Finite Group Databasebeta

← #10692032

Subgroups — #10692032

Order 768.

512 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79057C211yesmaximal, characteristic
2#79013C211yesmaximal, characteristic
3#70529C211yesmaximal, characteristic
4#70506C211yesmaximal, characteristic, complemented
5#70460C211yesmaximal, characteristic, complemented
6C3×#1292C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#9316—31nomaximal, Sylow
9C2×#909—21no—
10C4×#235C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×#825—21no—
16#5120—21no—
17C8×#67—22no—
18C4×C4⋊Dic3C2211yescharacteristic
19#4925C2211yescharacteristic
20#4849—21no—
21#4849C412yescomplemented
22#4841—21no—
23#4841C412yescomplemented
24#4834—21no—
25#4833—21no—
26D3×C4⋊C8—21no—
27#4830—21no—
28#4555—21no—
29#4551—22no—
30#4482—21no—
31#4482—21no—
32#3039—31no—
33#2988—31no—
34#1988—31no—
35#1806—31no—
36#1546—31no—
37#1303—31no—
38#1292D311yescharacteristic, complemented
39C2×#235D411yescharacteristic
40C22×Dic6—21no—
41C2×C3×M16—21no—
42C2×C3×M16—21no—
43C22×C3×C8D412yes—
44C2×C3×C42C2311yescharacteristic
45C4×#67—21no—
46C4×#67—21no—
47#926D411yescharacteristic
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4×Dic3—21no—
52#909—42no—
53#909—42no—
54C2×Dic12—22no—
55C2×#204—22no—
56C2×D3×C8—22no—
57C4×D12C2×C411yescharacteristic
58C4×D12D411yescharacteristic
59#872—21no—
60C42×D3—21no—
61#869—21no—
62#869—21no—
63#869—21no—
64#869C2×C412yes—
65C4×Dic6C2×C411yescharacteristic
66C4×Dic6D411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8C2×C412yes—
73C3⋊#95—21no—
74#830—21no—
75#825—42no—
76(C2×D3)⋊C8—22no—
77#816—22no—
78#816—22no—
79Dic3⋊C8—22no—
80C8×Dic3—22no—
81C4⋊(C3⋊C8)D411yescharacteristic
82C4⋊(C3⋊C8)D411yescharacteristic
83C4×C3⋊C8—21no—
84C2×#137—61no—
85C2×C4⋊Q8—31no—
86C4×#41—31no—
87#462—32no—
88#462—61no—
89#461—32no—
90#461—61no—
91C8×Q8—61no—
92C8×D4—61no—
93C8×D4—62no—
94#431—31no—
95#431—61no—
96#423C2×D311yescharacteristic
97#423C2×D311yescharacteristic
98C2×C4⋊C8—61no—
99C2×C4⋊C8C2×D311yescharacteristic
100C2×#104—61no—
101#419—61no—
102C4×C4⋊C4—31no—
103#340—62no—
104#339—61no—
105#332—61no—
106#332—61no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C2×#67—21no—
111C22×Dic3—41no—
112C22×Dic3—41no—
113#235—21no—
114#235—21no—
115#235D812yes—
116#235SD1612yes—
117C2×D12C22⋊C411yescharacteristic
118C2×C4×D3—21no—
119C2×C4×D3—21no—
120C2×C4×D3—21no—
121C2×Dic6—22no—
122C2×Dic6—41no—
123C2×Dic6—41no—
124C2×Dic6—41no—
125C2×Dic6—41no—
126C2×Dic6—41no—
127C2×Dic6C22⋊C411yescharacteristic
128C3×M16—41no—
129C3×M16—41no—
130C3×M16—41no—
131C3×M16—41no—
132C2×C3×C8—22no—
133C2×C3×C8—22no—
134C2×C3×C8—42no—
135C2×C3×C8—41no—
136C3×C42C22×C411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140#217—21no—
141(C2×D3)⋊C4—21no—
142(C2×D3)⋊C4—21no—
143C4⋊Dic3—22no—
144C4⋊Dic3—22no—
145C4⋊Dic3—22no—
146C4⋊Dic3—41no—
147C4⋊Dic3—41no—
148C4⋊Dic3—41no—
149C4⋊Dic3C22⋊C411yescharacteristic
150C4⋊Dic3C22⋊C411yescharacteristic
151Dic3⋊C4—21no—
152Dic3⋊C4—21no—
153C4×Dic3—21no—
154C4×Dic3—21no—
155C4×Dic3—21no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—21no—
160Dic12—44no—
161#204—44no—
162D3×C8—82no—
163C2×#41—31no—
164C22×Q8—61no—
165#137—122no—
166#137—122no—
167C2×Q16—62no—
168C2×SD16—62no—
169C2×M16—21no—
170C2×M16—21no—
171C22×C8—62no—
172C22×C8D1212yes—
173C4⋊Q8—32no—
174C4⋊Q8—61no—
175C4⋊Q8—61no—
176C4⋊Q8—61no—
177C4×Q8—31no—
178C4×Q8—31no—
179C4×D4—31no—
180C4×D4—31no—
181C4×D4—61no—
182C4×D4—61no—
183#117—31no—
184#117—61no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C42—61no—
189C2×C42—61no—
190C2×C42C22×D311yescharacteristic
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—31no—
197C4⋊C8—31no—
198C4⋊C8—62no—
199C4⋊C8C4×D312yes—
200#104—122no—
201#103—62no—
202#103—62no—
203C22⋊C8—62no—
204C22⋊C8—61no—
205C4×C8—62no—
206C4×C8—61no—
207#95—61no—
208C23×C3#12111yescharacteristic
209C22×D3—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—41no—
213C2×C3×C4C2×C22⋊C411yescharacteristic
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12311yescharacteristic
219C2×C3×C4C2×D811yescharacteristic
