C2×C8—121no—
272C2×C8—121no—
273C2×C8—121no—
274C2×C8—241no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—62no—
284C4⋊C4—121no—
285C4⋊C4#21711yescharacteristic
286C4⋊C4#21711yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C42—21no—
293C42—241no—
294C42D3×D411yescharacteristic
295C42#23711yescharacteristic
296C42C22×Dic311yescharacteristic
297C42C2×#6711yescharacteristic
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3#38611yescharacteristic
304C22×C3#47111yescharacteristic
305C22×C3#47311yescharacteristic
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—41no—
314C3×C4—82no—
315C3×C4#45311yescharacteristic
316C3×C4#45411yescharacteristic
317Dic3—82no—
318C23—21no—
319C23#93911yescharacteristic
320Q8—41no—
321Q8—42no—
322Q8—42no—
323Q8—81no—
324Q8—81no—
325D4—41no—
326D4—41no—
327D4—41no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—122no—
342C2×C4—241no—
343C2×C4—241no—
344C2×C4D4×Dic311yescharacteristic
345C2×C4D4×Dic311yescharacteristic
346C2×C4#93511yescharacteristic
347C2×C4#93711yescharacteristic
348C2×C4#93911yescharacteristic
349C2×C4C2×#21711yescharacteristic
350C8—42no—
351C8—121no—
352C8—121no—
353C8—121no—
354C8—121no—
355C8—241no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—81no—
359C2×C3#188311yescharacteristic
360C2×C3#190811yescharacteristic
361C2×C3#190911yescharacteristic
362C22—21no—
363C22—21no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22#520311yescharacteristic
368C22#520411yescharacteristic
369C22#522011yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—82no—
379C4—242no—
380C4#519811yescharacteristic
381C4#520111yescharacteristic
382C3#937711yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—81no—
386C2#7186511yescharacteristic
387C2#7208811yescharacteristic
388C2#7210311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3239C2
socleC22×C3#386
prime radicalC3×#121D4
cosocleC2×C3×C42C23
hypercenter#3239D3
hypocenterC3#9377
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3239

Complements

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

NH ≅ G/Ntype
3. #71778C2semidirect
4. #71776C2semidirect
6. #67755C2semidirect
7. #67749C2semidirect
11. C3×#501C4semidirect
12. C3×#489C4semidirect
92. #501Dic3semidirect
93. #489Dic3semidirect
382. C3#9377semidirect

Maximal subgroups

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

subgroupindex
C3×#32392
#831302
#717782
#717762
#717552
#677552
#677492
#93773