Finite Group Databasebeta

← #10691928

Subgroups — #10691928

Order 768.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2331C211yesmaximal, characteristic, complemented
2#83213C211yesmaximal, characteristic
3#71774C211yesmaximal, characteristic, complemented
4#69545C211yesmaximal, characteristic
5#69440C211yesmaximal, characteristic
6#67631C211yesmaximal, characteristic, complemented
7#67614C211yesmaximal, characteristic, complemented
8#9301—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#391—22no—
12C3×#389—21no—
13C3×#387—22no—
14C3×#382C2211yescharacteristic
15C3×C4×C4⋊C4—21no—
16C3×#376C2211yescharacteristic
17#5227—22no—
18#5085C412yescomplemented
19#5083—22no—
20#5079C412yescomplemented
21C2×#805—21no—
22C2×#805—21no—
23#4925C2211yescharacteristic
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26#4546—22no—
27#4451—22no—
28#3206—31no—
29#2331D311yescharacteristic, complemented
30#1963—31no—
31#1548—31no—
32#1517—31no—
33#1286—31no—
34#1274—31no—
35C22×C3×Q8D412yes—
36C2×#235—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8C2×C412yes—
40C3×C4⋊Q8D412yes—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—22no—
45C2×C3×C42—22no—
46C2×C3×C42D411yescharacteristic
47C2×C3×C42D411yescharacteristic
48C2×C3×C42C2311yescharacteristic
49C2×#215—24no—
50#930—21no—
51#930—21no—
52C2×#208—21no—
53#875—21no—
54#874C2×C411yescharacteristic
55C4×D12—21no—
56C4×D12—21no—
57#869C2×C411yescharacteristic
58C3×#95—22no—
59C3×#95—22no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95—21no—
63C3×#95D412yes—
64#835—24no—
65#830—21no—
66#830—21no—
67#805—42no—
68#805—42no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4×C3⋊C8—21no—
74C4×C3⋊C8C2×C412yes—
75#532—31no—
76C2×C4⋊Q8C2×D311yescharacteristic
77#492—32no—
78#487—32no—
79#461—62no—
80#448—62no—
81#431—31no—
82#431—31no—
83#423—31no—
84C2×#104—61no—
85C2×#104—61no—
86#391—22no—
87#389—21no—
88#387—22no—
89#382C2×D311yescharacteristic
90C4×C4⋊C4—21no—
91#376C2×D311yescharacteristic
92#333—62no—
93#327—62no—
94C2×C3×Q8—22no—
95C2×C3×Q8—82no—
96C2×C3×Q8C22⋊C414yes—
97C22×C3×C4—21no—
98C22×C3×C4—22no—
99C22×C3×C4—21no—
100C22×C3×C4—22no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C2×#67—41no—
105#235—41no—
106#235—41no—
107#235—41no—
108#235—41no—
109C2×D12—21no—
110C2×D12—21no—
111C2×D12—21no—
112C2×C4×D3—41no—
113C2×Dic6—21no—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—82no—
121C3×C42—21no—
122C3×C42—21no—
123C3×C42—21no—
124C3×C42—82no—
125C3×C42SD1612yescomplemented
126C3×C42C22×C411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42#4111yescharacteristic
129C3×C42#4111yescharacteristic
130#215—84no—
131(C2×D3)⋊C4—41no—
132(C2×D3)⋊C4—41no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135#208—41no—
136#208—41no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140C2×#41—61no—
141C22×Q8#6712yes—
142C2×SD16—64no—
143C2×M16—61no—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—31no—
147C4⋊Q8C4×D312yes—
148C4⋊Q8D1212yes—
149#127—31no—
150#124—61no—
151#121—61no—
152#121—61no—
153C4×D4—61no—
154C4×D4—61no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C42—22no—
160C2×C42#6711yescharacteristic
161C2×C42#6711yescharacteristic
162C2×C42C22×D311yescharacteristic
163C4⋊C8—61no—
164C4⋊C8—61no—
165C4⋊C8—61no—
166C4⋊C8—61no—
167#104—122no—
168#104—122no—
169#103—64no—
170C22⋊C8—61no—
171C22⋊C8—61no—
172C4×C8—32no—
173C4×C8—61no—
174#95—22no—
175#95—22no—
176#95—21no—
177#95—21no—
178#95—21no—
179#95#6712yes—
180C23×C3#12011yescharacteristic
181C22×D3—41no—
182C3×Q8—44no—
183C3×Q8—82no—
184C2×C3×C4—24no—
185C2×C3×C4—22no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—41no—
192C2×C3×C4—42no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—42no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4C2×C22⋊C411yescharacteristic
