Finite Group Databasebeta

← #10690414

Subgroups — #10690414

Order 768.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79008C211yesmaximal, characteristic, complemented
2#78468C211yesmaximal, characteristic
3#70426C211yesmaximal, characteristic
4#69492C211yesmaximal, characteristic
5#69471C211yesmaximal, characteristic, complemented
6C3×#1279C211yesmaximal, characteristic, complemented
7#67656C211yesmaximal, characteristic, complemented
8#9054—31nomaximal, Sylow
9#5477C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C3×C4⋊C8C2211yescharacteristic
12C2×C3×C4×C8C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5114—22no—
15#5102—21no—
16#5101—21no—
17C2×#805—21no—
18C2×#805—21no—
19#4925C2211yescharacteristic
20C4×#208C2211yescharacteristic
21#4831—21no—
22#4830—21no—
23#4828—22no—
24#4694C411yescharacteristic, complemented
25#4693C411yescharacteristic, complemented
26C4×#204—22no—
27#4681C411yescharacteristic, complemented
28#4680C411yescharacteristic, complemented
29#4483—21no—
30#4483—21no—
31#4452—22no—
32#3167—31no—
33#2962—31no—
34#1980—31no—
35#1503—31no—
36#1480—31no—
37#1279D311yescharacteristic, complemented
38#1278—31no—
39C2×#235—21no—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C42—21no—
44C2×C3×C42C2311yescharacteristic
45#930—21no—
46#930—21no—
47#926—21no—
48#924—21no—
49#924D412yes—
50C2×#208D411yescharacteristic
51C2×#208D411yescharacteristic
52C2×#204—22no—
53#875—21no—
54#874C2×C411yescharacteristic
55C4×D12—21no—
56C4×D12—21no—
57#870—21no—
58#869C2×C411yescharacteristic
59#869C2×C411yescharacteristic
60#869C2×C411yescharacteristic
61C4×Dic6—21no—
62C4×Dic6D412yes—
63C3×C4⋊C8—21no—
64C3×C4⋊C8D411yescharacteristic
65C3×C4⋊C8D411yescharacteristic
66C3×C4×C8—21no—
67C3×C4×C8C2×C411yescharacteristic
68C3×C4×C8C2×C411yescharacteristic
69C3×#95—21no—
70#830—21no—
71#821—22no—
72#817—22no—
73#816—22no—
74#815—22no—
75Dic3⋊C8—22no—
76Dic3⋊C8—22no—
77C8×Dic3—22no—
78#805—42no—
79#805—42no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84#533—31no—
85#532—31no—
86#498—31no—
87#492—31no—
88#487—31no—
89#486—31no—
90#465—62no—
91#446—62no—
92#446—61no—
93C8×Q8—61no—
94C4×SD16—62no—
95#431—31no—
96#424—61no—
97C2×C4⋊C8C2×D311yescharacteristic
98C2×#104—61no—
99C2×#104—61no—
100#405—61no—
101C4×M16—31no—
102C2×C4×C8C2×D311yescharacteristic
103C4×C4⋊C4C2×D311yescharacteristic
104#328—62no—
105D4⋊C8—61no—
106D4⋊C8—61no—
107C22×C3×C4—41no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C2×#67—41no—
112#235—41no—
113#235—41no—
114#235—41no—
115#235—41no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×C4×D3—41no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6C22⋊C412yes—
123C2×C3×C8—21no—
124C2×C3×C8—21no—
125C2×C3×C8—21no—
126C2×C3×C8—21no—
127C2×C3×C8—42no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—41no—
132C3×C42—41no—
133C3×C42—41no—
134C3×C42C22×C411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138#217—22no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3C22⋊C412yes—
147Dic3⋊C4—22no—
148Dic3⋊C4—22no—
149C4×Dic3—22no—
150#208—21no—
151#208—21no—
152#208—21no—
153#208SD1612yescomplemented
154C2×C3⋊C8—21no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—41no—
157#204—44no—
158C2×#41—61no—
159C2×SD16—62no—
160C2×M16—31no—
161C2×M16—31no—
162C22×C8—21no—
163C22×C8—21no—
164C4⋊Q8—31no—
165C4⋊Q8—31no—
166C4⋊Q8—31no—
167#127—31no—
168#125—61no—
169#124—61no—
170C22⋊Q8—32no—
171C22⋊Q8—61no—
172#121—61no—
173#121—61no—
174C4×Q8—32no—
175C4×Q8—61no—
176C4×D4—61no—
177C4×D4—61no—
178#117—61no—
179C2×C4⋊C4—21no—
180C2×C42—21no—
181C2×C42C22×D311yescharacteristic
182#106—62no—
183C4⋊C8—21no—
184C4⋊C8—62no—
185C4⋊C8—62no—
186C4⋊C8—61no—
187C4⋊C8D1211yescharacteristic
188C4⋊C8D1211yescharacteristic
189#104—122no—
190#104—122no—
191#103—62no—
192#102—62no—
193C22⋊C8—61no—
194#97—62no—
195#97—61no—
196C4×C8—21no—
197C4×C8—62no—
198C4×C8—61no—
199C4×C8—61no—
200C4×C8C4×D311yescharacteristic
201C4×C8C4×D311yescharacteristic
202#95—21no—
203C23×C3#12111yescharacteristic
204C22×D3—41no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—41no—
208C2×C3×C4—42no—
209C2×C3×C4—81no—
