Finite Group Databasebeta

← #10694552

Subgroups — #10694552

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#83334C211yesmaximal, characteristic
2C3×#1503C211yesmaximal, characteristic, complemented
3#71824C211yesmaximal, characteristic, complemented
4#70758C211yesmaximal, characteristic
5#70745C211yesmaximal, characteristic, complemented
6#69477C211yesmaximal, characteristic, complemented
7#69469C211yesmaximal, characteristic
8#9754—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10#5477C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×C4×C4⋊C4C2211yescharacteristic
13#5093C411yescharacteristic, complemented
14#5093C411yescharacteristic, complemented
15#5088C411yescharacteristic, complemented
16#5088C411yescharacteristic, complemented
17#5068—21no—
18#5068—21no—
19#5002—21no—
20#5002—21no—
21#4925C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×#803—21no—
24C2×C4×C3⋊C8C2211yescharacteristic
25C3×#328—22no—
26#4535—22no—
27#4524—21no—
28#4480—21no—
29#4479—21no—
30#4459—22no—
31#4451—22no—
32#3131—31no—
33#1770—31no—
34#1564—31no—
35#1503D311yescharacteristic, complemented
36#1502—31no—
37#1481—31no—
38#1480—31no—
39C2×#235—21no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×#125—21no—
43C3×C22⋊Q8—21no—
44C3×C22⋊Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×#117—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42—21no—
50C2×C3×C42C2311yescharacteristic
51#930—21no—
52#930—21no—
53C22×C3⋊C8—21no—
54C22×C3⋊C8D411yescharacteristic
55C22×C3⋊C8D411yescharacteristic
56#875—21no—
57#874C2×C411yescharacteristic
58C4×D12—21no—
59C4×D12—21no—
60#869C2×C411yescharacteristic
61C3×C4⋊C8—21no—
62C3×C22⋊C8—21no—
63C3×C4×C8—21no—
64C3×#95D412yes—
65#830—21no—
66#810—21no—
67#810—21no—
68#810—21no—
69#810—21no—
70#809—21no—
71#809—21no—
72#809—21no—
73#809—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#533C2×D311yescharacteristic
84#532—31no—
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—22no—
103#327—62no—
104Q8⋊C8—62no—
105D4⋊C8—62no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C22×C3×C4—22no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C2×#67—41no—
113#235—41no—
114#235—41no—
115#235—41no—
116#235—41no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×C4×D3—41no—
121C2×Dic6—21no—
122C2×C3×C8—41no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—42no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C42—41no—
132C3×C42—41no—
133C3×C42—41no—
134C3×C42C22×C411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138(C2×D3)⋊C4—41no—
139(C2×D3)⋊C4—41no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
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—61no—
155C22×C8—31no—
156C22×C8—31no—
157C22×C8—61no—
158C4⋊Q8—31no—
159C4⋊Q8C4×D311yescharacteristic
160C4⋊Q8C4×D311yescharacteristic
161#127—31no—
162#125—21no—
163#124—61no—
164C22⋊Q8—21no—
165C22⋊Q8—21no—
166#121—61no—
167#121—61no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×D4—61no—
171C4×D4—61no—
172#117—21no—
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#12111yescharacteristic
199C22×D3—41no—
200C3×Q8—42no—
201C3×Q8—81no—
202C2×C3×C4—22no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—42no—
209C2×C3×C4C2×C22⋊C411yescharacteristic
210C2×C3×C4C4×D411yescharacteristic
211C2×C3×C4C4×D411yescharacteristic
212C2×C3×C4#12111yescharacteristic
213C2×C3×C4#12411yescharacteristic
214C2×C3×C4#12711yescharacteristic
215C2×C3×C4#13511yescharacteristic
216C2×C3×C4#13511yescharacteristic
217C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
218C2×C3×C4#13711yescharacteristic
219#67—81no—
220#67—81no—
221C2×Dic3—41no—
222D12—41no—
223D12—41no—
224D12—42no—
225D12—81no—
226C4×D3—81no—
227C4×D3—81no—
228Dic6—41no—
229Dic6—41no—
230C3×C8—81no—
231C3⋊C8—21no—
232C3⋊C8—21no—
233C3⋊C8—21no—
234C3⋊C8—21no—
235C3⋊C8—42no—
236C3⋊C8—81no—
237#41—121no—
238#41—121no—
239#41—121no—
240#41—121no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—61no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—121no—
248C22×C4—22no—
249C22×C4—121no—
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—21no—
269C4⋊C4—21no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C22⋊C4—41no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C42—41no—
279C42—41no—
280C42—41no—
281C42C2×C4×D311yescharacteristic
282C42#23511yescharacteristic
283C42D3×D411yescharacteristic
284C42D3×D411yescharacteristic
285C22×C3—21no—
286C22×C3—21no—
287C22×C3#39011yescharacteristic
288C22×C3#46311yescharacteristic
289C22×C3#46411yescharacteristic
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—82no—
297C3×C4—81no—
298C3×C4—81no—
299C3×C4#41611yescharacteristic
300C3×C4#41711yescharacteristic
301C3×C4#44412yes—
302C3×C4#46311yescharacteristic
303C3×C4#46411yescharacteristic
304Dic3—81no—
305C23—121no—
306C23#93911yescharacteristic
307Q8—42no—
308Q8—81no—
309Q8—121no—
310Q8—121no—
311D4—121no—
312D4—121no—
313D4—122no—
314D4—241no—
315D4—241no—
316D4—241no—
317C2×C4—22no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—121no—
325C2×C4—241no—
326C2×C4—241no—
327C2×C4D3×C22⋊C411yescharacteristic
328C2×C4#67⋊C411yescharacteristic
329C2×C4#88411yescharacteristic
330C2×C4#88511yescharacteristic
331C2×C4#91411yescharacteristic
332C2×C4#91611yescharacteristic
333C2×C4#91811yescharacteristic
334C2×C4#91911yescharacteristic
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#197511yescharacteristic
347C2×C3#197611yescharacteristic
348C2×C3#198311yescharacteristic
349D3—161no—
350C22—21no—
351C22—21no—
352C22—241no—
353C22—241no—
354C22—241no—
355C22#495711yescharacteristic
356C22#516511yescharacteristic
357C22#518511yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—82no—
362C4—81no—
363C4—81no—
364C4—241no—
365C4#479811yescharacteristic
366C4#480311yescharacteristic
367C4#480411yescharacteristic
368C4#481911yescharacteristic
369C4#482312yes—
370C3#975411yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—481no—
374C2#7004211yescharacteristic
375C2#7023011yescharacteristic
376C2#7037411yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#390
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1503D3
hypocenterC3#9754
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
2. C3×#1503C2semidirect
3. #71824C2semidirect
5. #70745C2semidirect
6. #69477C2semidirect
13. #5093C4semidirect
14. #5093C4semidirect
15. #5088C4semidirect
16. #5088C4semidirect
35. #1503D3semidirect
370. C3#9754semidirect

Maximal subgroups

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

subgroupindex
#833342
C3×#15032
#718242
#707582
#707452
#694772
#694692
#97543