Finite Group Databasebeta

← #10690070

Subgroups — #10690070

Order 768.

430 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83349C211yesmaximal, characteristic
2#79000C211yesmaximal, characteristic
3C3×#1481C211yesmaximal, characteristic, complemented
4#71826C211yesmaximal, characteristic, complemented
5#70754C211yesmaximal, characteristic
6#67745C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#8994—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C4×#235C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13#5110—21no—
14#5101—21no—
15#5094C411yescharacteristic
16#5091C411yescharacteristic
17#5087C411yescharacteristic
18#5084C411yescharacteristic
19#5066—21no—
20#5063—21no—
21#5002—21no—
22#4998—21no—
23#4924C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26#4840—22no—
27#4835—21no—
28#4833—21no—
29C3×Q8⋊C8—21no—
30C3×Q8⋊C8—21no—
31#4535—22no—
32#4484—21no—
33#4483—21no—
34#3076—31no—
35#2925—31no—
36#1493—31no—
37#1481D311yescharacteristic, complemented
38#1476—31no—
39#1301—31no—
40#1277—31no—
41C2×#235D411yescharacteristic
42C22×C3×C8—21no—
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×C4⋊Q8C2×C411yescharacteristic
45C3×#125—21no—
46C3×C22⋊Q8—21no—
47C3×C22⋊Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×#117—21no—
51C2×C3×C42C2311yescharacteristic
52C4×#67—21no—
53C4×#67—21no—
54#926D411yescharacteristic
55C2×#208—21no—
56C22×C3⋊C8—21no—
57C22×C3⋊C8—21no—
58C4×D12C2×C411yescharacteristic
59C4×D12D411yescharacteristic
60#872—21no—
61C42×D3—21no—
62C4×Dic6C2×C411yescharacteristic
63C4×Dic6D411yescharacteristic
64C3×C4⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8D411yescharacteristic
67C3×C4⋊C8D411yescharacteristic
68C3×C22⋊C8—21no—
69C3×C4×C8—21no—
70(C2×D3)⋊C8—22no—
71(C2×D3)⋊C8—22no—
72#815—22no—
73C8×Dic3—22no—
74#810—21no—
75#810—21no—
76#809—21no—
77#809—21no—
78#808—21no—
79#808—21no—
80#807—21no—
81#807—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)C2×C411yescharacteristic
85C4⋊(C3⋊C8)C2×C411yescharacteristic
86C4⋊(C3⋊C8)D412yes—
87C4×C3⋊C8—21no—
88C4×C3⋊C8—21no—
89#533C2×D311yescharacteristic
90C4×#41—31no—
91#479—61no—
92#478—61no—
93#477—31no—
94#476—31no—
95#475—31no—
96#474—31no—
97#433—62no—
98#432—61no—
99#431—61no—
100#431C2×D311yescharacteristic
101#427—61no—
102#423—31no—
103C2×C4⋊C8—31no—
104C2×C4⋊C8C2×D311yescharacteristic
105#416—61no—
106#408—61no—
107#405—61no—
108C2×C4×C8—31no—
109Q8⋊C8—21no—
110Q8⋊C8—21no—
111Q8⋊C8—61no—
112Q8⋊C8—62no—
113D4⋊C8—61no—
114C2×C3×Q8—21no—
115C2×C3×Q8—21no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C2×#67—21no—
120#235—21no—
121#235—21no—
122#235—21no—
123#235—21no—
124C2×D12C22⋊C411yescharacteristic
125C2×C4×D3—21no—
126C2×C4×D3—21no—
127C2×C4×D3—21no—
128C2×Dic6C22⋊C411yescharacteristic
129C2×C3×C8—21no—
130C2×C3×C8—21no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42—41no—
142C3×C42C22×C411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146#217—21no—
147(C2×D3)⋊C4—21no—
148(C2×D3)⋊C4—21no—
149C4⋊Dic3C22⋊C411yescharacteristic
150C4⋊Dic3C22⋊C411yescharacteristic
151Dic3⋊C4—21no—
152Dic3⋊C4—21no—
153C4×Dic3—21no—
154C4×Dic3—21no—
155C4×Dic3—21no—
156#208—41no—
157#208—41no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C2×#41—31no—
167C2×M16—61no—
168C22×C8—21no—
169C22×C8—61no—
170C22×C8—61no—
171C4⋊Q8C4×D311yescharacteristic
172C4⋊Q8C4×D311yescharacteristic
173#125—21no—
174C22⋊Q8—21no—
175C22⋊Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—31no—
179C4×Q8—31no—
180C4×D4—31no—
181C4×D4—31no—
182C4×D4—61no—
183C4×D4—61no—
184#117—21no—
185#117—31no—
186#117—61no—
187C2×C42—61no—
188C2×C42C22×D311yescharacteristic
189#107—61no—
190#107—61no—
191#106—61no—
192#106—61no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—31no—
196C4⋊C8—31no—
197C4⋊C8—32no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200C4⋊C8#6711yescharacteristic
201C4⋊C8#6711yescharacteristic
202#103—61no—
203#103—61no—
204#102—61no—
205#102—61no—
206C22⋊C8—21no—
207C22⋊C8—62no—
208C22⋊C8—62no—
209#97—62no—
