Finite Group Databasebeta

← #10690696

Subgroups — #10690696

Order 768.

546 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79263C211yesmaximal, characteristic, complemented
2#73387C211yesmaximal, characteristic
3#70786C211yesmaximal, characteristic, complemented
4#70757C211yesmaximal, characteristic
5#70756C211yesmaximal, characteristic
6C3×#1274C211yesmaximal, characteristic, complemented
7#67614C211yesmaximal, characteristic, complemented
8#9102—31nomaximal, Sylow
9C2×#875C411yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#376C2211yescharacteristic
15#4939C411yescharacteristic
16#4935C411yescharacteristic
17#4929C411yescharacteristic
18#4925C2211yescharacteristic
19#4925C2211yescharacteristic
20#4830—22no—
21#4718—22no—
22C8×Dic6—22no—
23#4670—22no—
24#4668—22no—
25#4483—22no—
26#4469—22no—
27#4459—22no—
28#2982—31no—
29#2326—31no—
30#1478—31no—
31#1478—31no—
32#1461—31no—
33#1274—31no—
34#1274D311yescharacteristic, complemented
35C22×D12C2×C411yescharacteristic
36C22×C4×D3—21no—
37C22×Dic6C2×C411yescharacteristic
38C2×C3×C42C2×C411yescharacteristic
39C2×C3×C42C2×C411yescharacteristic
40C2×C3×C42C2311yescharacteristic
41C2×(C2×D3)⋊C4—22no—
42C2×(C2×D3)⋊C4—21no—
43C2×(C2×D3)⋊C4—22no—
44C2×C4⋊Dic3—22no—
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×Dic3⋊C4D412yes—
48C2×C4×Dic3D412yes—
49#875—22no—
50#875C814yescomplemented
51C4×D12—21no—
52C4×D12—21no—
53C4×D12—21no—
54C4×D12—21no—
55C4×Dic6—21no—
56C4×Dic6—21no—
57C4×Dic6—21no—
58C4×Dic6—21no—
59C3×C4⋊C8—21no—
60C3×C4⋊C8—21no—
61C3×C22⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×C4×C8—21no—
64C3×C4×C8D412yes—
65C3×#95—21no—
66C3×#95—21no—
67C3⋊#95—22no—
68#830—21no—
69#830—21no—
70Dic3⋊C8—24no—
71C8×Dic3—24no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8D412yes—
76C2×#124—31no—
77C2×C4×Q8—31no—
78C2×C4×D4—31no—
79C8×Q8—62no—
80C8×Q8—62no—
81#431—31no—
82#431—31no—
83#431—62no—
84#431C2×D311yescharacteristic
85#431C2×D311yescharacteristic
86#391—31no—
87#390—31no—
88#388—62no—
89#385—62no—
90#384—31no—
91#376C2×D311yescharacteristic
92D4⋊C8—62no—
93D4⋊C8—62no—
94#324—62no—
95C23×D3—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×Dic3—21no—
102C22×Dic3C22⋊C412yes—
103C22×Dic3C22⋊C412yes—
104C2×D12—21no—
105C2×D12—21no—
106C2×D12—21no—
107C2×D12—21no—
108C2×D12—21no—
109C2×D12C2×C812yes—
110C2×C4×D3—41no—
111C2×C4×D3—41no—
112C2×C4×D3—82no—
113C2×Dic6—21no—
114C2×Dic6—21no—
115C2×Dic6—21no—
116C2×Dic6—21no—
117C2×Dic6C2×C812yes—
118C2×Dic6M1612yes—
119C2×C3×C8—22no—
120C2×C3×C8—41no—
121C3×C42—21no—
122C3×C42—21no—
123C3×C42—21no—
124C3×C42C2×C812yes—
125C3×C42C2×D411yescharacteristic
126C3×C42C2×D411yescharacteristic
127C3×C42#4111yescharacteristic
128C3×C42#4111yescharacteristic
129(C2×D3)⋊C4—28no—
130(C2×D3)⋊C4—41no—
131(C2×D3)⋊C4—41no—
132(C2×D3)⋊C4—82no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—82no—
138Dic3⋊C4—22no—
139Dic3⋊C4—22no—
140C4×Dic3—22no—
141C4×Dic3—22no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—41no—
144C22×Q8—31no—
145C22×D4—31no—
146C23×C4—61no—
147#124—34no—
148#124—62no—
149C4×Q8—61no—
150C4×Q8—61no—
151C4×Q8—61no—
152C4×Q8—61no—
153C4×D4—61no—
154C4×D4—61no—
155C4×D4—61no—
156C4×D4—61no—
157C2×C4⋊C4—31no—
158C2×C4⋊C4—32no—
159C2×C4⋊C4—31no—
160C2×C4⋊C4—62no—
161C2×C22⋊C4—62no—
162C2×C22⋊C4—61no—
163C2×C22⋊C4—62no—
164C2×C42—32no—
165C2×C42C4×D311yescharacteristic
166C2×C42C4×D311yescharacteristic
167C2×C42C22×D311yescharacteristic
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—64no—
171C4⋊C8—61no—
172C4⋊C8—61no—
173C22⋊C8—21no—
174C22⋊C8—21no—
175C22⋊C8—61no—
176C22⋊C8—61no—
177C4×C8—21no—
178C4×C8—32no—
179C4×C8—64no—
180C4×C8—61no—
181C4×C8D1212yes—
182#95—21no—
183#95—21no—
184#95—62no—
185C23×C3C2×C22⋊C411yescharacteristic
186C22×D3—22no—
187C22×D3—42no—
188C22×D3—42no—
189C22×D3—41no—
190C22×D3—41no—
191C2×C3×C4—22no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4C4×D411yescharacteristic
200C2×C3×C4C4×D411yescharacteristic
