Finite Group Databasebeta

← #10691141

Subgroups — #10691141

Order 768.

500 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82463C211yesmaximal, characteristic, complemented
2#82441C211yesmaximal, characteristic
3C3×#1479C211yesmaximal, characteristic, complemented
4#70790C211yesmaximal, characteristic
5#70789C211yesmaximal, characteristic
6#67746C211yesmaximal, characteristic, complemented
7#67738C211yesmaximal, characteristic, complemented
8#9182—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×#942—21no—
11#5477C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#431C2211yescharacteristic
14#5229—21no—
15#5031C412yescomplemented
16#5028—22no—
17#5027C412yescomplemented
18#4980—22no—
19Dic3×C4⋊C4—21no—
20#4966—21no—
21#4937—21no—
22#4936C2211yescharacteristic
23C4×C4⋊Dic3C2211yescharacteristic
24#4932—21no—
25#4924C2211yescharacteristic
26C3×Q8⋊C8—22no—
27#4470—21no—
28#4470—21no—
29#4470—22no—
30#3037—31no—
31#2356—31no—
32#1517—31no—
33#1502—31no—
34#1479D311yescharacteristic, complemented
35#1303—31no—
36#1286—31no—
37C22×C3×Q8D411yescharacteristic
38C2×#235—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8C2×C412yes—
42C3×C4×Q8D412yes—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4D411yescharacteristic
45C2×C3×C42—21no—
46C2×C3×C42C2311yescharacteristic
47C2×#216—22no—
48#942—42no—
49#942—42no—
50C2×#215—22no—
51#930—21no—
52#930—21no—
53C2×C4⋊Dic3—21no—
54C2×C4⋊Dic3—21no—
55C2×C4×Dic3—22no—
56C2×C4×Dic3—21no—
57C2×#208—21no—
58#875—21no—
59#874C2×C411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#869C2×C411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C4⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×C22⋊C8—21no—
67C3×C4×C8—21no—
68C3×C4×C8—21no—
69C3⋊#95—22no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95—21no—
73C3⋊#95D411yescharacteristic
74C3⋊#95D411yescharacteristic
75C3⋊#95D412yes—
76#810—22no—
77#809—22no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)C2×C412yes—
80C2×#137—61no—
81#532—31no—
82C2×C4×Q8C2×D311yescharacteristic
83#464—62no—
84#462—32no—
85#460—32no—
86#431C2×D311yescharacteristic
87#431C2×D311yescharacteristic
88#423—31no—
89#417—61no—
90#391—61no—
91#387—62no—
92#383—61no—
93#382—31no—
94C4×C4⋊C4—31no—
95C4×C4⋊C4—61no—
96#376—61no—
97#327—61no—
98#327—61no—
99#327—62no—
100Q8⋊C8—22no—
101C2×C3×Q8—21no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8—21no—
105C2×C3×Q8—21no—
106C2×C3×Q8C22⋊C412yes—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C2×#67—41no—
113C22×Dic3—21no—
114C22×Dic3—22no—
115C22×Dic3—21no—
116#235—41no—
117#235—41no—
118#235—41no—
119#235—41no—
120C2×D12—21no—
121C2×D12—21no—
122C2×D12—21no—
123C2×C4×D3—41no—
124C2×Dic6—21no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4C22⋊C412yes—
131C3×C42—22no—
132C3×C42—22no—
133C3×C42C22×C411yescharacteristic
134C3×C42C2×D411yescharacteristic
135C3×C42#4111yescharacteristic
136C3×C42#4111yescharacteristic
137#216—44no—
138#215—44no—
139(C2×D3)⋊C4—41no—
140(C2×D3)⋊C4—41no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—42no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146C4×Dic3—44no—
147C4×Dic3—41no—
148C4×Dic3—41no—
149#208—41no—
150#208—41no—
151C2×C3⋊C8—22no—
152C2×#41—61no—
153C22×Q8D1211yescharacteristic
154#137—122no—
155#137—122no—
156C2×Q16—62no—
157C2×SD16—62no—
158C2×M16—61no—
159C4⋊Q8—31no—
160#127—31no—
161#124—61no—
162#121—61no—
163#121—61no—
164C4×Q8—21no—
165C4×Q8—21no—
166C4×Q8C4×D312yes—
167C4×Q8#6712yes—
168C4×D4—61no—
169C4×D4—61no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4D1211yescharacteristic
174C2×C42—21no—
175C2×C42—62no—
176C2×C42—61no—
177C2×C42C22×D311yescharacteristic
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—32no—
181C4⋊C8—61no—
182#103—62no—
183#102—62no—
184C22⋊C8—21no—
185C22⋊C8—21no—
186C4×C8—21no—
187C4×C8—21no—
188#95—31no—
189#95—31no—
190#95—32no—
191#95—62no—
192#95—62no—
193#95—62no—
194#95—61no—
195C23×C3#12011yescharacteristic
196C22×D3—41no—
197C3×Q8—22no—
198C3×Q8—22no—
199C3×Q8—41no—
200C3×Q8—41no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4C2×C22⋊C411yescharacteristic