210C4×C8—21no—
211C4×C8—62no—
212C4×C8—61no—
213C4×C8—61no—
214C23×C3#12311yescharacteristic
215C22×D3—21no—
216C3×Q8—42no—
217C3×Q8—41no—
218C3×Q8—41no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4C2×C22⋊C411yescharacteristic
225C2×C3×C4C4×D411yescharacteristic
226C2×C3×C4C4×D411yescharacteristic
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#13511yescharacteristic
231C2×C3×C4#13511yescharacteristic
232C2×C3×C4#13711yescharacteristic
233C2×C3×C4#13711yescharacteristic
234#67—41no—
235#67—41no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239D12—21no—
240D12—21no—
241C4×D3—41no—
242C4×D3—41no—
243C4×D3—41no—
244C4×D3—41no—
245C4×D3—81no—
246Dic6—21no—
247Dic6—21no—
248C3×C8—41no—
249C3×C8—41no—
250C3×C8—81no—
251C3⋊C8—42no—
252C3⋊C8—41no—
253C3⋊C8—41no—
254C3⋊C8—42no—
255#41—61no—
256#41—61no—
257#41—61no—
258#41—61no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—31no—
262C2×D4—31no—
263C2×D4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4#23511yescharacteristic
268C22×C4D3×D411yescharacteristic
269C22×C4#23711yescharacteristic
270M16—121no—
271M16—121no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—62no—
278C2×C8—61no—
279C2×C8—61no—
280C2×C8—62no—
281C2×C8—121no—
282C2×C8—121no—
283C2×C8—121no—
284C2×C8—121no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—31no—
290C4⋊C4—31no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C22⋊C4—41no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C42—41no—
300C42—61no—
301C42—61no—
302C42—61no—
303C42C2×C4×D311yescharacteristic
304C42D3×D411yescharacteristic
305C42D3×D411yescharacteristic
306C42C2×#6711yescharacteristic
307C22×C3—21no—
308C22×C3—21no—
309C22×C3#38611yescharacteristic
310C22×C3#48511yescharacteristic
311C22×C3#48511yescharacteristic
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—41no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—81no—
319C3×C4—81no—
320C3×C4#41611yescharacteristic
321C3×C4#41911yescharacteristic
322C3×C4#44311yescharacteristic
323C3×C4#44411yescharacteristic
324C3×C4#45211yescharacteristic
325C3×C4#45211yescharacteristic
326Dic3—41no—
327Dic3—41no—
328Dic3—41no—
329C23—61no—
330C23#88311yescharacteristic
331Q8—42no—
332Q8—41no—
333Q8—41no—
334Q8—61no—
335Q8—61no—
336D4—61no—
337D4—61no—
338D4—121no—
339D4—121no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—241no—
353C2×C4D3×C22⋊C411yescharacteristic
354C2×C4#67⋊C411yescharacteristic
355C2×C4#88411yescharacteristic
356C2×C4#91511yescharacteristic
357C2×C4#91611yescharacteristic
358C2×C4#91811yescharacteristic
359C2×C4#91911yescharacteristic
360C2×C4C4×#6711yescharacteristic
361C2×C4#93711yescharacteristic
362C2×C4#93811yescharacteristic
363C8—41no—
364C8—41no—
365C8—81no—
366C8—122no—
367C8—122no—
368C8—121no—
369C8—121no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3#187311yescharacteristic
373C2×C3#187611yescharacteristic
374C2×C3#188111yescharacteristic
375D3—81no—
376C22—21no—
377C22—21no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22#481411yescharacteristic
382C22#481511yescharacteristic
383C22#495611yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—81no—
388C4—81no—
389C4—121no—
390C4—121no—
391C4—121no—
392C4#480211yescharacteristic
393C4#480411yescharacteristic
394C4#514411yescharacteristic
395C4#514511yescharacteristic
396C4#517311yescharacteristic
397C4#518711yescharacteristic
398C3#899411yescharacteristic, complemented, Sylow
399C2—21no—
400C2—21no—
401C2—241no—
402C2#7001111yescharacteristic
403C2#7026911yescharacteristic
404C2#7041611yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1481C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1481D3
hypocenterC3#8994
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
3. C3×#1481C2semidirect
4. #71826C2semidirect
6. #67745C2semidirect
7. #67703C2semidirect
37. #1481D3semidirect
398. C3#8994semidirect

Maximal subgroups

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

subgroupindex
#833492
#790002
C3×#14812
#718262
#707542
#677452
#677032
#89943