201C2×C3×C4#12011yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12111yescharacteristic
204C2×C3×C4#12311yescharacteristic
205C2×C3×C4C22×C811yescharacteristic
206C2×C3×C4C2×M1611yescharacteristic
207C2×C3×C4#13111yescharacteristic
208C2×C3×C4#13111yescharacteristic
209C2×Dic3—22no—
210C2×Dic3—22no—
211C2×Dic3—22no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214C2×Dic3—41no—
215C2×Dic3C22⋊C814yes—
216D12—22no—
217D12—22no—
218D12—41no—
219D12—41no—
220C4×D3—82no—
221C4×D3—82no—
222Dic6—22no—
223Dic6—22no—
224Dic6—22no—
225Dic6—22no—
226C3×C8—42no—
227C3×C8—81no—
228C3⋊C8—42no—
229C3⋊C8—81no—
230C24—61no—
231C2×Q8—32no—
232C2×Q8—32no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×Q8—61no—
236C2×Q8—61no—
237C2×D4—32no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—32no—
246C22×C4—32no—
247C22×C4—61no—
248C22×C4—121no—
249C22×C4—121no—
250C22×C4—242no—
251C22×C4C2×C4×D311yescharacteristic
252C22×C4D3×D411yescharacteristic
253C22×C4D3×D411yescharacteristic
254C2×C8—22no—
255C2×C8—41no—
256C2×C8—62no—
257C2×C8—121no—
258C4⋊C4—61no—
259C4⋊C4—62no—
260C4⋊C4—61no—
261C4⋊C4—62no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—242no—
265C22⋊C4—68no—
266C22⋊C4—121no—
267C22⋊C4—121no—
268C22⋊C4—242no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—62no—
273C42—62no—
274C42D3×C812yes—
275C42C2×D1211yescharacteristic
276C42#23511yescharacteristic
277C42D3×D411yescharacteristic
278C42#23911yescharacteristic
279C22×C3—21no—
280C22×C3—21no—
281C22×C3#38611yescharacteristic
282C22×C3C2×C22⋊C811yescharacteristic
283C22×C3#40811yescharacteristic
284C2×D3—44no—
285C2×D3—42no—
286C2×D3—82no—
287C2×D3—81no—
288C2×D3—81no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—42no—
294C3×C4—81no—
295C3×C4C8×D411yescharacteristic
296C3×C4C8×D411yescharacteristic
297C3×C4#43511yescharacteristic
298C3×C4#43511yescharacteristic
299Dic3—24no—
300Dic3—42no—
301Dic3—81no—
302C23—62no—
303C23—122no—
304C23—122no—
305C23—121no—
306C23—121no—
307C23D3×C22⋊C411yescharacteristic
308Q8—62no—
309Q8—62no—
310Q8—62no—
311Q8—62no—
312D4—62no—
313D4—62no—
314D4—121no—
315D4—121no—
316C2×C4—22no—
317C2×C4—34no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—62no—
326C2×C4—62no—
327C2×C4—62no—
328C2×C4—122no—
329C2×C4—122no—
330C2×C4—121no—
331C2×C4—242no—
332C2×C4—242no—
333C2×C4C4×D1211yescharacteristic
334C2×C4#88211yescharacteristic
335C2×C4#88411yescharacteristic
336C2×C4#89311yescharacteristic
337C2×C4#89411yescharacteristic
338C2×C4#89511yescharacteristic
339C2×C4C2×D3×C811yescharacteristic
340C2×C4#90111yescharacteristic
341C2×C4D3×M1611yescharacteristic
342C2×C4#90711yescharacteristic
343C8—42no—
344C8—81no—
345C8—122no—
346C8—241no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3#186911yescharacteristic
350C2×C3#191011yescharacteristic
351C2×C3#191211yescharacteristic
352D3—82no—
353C22—21no—
354C22—21no—
355C22—124no—
356C22—122no—
357C22—242no—
358C22—241no—
359C22—241no—
360C22#466511yescharacteristic
361C22D3×C22⋊C811yescharacteristic
362C22#472711yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—42no—
368C4—64no—
369C4—81no—
370C4—122no—
371C4—241no—
372C4C8×D1211yescharacteristic
373C4#470611yescharacteristic
374C4#483411yescharacteristic
375C4#483611yescharacteristic
376C3#910211yescharacteristic, complemented, Sylow
377C2—21no—
378C2—21no—
379C2—242no—
380C2#6866811yescharacteristic
381C2#6870811yescharacteristic
382C2#6871311yescharacteristic
subgroupG/H
centerC22#4665
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×#1274C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1274D3
hypocenterC3#9102
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
1. #79263C2semidirect
3. #70786C2semidirect
6. C3×#1274C2semidirect
7. #67614C2semidirect
34. #1274D3semidirect
50. #875C8semidirect
376. C3#9102semidirect

Maximal subgroups

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

subgroupindex
#792632
#733872
#707862
#707572
#707562
C3×#12742
#676142
#91023