Finite Group Databasebeta

← #10691267

Subgroups — #10691267

Order 768.

478 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83349C211yesmaximal, characteristic
2#79276C211yesmaximal, characteristic
3C3×#1503C211yesmaximal, characteristic, complemented
4#71826C211yesmaximal, characteristic, complemented
5#69494C211yesmaximal, characteristic
6#67745C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#9207—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C4×#235C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#5094C411yescharacteristic, complemented
14#5091C411yescharacteristic, complemented
15#5087C411yescharacteristic, complemented
16#5084C411yescharacteristic, complemented
17#5066—21no—
18#5063—21no—
19#5002—21no—
20#4998—21no—
21C2×#805—21no—
22#4924C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24C2×C4×C3⋊C8C2211yescharacteristic
25#4834—21no—
26#4830—21no—
27#67⋊C8—22no—
28#4685—21no—
29C4×D3×C8—21no—
30C3×#328—22no—
31#4535—21no—
32#4535—21no—
33#4523—21no—
34#3076—31no—
35C8×#41—31no—
36#1806—31no—
37#1503D311yescharacteristic, complemented
38#1493—31no—
39#1301—31no—
40#1279—31no—
41C2×#235D411yescharacteristic
42C3×C4⋊Q8C2×C411yescharacteristic
43C3×C4⋊Q8C2×C411yescharacteristic
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×#117—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C4×#67—21no—
54C4×#67—21no—
55#926D411yescharacteristic
56C2×#208—21no—
57C22×C3⋊C8—21no—
58C22×C3⋊C8—21no—
59C2×D3×C8—22no—
60C4×D12C2×C411yescharacteristic
61C4×D12D411yescharacteristic
62#872—21no—
63C42×D3—21no—
64C4×Dic6C2×C411yescharacteristic
65C4×Dic6D411yescharacteristic
66C3×C4⋊C8—21no—
67C3×C22⋊C8D412yes—
68C3×C4×C8—21no—
69C3×#95—21no—
70(C2×D3)⋊C8—22no—
71Dic3⋊C8—22no—
72C8×Dic3—22no—
73#810—21no—
74#810—21no—
75#809—21no—
76#809—21no—
77#808—21no—
78#808—21no—
79#807—21no—
80#807—21no—
81#805—42no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)C2×C411yescharacteristic
86C4⋊(C3⋊C8)C2×C411yescharacteristic
87C4×C3⋊C8—21no—
88C4×C3⋊C8D411yescharacteristic
89C4×C3⋊C8D411yescharacteristic
90#533C2×D311yescharacteristic
91C4×#41—31no—
92#479—61no—
93#478—61no—
94#477—31no—
95#476—31no—
96#475—31no—
97#474—31no—
98C8×Q8—61no—
99C8×D4—61no—
100C8×D4—62no—
101#431—61no—
102#431C2×D311yescharacteristic
103#427—61no—
104#423—31no—
105C2×C4⋊C8—31no—
106C2×#104—61no—
107#416—61no—
108C2×C4×C8—31no—
109C2×C4×C8—61no—
110C4×C4⋊C4C2×D311yescharacteristic
111#339—61no—
112#328—22no—
113Q8⋊C8—61no—
114Q8⋊C8—61no—
115C2×C3×Q8—21no—
116C2×C3×Q8—21no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C2×#67—21no—
122#235—21no—
123#235—21no—
124#235SD1612yes—
125#235Q1612yes—
126C2×D12C22⋊C411yescharacteristic
127C2×C4×D3—21no—
128C2×C4×D3—21no—
129C2×C4×D3—21no—
130C2×Dic6C22⋊C411yescharacteristic
131C2×C3×C8—22no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C22⋊C4—41no—
141C3×C42—41no—
142C3×C42—41no—
143C3×C42—41no—
144C3×C42C22×C411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42#4111yescharacteristic
148#217—21no—
149(C2×D3)⋊C4—21no—
150(C2×D3)⋊C4—21no—
151C4⋊Dic3C22⋊C411yescharacteristic
152C4⋊Dic3C22⋊C411yescharacteristic
153Dic3⋊C4—21no—
154Dic3⋊C4—21no—
155C4×Dic3—21no—
156C4×Dic3—21no—
157C4×Dic3—21no—
158#208—41no—
159#208—41no—
160C2×C3⋊C8—21no—
161C2×C3⋊C8—21no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—42no—
165C2×C3⋊C8—41no—
166C2×C3⋊C8—41no—
167C2×C3⋊C8—41no—
168D3×C8—44no—
169C2×#41—31no—
170C2×M16—61no—
171C22×C8—62no—
172C22×C8—61no—
173C22×C8—61no—
174C4⋊Q8C4×D311yescharacteristic
175C4⋊Q8C4×D311yescharacteristic
176#125—21no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—31no—
182C4×Q8—31no—
183C4×D4—31no—
184C4×D4—31no—
185C4×D4—61no—
186C4×D4—61no—
187#117—21no—
188#117—31no—
189#117—61no—
190C2×C4⋊C4—21no—
191C2×C42—21no—
192C2×C42—61no—
193C2×C42C22×D311yescharacteristic
194#107—61no—
195#107—61no—
196#106—61no—
197#106—61no—
198C4⋊C8—21no—
199C4⋊C8—31no—
200C4⋊C8—31no—
201C4⋊C8—62no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204C4⋊C8—61no—
205#104—122no—
206#103—61no—
207#103—61no—
208#102—61no—
209#102—61no—
210C22⋊C8—62no—
211C22⋊C8#6712yes—
212C4×C8—21no—
213C4×C8—31no—
214C4×C8—31no—
215C4×C8—62no—
216C4×C8—61no—
217#95—21no—
218C23×C3#12111yescharacteristic
