Finite Group Databasebeta

← #10691017

Subgroups — #10691017

Order 768.

458 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2984C211yesmaximal, characteristic, complemented
2#82444C211yesmaximal, characteristic
3#71631C211yesmaximal, characteristic, complemented
4#70806C211yesmaximal, characteristic
5#70789C211yesmaximal, characteristic
6#67749C211yesmaximal, characteristic, complemented
7#67742C211yesmaximal, characteristic, complemented
8#9164—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#446—21no—
12C3×#446—22no—
13C3×C8×Q8—21no—
14C3×#432—21no—
15C3×#432C2211yescharacteristic
16C3×#431—21no—
17C3×#431C2211yescharacteristic
18#5227—21no—
19C2×#835—21no—
20#5027C411yescharacteristic, complemented
21#5027C411yescharacteristic, complemented
22#5026—21no—
23#5026—21no—
24#5021C411yescharacteristic, complemented
25#5021C411yescharacteristic, complemented
26#4967—21no—
27C4×C4⋊Dic3C2211yescharacteristic
28#4926C2211yescharacteristic
29C2×#803C2211yescharacteristic
30#4529—21no—
31#4470—21no—
32#4470—21no—
33#4464—22no—
34#3112—31no—
35#2984D311yescharacteristic, complemented
36#1763—31no—
37#1530—31no—
38#1502—31no—
39#1302—31no—
40#1285—31no—
41C22×C3×Q8D411yescharacteristic
42C2×#235—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8C2×C411yescharacteristic
46C3×C4×Q8C2×C411yescharacteristic
47C3×C4×Q8D411yescharacteristic
48C3×C4×Q8D411yescharacteristic
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#215—21no—
54C2×#215—21no—
55#930—21no—
56#930—21no—
57C2×C4⋊Dic3—21no—
58C2×C4×Dic3—21no—
59C22×C3⋊C8—21no—
60#875—21no—
61#874C2×C411yescharacteristic
62C4×D12—21no—
63C4×D12—21no—
64#869C2×C411yescharacteristic
65C3×C4⋊C8—22no—
66C3×C4⋊C8—22no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8D411yescharacteristic
71C3×C22⋊C8D411yescharacteristic
72C3×#97—22no—
73C3×#97D412yes—
74C3×C4×C8—22no—
75C3×C4×C8—21no—
76#835—24no—
77#835—21no—
78#835—21no—
79C3⋊#95—21no—
80#830—21no—
81#809—21no—
82#809—21no—
83#807—21no—
84#807—21no—
85C4⋊(C3⋊C8)C2×C411yescharacteristic
86C4⋊(C3⋊C8)C2×C411yescharacteristic
87#803—21no—
88#803—21no—
89#803—21no—
90#532—31no—
91C2×C4×Q8C2×D311yescharacteristic
92#480—61no—
93#477—31no—
94#477—31no—
95#460—31no—
96#460—31no—
97#448—61no—
98#446—21no—
99#446—22no—
100C8×Q8—21no—
101#440—61no—
102#440—61no—
103#432—21no—
104#432—31no—
105#432C2×D311yescharacteristic
106#431—21no—
107#431C2×D311yescharacteristic
108C2×#103—61no—
109C2×#97—31no—
110C4×C4⋊C4—31no—
111#340—61no—
112#329—62no—
113#327—61no—
114#327—61no—
115C2×C3×Q8—21no—
116C2×C3×Q8—22no—
117C2×C3×Q8—41no—
118C2×C3×Q8C22⋊C411yescharacteristic
119C2×C3×Q8C22⋊C411yescharacteristic
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C2×#67—41no—
126C22×Dic3—41no—
127#235—81no—
128#235—81no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×C4×D3—41no—
133C2×Dic6—21no—
134C2×C3×C8—21no—
135C2×C3×C8—21no—
136C2×C3×C8—22no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4C22⋊C411yescharacteristic
143C3×C4⋊C4C22⋊C411yescharacteristic
144C3×C42—21no—
145C3×C42—21no—
146C3×C42—21no—
147C3×C42—21no—
148C3×C42C22×C411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152#215—42no—
153#215—42no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—22no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—41no—
160C4×Dic3—81no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—41no—
166C2×#41—61no—
167C22×Q8D1211yescharacteristic
168C2×SD16—61no—
169C2×SD16—61no—
170C22×C8—61no—
171C4⋊Q8—31no—
172#127—31no—
173#124—61no—
174#121—61no—
175#121—61no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8C4×D311yescharacteristic
179C4×Q8C4×D311yescharacteristic
180C4×Q8#6711yescharacteristic
181C4×Q8#6711yescharacteristic
182C4×D4—61no—
183C4×D4—61no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4D1211yescharacteristic
187C2×C42—21no—
188C2×C42—61no—
189C2×C42C22×D311yescharacteristic
190#107—61no—
191#107—61no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—31no—
197C4⋊C8—31no—
198#103—64no—
199#103—61no—
200#103—61no—
201#102—61no—
202#102—61no—
203C22⋊C8—21no—