209C2×C3×C4C4×D411yescharacteristic
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12311yescharacteristic
214C2×C3×C4#13511yescharacteristic
215C2×C3×C4#13511yescharacteristic
216C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
217C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
218#67—81no—
219#67—81no—
220C2×Dic3—22no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—42no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—42no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—42no—
230D12—41no—
231D12—41no—
232D12—81no—
233D12—81no—
234C4×D3—81no—
235C4×D3—81no—
236Dic6—41no—
237Dic6—41no—
238C3×C8—81no—
239C3×C8—81no—
240C3⋊C8—42no—
241#41—121no—
242#41—121no—
243#41—121no—
244#41—121no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—61no—
251C2×Q8(C2×D3)⋊C412yes—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—121no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—61no—
259C22×C4—62no—
260C22×C4—61no—
261C22×C4—121no—
262C22×C4C2×D1211yescharacteristic
263C22×C4D3×D411yescharacteristic
264C22×C4D3×D411yescharacteristic
265Q16—124no—
266SD16—124no—
267M16—121no—
268M16—121no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—62no—
272C4⋊C4—22no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—122no—
278C4⋊C4—121no—
279C4⋊C4—121no—
280C4⋊C4(C2×D3)⋊C412yes—
281C22⋊C4—121no—
282C22⋊C4—121no—
283C42—22no—
284C42—22no—
285C42—124no—
286C42—121no—
287C42—121no—
288C42C2×C4×D311yescharacteristic
289C42#23711yescharacteristic
290C42#23711yescharacteristic
291C42C2×#6711yescharacteristic
292C22×C3—21no—
293C22×C3—21no—
294C22×C3#38611yescharacteristic
295C22×C3#44911yescharacteristic
296C22×C3#44911yescharacteristic
297C2×D3—81no—
298C2×D3—81no—
299C2×D3—81no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—42no—
305C3×C4—42no—
306C3×C4#41611yescharacteristic
307C3×C4#41811yescharacteristic
308C3×C4#48211yescharacteristic
309C3×C4#48211yescharacteristic
310Dic3—42no—
311Dic3—81no—
312Dic3—82no—
313Dic3—81no—
314C23—121no—
315C23#88211yescharacteristic
316Q8—22no—
317Q8—22no—
318Q8—41no—
319Q8—41no—
320Q8—121no—
321Q8—121no—
322D4—121no—
323D4—121no—
324D4—241no—
325D4—241no—
326D4—241no—
327D4—241no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—62no—
336C2×C4—121no—
337C2×C4—121no—
338C2×C4—122no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—122no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—122no—
345C2×C4—241no—
346C2×C4—241no—
347C2×C4#67⋊C411yescharacteristic
348C2×C4#67⋊C411yescharacteristic
349C2×C4#88611yescharacteristic
350C2×C4#90411yescharacteristic
351C2×C4#90411yescharacteristic
352C2×C4#90811yescharacteristic
353C2×C4#90811yescharacteristic
354C2×C4C2×(C2×D3)⋊C411yescharacteristic
355C2×C4#93911yescharacteristic
356C2×C4#93911yescharacteristic
357C8—81no—
358C8—81no—
359C8—122no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3#187511yescharacteristic
363C2×C3#190511yescharacteristic
364C2×C3#190611yescharacteristic
365D3—161no—
366C22—21no—
367C22—21no—
368C22—241no—
369C22—241no—
370C22—241no—
371C22#473411yescharacteristic
372C22#473411yescharacteristic
373C22#495811yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—42no—
379C4—42no—
380C4—122no—
381C4—241no—
382C4—242no—
383C4—241no—
384C4#473711yescharacteristic
385C4#473711yescharacteristic
386C4#511311yescharacteristic
387C4#513111yescharacteristic
388C3#918211yescharacteristic, complemented, Sylow
389C2—21no—
390C2—21no—
391C2—481no—
392C2#6993311yescharacteristic
393C2#7033411yescharacteristic
394C2#7039511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1479C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1479D3
hypocenterC3#9182
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
1. #82463C2semidirect
3. C3×#1479C2semidirect
6. #67746C2semidirect
7. #67738C2semidirect
15. #5031C4semidirect
17. #5027C4semidirect
34. #1479D3semidirect
388. C3#9182semidirect

Maximal subgroups

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

subgroupindex
#824632
#824412
C3×#14792
#707902
#707892
#677462
#677382
#91823