219C22×D3—21no—
220C3×Q8—41no—
221C3×Q8—41no—
222C3×Q8—81no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—81no—
230C2×C3×C4—81no—
231C2×C3×C4C2×C22⋊C411yescharacteristic
232C2×C3×C4C4×D411yescharacteristic
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12311yescharacteristic
237C2×C3×C4C2×SD1611yescharacteristic
238C2×C3×C4C2×Q1611yescharacteristic
239C2×C3×C4#13511yescharacteristic
240C2×C3×C4#13511yescharacteristic
241#67—22no—
242#67—22no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246D12—21no—
247D12#10312yes—
248C4×D3—22no—
249C4×D3—22no—
250C4×D3—42no—
251C4×D3—41no—
252C4×D3—41no—
253Dic6—21no—
254Dic6#10312yes—
255C3×C8—42no—
256C3⋊C8—22no—
257C3⋊C8—41no—
258C3⋊C8—41no—
259C3⋊C8—41no—
260C3⋊C8—81no—
261#41—32no—
262#41—32no—
263#41—61no—
264#41—61no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—31no—
268C2×D4—31no—
269C2×D4—61no—
270C22×C4—41no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4D3×D411yescharacteristic
275C22×C4#23711yescharacteristic
276C22×C4C2×#6711yescharacteristic
277M16—121no—
278M16—121no—
279C2×C8—22no—
280C2×C8—61no—
281C2×C8—61no—
282C2×C8—61no—
283C2×C8—61no—
284C2×C8—124no—
285C2×C8—122no—
286C2×C8—121no—
287C2×C8—121no—
288C2×C8—121no—
289C4⋊C4—22no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—31no—
295C4⋊C4—31no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C22⋊C4—41no—
302C22⋊C4—61no—
303C22⋊C4—61no—
304C22⋊C4—61no—
305C42—41no—
306C42—41no—
307C42—41no—
308C42—61no—
309C42—61no—
310C42—61no—
311C42C2×C4×D311yescharacteristic
312C42#23511yescharacteristic
313C42D3×D411yescharacteristic
314C42D3×D411yescharacteristic
315C22×C3—21no—
316C22×C3—21no—
317C22×C3#38611yescharacteristic
318C22×C3#46511yescharacteristic
319C22×C3#46711yescharacteristic
320C2×D3—22no—
321C2×D3—41no—
322C2×D3—41no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—82no—
327C3×C4—81no—
328C3×C4—81no—
329C3×C4C2×#10311yescharacteristic
330C3×C4#41611yescharacteristic
331C3×C4#44311yescharacteristic
332C3×C4#44411yescharacteristic
333C3×C4#46511yescharacteristic
334C3×C4#46711yescharacteristic
335Dic3—22no—
336Dic3—41no—
337Dic3—41no—
338C23—61no—
339C23#93811yescharacteristic
340Q8—32no—
341Q8—41no—
342Q8—41no—
343Q8—61no—
344Q8—81no—
345D4—32no—
346D4—62no—
347D4—62no—
348D4—61no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—61no—
356C2×C4—62no—
357C2×C4—62no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—122no—
363C2×C4—121no—
364C2×C4—121no—
365C2×C4D3×C22⋊C411yescharacteristic
366C2×C4#67⋊C411yescharacteristic
367C2×C4#88311yescharacteristic
368C2×C4#88411yescharacteristic
369C2×C4D3×SD1611yescharacteristic
370C2×C4#91611yescharacteristic
371C2×C4D3×Q1611yescharacteristic
372C2×C4#91911yescharacteristic
373C2×C4C4×#6711yescharacteristic
374C2×C4#93711yescharacteristic
375C8—42no—
376C8—62no—
377C8—121no—
378C8—121no—
379C8—121no—
380C8—241no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3#187611yescharacteristic
384C2×C3#189211yescharacteristic
385C2×C3#190611yescharacteristic
386D3—42no—
387C22—21no—
388C22—21no—
389C22—62no—
390C22—121no—
391C22—121no—
392C22#495611yescharacteristic
393C22#517111yescharacteristic
394C22#518811yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—62no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—121no—
403C4—121no—
404C4D3×#10311yescharacteristic
405C4#480411yescharacteristic
406C4#480711yescharacteristic
407C4#481311yescharacteristic
408C4#482311yescharacteristic
409C4#482411yescharacteristic
410C3#920711yescharacteristic, complemented, Sylow
411C2—21no—
412C2—21no—
413C2—122no—
414C2#7005511yescharacteristic
415C2#7027011yescharacteristic
416C2#7039011yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#386
prime radicalC2×#235D4
cosocleC2×C3×C42C23
hypercenter#1503D3
hypocenterC3#9207
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
3. C3×#1503C2semidirect
4. #71826C2semidirect
6. #67745C2semidirect
7. #67619C2semidirect
13. #5094C4semidirect
14. #5091C4semidirect
15. #5087C4semidirect
16. #5084C4semidirect
37. #1503D3semidirect
410. C3#9207semidirect

Maximal subgroups

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

subgroupindex
#833492
#792762
C3×#15032
#718262
#694942
#677452
#676192
#92073