210C2×C3×C4—81no—
211C2×C3×C4C2×C22⋊C411yescharacteristic
212C2×C3×C4C4×D411yescharacteristic
213C2×C3×C4C4×D411yescharacteristic
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12111yescharacteristic
216C2×C3×C4#12311yescharacteristic
217C2×C3×C4C2×SD1611yescharacteristic
218C2×C3×C4#13511yescharacteristic
219C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
220C2×C3×C4#13711yescharacteristic
221#67—81no—
222#67—81no—
223C2×Dic3—22no—
224C2×Dic3—22no—
225C2×Dic3—41no—
226D12—42no—
227D12—41no—
228D12—41no—
229D12—41no—
230D12—41no—
231C4×D3—81no—
232C4×D3—81no—
233Dic6—41no—
234Dic6—41no—
235Dic6—42no—
236Dic6—41no—
237Dic6—41no—
238C3×C8—22no—
239C3×C8—41no—
240C3×C8—41no—
241C3×C8—41no—
242C3⋊C8—22no—
243C3⋊C8—41no—
244C3⋊C8—81no—
245#41—121no—
246#41—121no—
247#41—121no—
248#41—121no—
249C2×Q8—32no—
250C2×Q8—61no—
251C2×Q8—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—121no—
256C22×C4—41no—
257C22×C4—121no—
258C22×C4#23511yescharacteristic
259C22×C4D3×D411yescharacteristic
260C22×C4D3×D411yescharacteristic
261SD16—124no—
262M16—32no—
263M16—61no—
264M16—61no—
265M16—61no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—42no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—61no—
274C2×C8—61no—
275C2×C8—121no—
276C4⋊C4—22no—
277C4⋊C4—32no—
278C4⋊C4—41no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—62no—
283C4⋊C4—62no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C22⋊C4—62no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C42—41no—
290C42—41no—
291C42—62no—
292C42C2×C4×D311yescharacteristic
293C42C2×D1211yescharacteristic
294C42D3×D411yescharacteristic
295C42#23911yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3#38611yescharacteristic
299C22×C3#46011yescharacteristic
300C22×C3#46411yescharacteristic
301C2×D3—81no—
302C2×D3—81no—
303C2×D3—81no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—82no—
308C3×C4C4×SD1611yescharacteristic
309C3×C4#44011yescharacteristic
310C3×C4#44311yescharacteristic
311C3×C4#44411yescharacteristic
312C3×C4#44811yescharacteristic
313C3×C4#44911yescharacteristic
314Dic3—42no—
315Dic3—42no—
316Dic3—81no—
317C23—121no—
318C23#88411yescharacteristic
319Q8—122no—
320Q8—121no—
321Q8—121no—
322Q8—121no—
323Q8—121no—
324D4—122no—
325D4—121no—
326D4—121no—
327D4—121no—
328D4—121no—
329D4—241no—
330D4—241no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—121no—
340C2×C4—241no—
341C2×C4—241no—
342C2×C4C4×D1211yescharacteristic
343C2×C4D3×C22⋊C411yescharacteristic
344C2×C4#88211yescharacteristic
345C2×C4#89311yescharacteristic
346C2×C4#89411yescharacteristic
347C2×C4#89511yescharacteristic
348C2×C4#91111yescharacteristic
349C2×C4D3×SD1611yescharacteristic
350C2×C4#91611yescharacteristic
351C2×C4#91811yescharacteristic
352C8—22no—
353C8—41no—
354C8—41no—
355C8—41no—
356C8—62no—
357C8—121no—
358C8—241no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3#187611yescharacteristic
362C2×C3#188011yescharacteristic
363C2×C3#188411yescharacteristic
364D3—161no—
365C22—21no—
366C22—21no—
367C22—241no—
368C22—241no—
369C22—241no—
370C22#466511yescharacteristic
371C22#476211yescharacteristic
372C22#479811yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—82no—
377C4—122no—
378C4—122no—
379C4—241no—
380C4#470011yescharacteristic
381C4#471711yescharacteristic
382C4#478111yescharacteristic
383C4#480611yescharacteristic
384C4#485211yescharacteristic
385C4#488911yescharacteristic
386C3#905411yescharacteristic, complemented, Sylow
387C2—21no—
388C2—21no—
389C2—481no—
390C2#6869511yescharacteristic
391C2#6877711yescharacteristic
392C2#6890911yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1279D3
hypocenterC3#9054
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
1. #79008C2semidirect
5. #69471C2semidirect
6. C3×#1279C2semidirect
7. #67656C2semidirect
24. #4694C4semidirect
25. #4693C4semidirect
27. #4681C4semidirect
28. #4680C4semidirect
37. #1279D3semidirect
153. #208SD16semidirect
386. C3#9054semidirect

Maximal subgroups

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

subgroupindex
#790082
#784682
#704262
#694922
#694712
C3×#12792
#676562
#90543