Finite Group Databasebeta

← #10691127

Subgroups — #10691127

Order 768.

544 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82460C211yesmaximal, characteristic, complemented
2#82441C211yesmaximal, characteristic
3C3×#1479C211yesmaximal, characteristic, complemented
4#70790C211yesmaximal, characteristic
5#70789C211yesmaximal, characteristic
6#67746C211yesmaximal, characteristic, complemented
7#67738C211yesmaximal, characteristic, complemented
8#9180—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×Q8×Dic3—21no—
11C2×#942—21no—
12#5477C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#431C2211yescharacteristic
15#5031C412yescomplemented
16#5028—22no—
17#5027C412yescomplemented
18#4984—21no—
19#4984—22no—
20#4966—21no—
21#4936—21no—
22#4936C2211yescharacteristic
23C4×C4⋊Dic3—21no—
24C4×C4⋊Dic3C2211yescharacteristic
25#4924C2211yescharacteristic
26C3×Q8⋊C8—22no—
27#4470—21no—
28#4470—21no—
29#4470—22no—
30#3037—31no—
31#2351—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
47Q8×Dic3—42no—
48Q8×Dic3—42no—
49C2×#216—22no—
50#942—42no—
51#942—42no—
52C2×#215—22no—
53#930—21no—
54#930—21no—
55C2×C4⋊Dic3—22no—
56C2×C4⋊Dic3—22no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3—21no—
59C2×C4×Dic3—22no—
60C2×C4×Dic3—21no—
61C2×#208—21no—
62#875—21no—
63#874C2×C411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66#869C2×C411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72C3×C4×C8—21no—
73C3⋊#95—22no—
74C3⋊#95—21no—
75C3⋊#95D411yescharacteristic
76C3⋊#95D411yescharacteristic
77C3⋊#95D412yes—
78#810—22no—
79#809—22no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)C2×C412yes—
82C2×#137—61no—
83#532—31no—
84C2×C4×Q8—61no—
85C2×C4×Q8C2×D311yescharacteristic
86#464—62no—
87#462—32no—
88#460—32no—
89#431C2×D311yescharacteristic
90#431C2×D311yescharacteristic
91#423—31no—
92#417—61no—
93#389—62no—
94#389—61no—
95#382—31no—
96#382—61no—
97C4×C4⋊C4—31no—
98C4×C4⋊C4—61no—
99#327—61no—
100#327—61no—
101#327—62no—
102Q8⋊C8—22no—
103C2×C3×Q8—21no—
104C2×C3×Q8—21no—
105C2×C3×Q8—21no—
106C2×C3×Q8C22⋊C412yes—
107C2×C3×Q8D812yes—
108C2×C3×Q8SD1612yes—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C2×#67—41no—
115C22×Dic3—21no—
116C22×Dic3—21no—
117C22×Dic3—22no—
118#235—41no—
119#235—41no—
120#235—41no—
121#235—41no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×C4×D3—41no—
126C2×Dic6—21no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4C22⋊C412yes—
133C3×C42—22no—
134C3×C42—22no—
135C3×C42C22×C411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138C3×C42#4111yescharacteristic
139#216—44no—
140#215—44no—
141(C2×D3)⋊C4—41no—
142(C2×D3)⋊C4—41no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—42no—
146C4⋊Dic3—44no—
147C4⋊Dic3—44no—
148C4⋊Dic3—41no—
149C4⋊Dic3—41no—
150C4×Dic3—44no—
151C4×Dic3—41no—
152C4×Dic3—41no—
153#208—41no—
154#208—41no—
155C2×C3⋊C8—22no—
156C2×#41—61no—
157C22×Q8D1211yescharacteristic
158#137—122no—
159#137—122no—
160C2×Q16—62no—
161C2×SD16—62no—
162C2×M16—61no—
163C4⋊Q8—31no—
164#127—31no—
165#124—61no—
166#121—61no—
167#121—61no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—122no—
171C4×Q8—122no—
172C4×Q8C4×D312yes—
173C4×Q8#6712yes—
174C4×D4—61no—
175C4×D4—61no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—62no—
178C2×C4⋊C4—62no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4—61no—
181C2×C4⋊C4D1211yescharacteristic
182C2×C42—21no—
183C2×C42—62no—
184C2×C42—61no—
185C2×C42C22×D311yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—32no—
189C4⋊C8—61no—
190#103—62no—
191#102—62no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C4×C8—21no—
195C4×C8—21no—
196#95—31no—
197#95—31no—
198#95—32no—
199#95—62no—
200#95—61no—
201C23×C3#12011yescharacteristic
202C22×D3—41no—
203C3×Q8—22no—
204C3×Q8—22no—
205C3×Q8—22no—
206C3×Q8#10214yes—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4C2×C22⋊C411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12311yescharacteristic