204C22⋊C8—61no—
205C22⋊C8#6711yescharacteristic
206C22⋊C8#6711yescharacteristic
207#97—22no—
208#97—61no—
209#97—61no—
210#97—61no—
211#97#6712yes—
212C4×C8—22no—
213C4×C8—21no—
214#95—61no—
215C23×C3#12111yescharacteristic
216C22×D3—41no—
217C3×Q8—21no—
218C3×Q8—21no—
219C3×Q8—21no—
220C3×Q8—42no—
221C3×Q8#10412yes—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4C2×C22⋊C411yescharacteristic
235C2×C3×C4C4×D411yescharacteristic
236C2×C3×C4C4×D411yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12311yescharacteristic
240#67—81no—
241#67—81no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—81no—
245C2×Dic3—81no—
246D12—41no—
247D12—41no—
248D12—81no—
249D12—81no—
250C4×D3—82no—
251Dic6—81no—
252C3×C8—41no—
253C3×C8—41no—
254C3×C8—42no—
255C3⋊C8—42no—
256C3⋊C8—41no—
257C3⋊C8—41no—
258#41—241no—
259#41—241no—
260C2×Q8—21no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—61no—
264C2×Q8(C2×D3)⋊C411yescharacteristic
265C2×Q8(C2×D3)⋊C411yescharacteristic
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—61no—
269C2×D4—121no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—121no—
273C22×C4—121no—
274C22×C4C2×D1211yescharacteristic
275C22×C4#23511yescharacteristic
276C22×C4C2×#6711yescharacteristic
277SD16—122no—
278SD16—122no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—62no—
283C2×C8—61no—
284C2×C8—61no—
285C2×C8—121no—
286C2×C8—121no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—62no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—121no—
296C4⋊C4(C2×D3)⋊C411yescharacteristic
297C4⋊C4(C2×D3)⋊C411yescharacteristic
298C22⋊C4—121no—
299C22⋊C4—121no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—241no—
305C42C2×C4×D311yescharacteristic
306C42#23511yescharacteristic
307C42C2×#6711yescharacteristic
308C42C2×#6711yescharacteristic
309C22×C3—21no—
310C22×C3—21no—
311C22×C3#38611yescharacteristic
312C22×C3#47111yescharacteristic
313C22×C3#47211yescharacteristic
314C2×D3—81no—
315C2×D3—81no—
316C2×D3—81no—
317C3×C4—21no—
318C3×C4—22no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4—41no—
326C3×C4C2×#10411yescharacteristic
327C3×C4#42111yescharacteristic
328Dic3—82no—
329Dic3—81no—
330C23—121no—
331C23#93011yescharacteristic
332Q8—21no—
333Q8—21no—
334Q8—21no—
335Q8—42no—
336Q8—241no—
337Q8#82512yes—
338D4—121no—
339D4—121no—
340D4—241no—
341D4—241no—
342D4—241no—
343D4—241no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—122no—
357C2×C4—121no—
358C2×C4—242no—
359C2×C4—241no—
360C2×C4—241no—
361C2×C4C2×(C2×D3)⋊C411yescharacteristic
362C2×C4C4×#6711yescharacteristic
363C2×C4C4×#6711yescharacteristic
364C2×C4#92911yescharacteristic
365C2×C4#93011yescharacteristic
366C2×C4#95311yescharacteristic
367C8—41no—
368C8—41no—
369C8—42no—
370C8—122no—
371C8—121no—
372C8—121no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3#187111yescharacteristic
376C2×C3#190511yescharacteristic
377C2×C3#190711yescharacteristic
378D3—161no—
379C22—21no—
380C22—21no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22#514111yescharacteristic
385C22#514211yescharacteristic
386C22#521311yescharacteristic
387C4—21no—
388C4—22no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—242no—
397C4—241no—
398C4C2×#82511yescharacteristic
399C4#513911yescharacteristic
400C3#916411yescharacteristic, complemented, Sylow
401C2—21no—
402C2—21no—
403C2—481no—
404C2#7166211yescharacteristic
405C2#7198411yescharacteristic
406C2#7201811yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2984C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#2984D3
hypocenterC3#9164
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2984

Complements

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

NH ≅ G/Ntype
1. C3×#2984C2semidirect
3. #71631C2semidirect
6. #67749C2semidirect
7. #67742C2semidirect
20. #5027C4semidirect
21. #5027C4semidirect
24. #5021C4semidirect
25. #5021C4semidirect
35. #2984D3semidirect
400. C3#9164semidirect

Maximal subgroups

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

subgroupindex
C3×#29842
#824442
#716312
#708062
#707892
#677492
#677422
#91643