Finite Group Databasebeta

← #10694556

Subgroups — #10694556

Order 768.

416 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83281C211yesmaximal, characteristic
2C3×#1502C211yesmaximal, characteristic
3#71800C211yesmaximal, characteristic
4#70768C211yesmaximal, characteristic, complemented
5#70745C211yesmaximal, characteristic, complemented
6#69477C211yesmaximal, characteristic, complemented
7#69471C211yesmaximal, characteristic, complemented
8#9754—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#5233—21no—
14#5233—21no—
15#5079C411yescharacteristic, complemented
16#5079C411yescharacteristic, complemented
17#5070C411yescharacteristic, complemented
18#5070C411yescharacteristic, complemented
19#5056—21no—
20#5056—21no—
21#4925C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×#803—21no—
24C2×C4×C3⋊C8C2211yescharacteristic
25C3×#327—22no—
26D4⋊(C3⋊C8)—22no—
27#4524—21no—
28#4480—21no—
29#4479—21no—
30#4456—22no—
31#4452—22no—
32#3131—31no—
33#1770—31no—
34#1564—31no—
35#1503—31no—
36#1502D311yescharacteristic
37#1481—31no—
38#1480—31no—
39C2×C3×#41—21no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#127C2×C411yescharacteristic
42C3×#124—21no—
43C3×#121—21no—
44C3×#121—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42—21no—
49C2×C3×C42C2311yescharacteristic
50#926—21no—
51#924—21no—
52#924—21no—
53C22×C3⋊C8—21no—
54C22×C3⋊C8D411yescharacteristic
55C22×C3⋊C8D411yescharacteristic
56#870—21no—
57#869C2×C411yescharacteristic
58#869C2×C411yescharacteristic
59C4×Dic6—21no—
60C4×Dic6—21no—
61C3×C4⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×C4×C8—21no—
64C3×#95D412yes—
65#835—21no—
66#835—21no—
67#835—21no—
68#835—21no—
69#832—21no—
70#832—21no—
71#832—21no—
72#832—21no—
73#830—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77#803—42no—
78C4×C3⋊C8—21no—
79C4×C3⋊C8C2×C411yescharacteristic
80C4×C3⋊C8C2×C411yescharacteristic
81C4×C3⋊C8D411yescharacteristic
82C4×C3⋊C8D411yescharacteristic
83#533—31no—
84#532C2×D311yescharacteristic
85#488—61no—
86#488—61no—
87#487—31no—
88#487—31no—
89#486—31no—
90#486—31no—
91#431—31no—
92#431C2×D311yescharacteristic
93C2×C4⋊C8—31no—
94#416—61no—
95#416—61no—
96C2×#97—61no—
97C2×C4×C8—31no—
98C4×C4⋊C4C2×D311yescharacteristic
99#336—61no—
100#335—61no—
101#334—61no—
102#328—62no—
103#327—22no—
104Q8⋊C8—62no—
105D4⋊C8—62no—
106C3×#41—41no—
107C3×#41—41no—
108C3×#41—41no—
109C3×#41—41no—
110C2×C3×Q8—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—41no—
115C22×C3×C4—22no—
116C22×C3×C4—41no—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×C3×C8—41no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—42no—
126C3×C22⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C42—41no—
129C3×C42—41no—
130C3×C42C22×C411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D411yescharacteristic
133C3×C42#4111yescharacteristic
134#217—41no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139Dic3⋊C4—41no—
140Dic3⋊C4—41no—
141C4×Dic3—41no—
142C2×C3⋊C8—21no—
143C2×C3⋊C8—21no—
144C2×C3⋊C8—21no—
145C2×C3⋊C8—21no—
146C2×C3⋊C8—22no—
147C2×C3⋊C8—41no—
148C2×C3⋊C8—41no—
149C2×C3⋊C8—41no—
150C2×C3⋊C8C22⋊C411yescharacteristic
151C2×C3⋊C8C22⋊C411yescharacteristic
152C2×C3⋊C8C22⋊C411yescharacteristic
153C2×C3⋊C8C22⋊C411yescharacteristic
154C2×#41—21no—
155C22×C8—31no—
156C22×C8—31no—
157C22×C8—61no—
158C4⋊Q8—31no—
159C4⋊Q8—31no—
160C4⋊Q8C4×D311yescharacteristic
161#127C4×D311yescharacteristic
162#125—61no—
163#124—21no—
164C22⋊Q8—61no—
165C22⋊Q8—61no—
166#121—21no—
167#121—21no—
168C4×Q8—61no—
169C4×Q8—61no—
170C4×D4—21no—
171C4×D4—21no—
172#117—61no—
173C2×C4⋊C4—21no—
174C2×C42—21no—
175C2×C42C22×D311yescharacteristic
176C4⋊C8—21no—
177C4⋊C8—32no—
178C4⋊C8—61no—
179C4⋊C8—61no—
180#103—61no—
181#103—61no—
182#103—61no—
183#103—61no—
184#102—61no—
185#102—61no—
186#102—61no—
187#102—61no—
188C22⋊C8—21no—
189C22⋊C8—61no—
190#97—122no—
191C4×C8—21no—
192C4×C8—31no—
193C4×C8—31no—
194C4×C8—31no—
195C4×C8—31no—