220C2×C3×C4C2×D811yescharacteristic
221C2×C3×C4C2×SD1611yescharacteristic
222C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
223C2×C3×C4#13711yescharacteristic
224#67—81no—
225#67—81no—
226C2×Dic3—22no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—41no—
230C2×Dic3—42no—
231C2×Dic3—41no—
232C2×Dic3—42no—
233C2×Dic3—41no—
234C2×Dic3—42no—
235C2×Dic3—41no—
236D12—41no—
237D12—41no—
238D12—81no—
239D12—81no—
240C4×D3—81no—
241C4×D3—81no—
242Dic6—41no—
243Dic6—41no—
244C3×C8—81no—
245C3×C8—81no—
246C3⋊C8—42no—
247#41—121no—
248#41—121no—
249#41—121no—
250#41—121no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—61no—
255C2×Q8#20412yes—
256C2×Q8D2412yes—
257C2×Q8(C2×D3)⋊C412yes—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—121no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—62no—
267C22×C4—121no—
268C22×C4C2×D1211yescharacteristic
269C22×C4D3×D411yescharacteristic
270C22×C4D3×D411yescharacteristic
271Q16—124no—
272SD16—124no—
273M16—121no—
274M16—121no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—62no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4—124no—
284C4⋊C4—122no—
285C4⋊C4—124no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C4⋊C4(C2×D3)⋊C412yes—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C42—22no—
292C42—22no—
293C42—124no—
294C42—121no—
295C42—121no—
296C42C2×C4×D311yescharacteristic
297C42#23711yescharacteristic
298C42#23711yescharacteristic
299C42C2×#6711yescharacteristic
300C22×C3—21no—
301C22×C3—21no—
302C22×C3#38611yescharacteristic
303C22×C3#44711yescharacteristic
304C22×C3#44811yescharacteristic
305C2×D3—81no—
306C2×D3—81no—
307C2×D3—81no—
308C3×C4—24no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—42no—
314C3×C4C2×#10211yescharacteristic
315C3×C4#41711yescharacteristic
316C3×C4#48011yescharacteristic
317C3×C4#48111yescharacteristic
318Dic3—42no—
319Dic3—82no—
320Dic3—81no—
321Dic3—81no—
322C23—121no—
323C23#88211yescharacteristic
324Q8—22no—
325Q8—22no—
326Q8—22no—
327Q8—121no—
328Q8—121no—
329Q8#82114yes—
330D4—121no—
331D4—121no—
332D4—241no—
333D4—241no—
334D4—241no—
335D4—241no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—62no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—122no—
348C2×C4—121no—
349C2×C4—122no—
350C2×C4—121no—
351C2×C4—122no—
352C2×C4—121no—
353C2×C4—241no—
354C2×C4—241no—
355C2×C4#67⋊C411yescharacteristic
356C2×C4#67⋊C411yescharacteristic
357C2×C4#88611yescharacteristic
358C2×C4C2×#20411yescharacteristic
359C2×C4C2×D2411yescharacteristic
360C2×C4#90811yescharacteristic
361C2×C4#90911yescharacteristic
362C2×C4C2×(C2×D3)⋊C411yescharacteristic
363C2×C4#93911yescharacteristic
364C2×C4#93911yescharacteristic
365C8—81no—
366C8—81no—
367C8—122no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3#187411yescharacteristic
371C2×C3#190511yescharacteristic
372C2×C3#190611yescharacteristic
373D3—161no—
374C22—21no—
375C22—21no—
376C22—241no—
377C22—241no—
378C22—241no—
379C22#473211yescharacteristic
380C22#473811yescharacteristic
381C22#495811yescharacteristic
382C4—24no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—42no—
388C4—122no—
389C4—242no—
390C4—241no—
391C4—241no—
392C4#473511yescharacteristic
393C4#473611yescharacteristic
394C4C2×#82111yescharacteristic
395C4#513311yescharacteristic
396C3#918011yescharacteristic, complemented, Sylow
397C2—21no—
398C2—21no—
399C2—481no—
400C2#6993211yescharacteristic
401C2#7033411yescharacteristic
402C2#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#9180
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. #82460C2semidirect
3. C3×#1479C2semidirect
6. #67746C2semidirect
7. #67738C2semidirect
15. #5031C4semidirect
17. #5027C4semidirect
34. #1479D3semidirect
396. C3#9180semidirect

Maximal subgroups

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

subgroupindex
#824602
#824412
C3×#14792
#707902
#707892
#677462
#677382
#91803