199C2×C3×C4C4×D411yescharacteristic
200C2×C3×C4C4×D411yescharacteristic
201C2×C3×C4#12111yescharacteristic
202C2×C3×C4#12111yescharacteristic
203C2×C3×C4#12311yescharacteristic
204C2×C3×C4C2×SD1611yescharacteristic
205C2×C3×C4#13511yescharacteristic
206C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
207C2×C3×C4#13711yescharacteristic
208#67—81no—
209#67—81no—
210C2×Dic3—41no—
211D12—42no—
212D12—41no—
213D12—41no—
214D12—81no—
215C4×D3—81no—
216C4×D3—81no—
217Dic6—41no—
218Dic6—41no—
219C3⋊C8—42no—
220C3⋊C8—81no—
221C3⋊C8—81no—
222#41—121no—
223#41—121no—
224#41—121no—
225#41—121no—
226C2×Q8—22no—
227C2×Q8—61no—
228C2×Q8—82no—
229C2×Q8(C2×D3)⋊C414yes—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—121no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—121no—
239C22×C4C2×#6711yescharacteristic
240C22×C4C2×#6711yescharacteristic
241C22×C4C2×#6711yescharacteristic
242SD16—244no—
243M16—121no—
244M16—121no—
245C2×C8—62no—
246C2×C8—121no—
247C2×C8—121no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—82no—
257C22⋊C4—121no—
258C22⋊C4—121no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—82no—
263C42#21512yescomplemented
264C42C2×C4×D311yescharacteristic
265C42C2×D1211yescharacteristic
266C42#23511yescharacteristic
267C42#23511yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3#38611yescharacteristic
271C22×C3#44811yescharacteristic
272C22×C3#44911yescharacteristic
273C2×D3—81no—
274C2×D3—81no—
275C2×D3—81no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—42no—
281C3×C4—44no—
282C3×C4—82no—
283C3×C4—81no—
284C3×C4C4×SD1611yescharacteristic
285C3×C4#44011yescharacteristic
286C3×C4#46011yescharacteristic
287C3×C4#46411yescharacteristic
288Dic3—81no—
289C23—121no—
290C23#95311yescharacteristic
291Q8—44no—
292Q8—82no—
293Q8—121no—
294Q8—121no—
295D4—122no—
296D4—121no—
297D4—121no—
298D4—241no—
299D4—241no—
300D4—241no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—121no—
316C2×C4—241no—
317C2×C4—241no—
318C2×C4C2×(C2×D3)⋊C411yescharacteristic
319C2×C4C4×#6711yescharacteristic
320C2×C4C4×#6711yescharacteristic
321C2×C4#92911yescharacteristic
322C2×C4#93011yescharacteristic
323C2×C4#93011yescharacteristic
324C2×C4C2×#21511yescharacteristic
325C2×C4#94211yescharacteristic
326C2×C4#94911yescharacteristic
327C2×C4#95011yescharacteristic
328C8—122no—
329C8—241no—
330C8—241no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3#188011yescharacteristic
334C2×C3#190511yescharacteristic
335C2×C3#191111yescharacteristic
336D3—161no—
337C22—21no—
338C22—21no—
339C22—241no—
340C22—241no—
341C22—241no—
342C22#521311yescharacteristic
343C22#522711yescharacteristic
344C22#523811yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—42no—
350C4—44no—
351C4—82no—
352C4—81no—
353C4—241no—
354C4C4×#21511yescharacteristic
355C4#502611yescharacteristic
356C4#502711yescharacteristic
357C4#502811yescharacteristic
358C3#930111yescharacteristic, complemented, Sylow
359C2—21no—
360C2—21no—
361C2—481no—
362C2#7163311yescharacteristic
363C2#7165211yescharacteristic
364C2#7166211yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2331C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#2331D3
hypocenterC3#9301
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2331

Complements

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

NH ≅ G/Ntype
1. C3×#2331C2semidirect
3. #71774C2semidirect
6. #67631C2semidirect
7. #67614C2semidirect
18. #5085C4semidirect
20. #5079C4semidirect
29. #2331D3semidirect
125. C3×C42SD16semidirect
263. C42#215semidirect
358. C3#9301semidirect

Maximal subgroups

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

subgroupindex
C3×#23312
#832132
#717742
#695452
#694402
#676312
#676142
#93013