220C2×C3×C4C2×SD1611yescharacteristic
221C2×C3×C4#13511yescharacteristic
222C2×C3×C4#13511yescharacteristic
223#67—22no—
224#67—22no—
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—81no—
231C2×Dic3—81no—
232C2×Dic3—81no—
233C2×Dic3—81no—
234D12—21no—
235D12#10212yes—
236C4×D3—22no—
237C4×D3—22no—
238C4×D3—41no—
239C4×D3—41no—
240C4×D3—81no—
241Dic6—21no—
242Dic6—42no—
243Dic6—42no—
244Dic6—81no—
245Dic6—81no—
246Dic6#10212yes—
247C3×C8—42no—
248C3×C8—42no—
249C3×C8—81no—
250C3⋊C8—41no—
251C3⋊C8—41no—
252#41—32no—
253#41—32no—
254#41—61no—
255#41—61no—
256C2×Q8—31no—
257C2×Q8—62no—
258C2×Q8—121no—
259C2×Q8—121no—
260C2×Q8—121no—
261C2×Q8—121no—
262C2×Q8—121no—
263C2×D4—31no—
264C2×D4—61no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—121no—
269C22×C4—121no—
270C22×C4C2×D1211yescharacteristic
271C22×C4D3×D411yescharacteristic
272C22×C4#23911yescharacteristic
273Q16—124no—
274SD16—124no—
275M16—41no—
276M16—41no—
277M16—41no—
278M16—41no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—42no—
282C2×C8—41no—
283C2×C8—61no—
284C2×C8—61no—
285C2×C8—242no—
286C4⋊C4—31no—
287C4⋊C4—31no—
288C4⋊C4—62no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—62no—
292C4⋊C4—62no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C42—61no—
300C42—61no—
301C42—61no—
302C42—121no—
303C42—121no—
304C42C2×C4×D311yescharacteristic
305C42#23511yescharacteristic
306C42D3×D411yescharacteristic
307C42D3×D411yescharacteristic
308C22×C3—21no—
309C22×C3—21no—
310C22×C3#38611yescharacteristic
311C22×C3#46511yescharacteristic
312C22×C3#46611yescharacteristic
313C2×D3—22no—
314C2×D3—41no—
315C2×D3—41no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4C2×#10211yescharacteristic
320C3×C4#41611yescharacteristic
321C3×C4#45511yescharacteristic
322C3×C4#45611yescharacteristic
323C3×C4#46511yescharacteristic
324C3×C4#46611yescharacteristic
325Dic3—22no—
326Dic3—41no—
327Dic3—41no—
328Dic3—82no—
329Dic3—82no—
330C23—61no—
331C23#89511yescharacteristic
332Q8—32no—
333Q8—61no—
334Q8—122no—
335Q8—122no—
336Q8—241no—
337Q8—241no—
338D4—32no—
339D4—62no—
340D4—62no—
341D4—61no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—61no—
346C2×C4—62no—
347C2×C4—62no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4—241no—
355C2×C4—241no—
356C2×C4—241no—
357C2×C4—241no—
358C2×C4—241no—
359C2×C4C4×D1211yescharacteristic
360C2×C4D3×C22⋊C411yescharacteristic
361C2×C4#88211yescharacteristic
362C2×C4#88411yescharacteristic
363C2×C4#89311yescharacteristic
364C2×C4#89411yescharacteristic
365C2×C4D3×D811yescharacteristic
366C2×C4D3×SD1611yescharacteristic
367C2×C4#91611yescharacteristic
368C2×C4#91911yescharacteristic
369C8—42no—
370C8—42no—
371C8—81no—
372C8—121no—
373C8—121no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3#188211yescharacteristic
377C2×C3#189311yescharacteristic
378C2×C3#190211yescharacteristic
379D3—42no—
380C22—21no—
381C22—21no—
382C22—62no—
383C22—121no—
384C22—121no—
385C22#466511yescharacteristic
386C22#486411yescharacteristic
387C22#488311yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—62no—
392C4—121no—
393C4—121no—
394C4—242no—
395C4—242no—
396C4#474611yescharacteristic
397C4#475211yescharacteristic
398C4D3×#10211yescharacteristic
399C4#477511yescharacteristic
400C4#480411yescharacteristic
401C4#480711yescharacteristic
402C3#931611yescharacteristic, complemented, Sylow
403C2—21no—
404C2—21no—
405C2—122no—
406C2#6872511yescharacteristic
407C2#6888311yescharacteristic
408C2#6900411yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1292C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1292D3
hypocenterC3#9316
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1292

Complements

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

NH ≅ G/Ntype
4. #70506C2semidirect
5. #70460C2semidirect
6. C3×#1292C2semidirect
7. #67715C2semidirect
21. #4849C4semidirect
23. #4841C4semidirect
38. #1292D3semidirect
402. C3#9316semidirect

Maximal subgroups

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

subgroupindex
#790572
#790132
#705292
#705062
#704602
C3×#12922
#677152
#93163