196C4×C8—61no—
197#95#6712yes—
198C23×C3—41no—
199C23×C3#12111yescharacteristic
200C3×Q8—41no—
201C3×Q8—41no—
202C3×D4—41no—
203C3×D4—41no—
204C3×D4—42no—
205C3×D4—81no—
206C3×D4—81no—
207C3×D4—81no—
208C2×C3×C4—22no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—41no—
213C2×C3×C4—42no—
214C2×C3×C4—81no—
215C2×C3×C4—81no—
216C2×C3×C4C2×C22⋊C411yescharacteristic
217C2×C3×C4C4×D411yescharacteristic
218C2×C3×C4C4×D411yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12411yescharacteristic
221C2×C3×C4#12711yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×C3×C4#13511yescharacteristic
224C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
225C2×C3×C4#13711yescharacteristic
226C2×Dic3—41no—
227C2×Dic3—41no—
228Dic6—42no—
229Dic6—81no—
230C3×C8—81no—
231C3⋊C8—21no—
232C3⋊C8—21no—
233C3⋊C8—21no—
234C3⋊C8—21no—
235C3⋊C8—42no—
236C3⋊C8—81no—
237#41—41no—
238#41—41no—
239#41—41no—
240#41—41no—
241C2×Q8—21no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4D3×D411yescharacteristic
251C22×C4#23711yescharacteristic
252C22×C4C2×#6711yescharacteristic
253C2×C8—31no—
254C2×C8—31no—
255C2×C8—31no—
256C2×C8—31no—
257C2×C8—41no—
258C2×C8—61no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—62no—
263C2×C8—121no—
264C2×C8—121no—
265C2×C8—121no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—42no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—61no—
272C4⋊C4—61no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—121no—
278C42—41no—
279C42—41no—
280C42—121no—
281C42C2×C4×D311yescharacteristic
282C42#23511yescharacteristic
283C42D3×D411yescharacteristic
284C42D3×D411yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—81no—
288C22×C3—81no—
289C22×C3—81no—
290C22×C3#39011yescharacteristic
291C22×C3#46311yescharacteristic
292C22×C3#46411yescharacteristic
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—82no—
297C3×C4—81no—
298C3×C4#41611yescharacteristic
299C3×C4#41711yescharacteristic
300C3×C4#44412yes—
301C3×C4#46311yescharacteristic
302C3×C4#46411yescharacteristic
303Dic3—81no—
304Dic3—81no—
305C23—41no—
306C23#93911yescharacteristic
307Q8—41no—
308Q8—41no—
309Q8—122no—
310Q8—241no—
311D4—41no—
312D4—41no—
313D4—42no—
314D4—81no—
315D4—81no—
316D4—81no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4D3×C22⋊C411yescharacteristic
328C2×C4#67⋊C411yescharacteristic
329C2×C4#88411yescharacteristic
330C2×C4#88511yescharacteristic
331C2×C4#91111yescharacteristic
332C2×C4#91211yescharacteristic
333C2×C4#91511yescharacteristic
334C2×C4#91611yescharacteristic
335C2×C4C4×#6711yescharacteristic
336C2×C4#94011yescharacteristic
337C8—61no—
338C8—61no—
339C8—61no—
340C8—61no—
341C8—81no—
342C8—122no—
343C8—241no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—161no—
347C2×C3#197511yescharacteristic
348C2×C3#197611yescharacteristic
349C2×C3#198311yescharacteristic
350C22—21no—
351C22—21no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22#495711yescharacteristic
356C22#515311yescharacteristic
357C22#516611yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—82no—
362C4—81no—
363C4—241no—
364C4—241no—
365C4#476711yescharacteristic
366C4#477111yescharacteristic
367C4#477211yescharacteristic
368C4#478711yescharacteristic
369C4#482312yes—
370C3#975411yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—161no—
374C2#7004211yescharacteristic
375C2#7016211yescharacteristic
376C2#7037411yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1502C2
socleC22×C3#390
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1502D3
hypocenterC3#9754
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
4. #70768C2semidirect
5. #70745C2semidirect
6. #69477C2semidirect
7. #69471C2semidirect
15. #5079C4semidirect
16. #5079C4semidirect
17. #5070C4semidirect
18. #5070C4semidirect
370. C3#9754semidirect

Maximal subgroups

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

subgroupindex
#832812
C3×#15022
#718002
#707682
#707452
#694772
#694712
#97543