Finite Group Databasebeta

← #10877383

Subgroups — #10877383

Order 768.

626 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83810C211yesmaximal, characteristic, complemented
2#78658C211yesmaximal, characteristic
3C3×#1868C211yesmaximal, characteristic, complemented
4#72021C211yesmaximal, characteristic
5#70549C212yesmaximal, complemented
6#70538C212yesmaximal, complemented
7#70045C211yesmaximal, characteristic
8#70041C211yesmaximal, characteristic, complemented
9#70002C211yesmaximal, characteristic, complemented
10#69032C212yesmaximal, complemented
11#69024C212yesmaximal, complemented
12#27694—31nomaximal, Sylow
13C3×#568C2211yescharacteristic
14#5832C2211yescharacteristic
15#5819C2211yescharacteristic
16C2×#948C2211yescharacteristic
17C2×#907C2211yescharacteristic
18C3×#424C2212yes—
19C3×#410C2211yescharacteristic
20C3×#408C2211yescharacteristic
21#5240C2211yescharacteristic
22#5139C2212yes—
23#5124—22no—
24#5124C2212yes—
25C2×#822—22no—
26C2×#822C2212yes—
27#5118C2211yescharacteristic
28#5110C2211yescharacteristic
29#5102C2212yes—
30C2×#815C2211yescharacteristic
31#5005C2212yes—
32#4998C2212yes—
33#4730—22no—
34#4728—22no—
35C3×#344C2212yes—
36#4559C2212yes—
37#4555C2212yes—
38#4532C2212yes—
39#4531C2212yes—
40#2932—31no—
41#2916—31no—
42#2102—32no—
43#2098—32no—
44#1946—32no—
45#1884—31no—
46#1884—31no—
47#1868—31no—
48#1868D311yescharacteristic, complemented
49#1814—32no—
50#1812—31no—
51C2×C3×#41C2311yescharacteristic
52C2×#235C2311yescharacteristic
53C3×#131—42no—
54C2×C3×M16—22no—
55C2×C3×M16C2312yes—
56C2×C3×M16C2311yescharacteristic
57C22×C3×C8C2311yescharacteristic
58C3×#117C2312yes—
59#948—22no—
60#948—22no—
61#948—42no—
62#947Q811yescharacteristic
63#944Q811yescharacteristic
64#940Q811yescharacteristic
65#939—22no—
66#936Q811yescharacteristic
67C4×#67—22no—
68#926C2312yes—
69C2×C4×Dic3C2311yescharacteristic
70C2×#208—22no—
71C2×#208D412yes—
72C2×#208C2312yes—
73C2×#208C2311yescharacteristic
74C22×C3⋊C8D412yes—
75C22×C3⋊C8C2311yescharacteristic
76#907—22no—
77#907—22no—
78C2×ΓL(1,𝔽25)—21no—
79C2×D3×C8—21no—
80C3×C4⋊C8—42no—
81C3×#99—22no—
82C3×C22⋊C8—22no—
83#834—22no—
84#825—42no—
85#822—22no—
86#822—24no—
87#822—22no—
88(C2×D3)⋊C8—22no—
89#815—21no—
90#815—21no—
91Dic3⋊C8—22no—
92Dic3⋊C8—42no—
93#808—22no—
94#807—22no—
95#805—42no—
96#568—31no—
97#568C2×D311yescharacteristic
98C2×#131—31no—
99C2×#131—31no—
100#532—31no—
101#435—62no—
102#435—62no—
103#430—32no—
104#430—62no—
105C2×#108—32no—
106C2×#108—62no—
107#427—32no—
108#424—32no—
109#424C2×D312yes—
110#423—31no—
111#421—32no—
112#421—32no—
113#410—31no—
114#410C2×D311yescharacteristic
115#408—31no—
116#408C2×D311yescharacteristic
117C2×#97—31no—
118#344—32no—
119#344C2×D312yes—
120#339—32no—
121#339—32no—
122H(3,ℤ/4)—32no—
123C3×#41—22no—
124C3×#41—22no—
125C2×C3×Q8C2×Q811yescharacteristic
126C2×C3×D4C2×Q811yescharacteristic
127C2×C3×D4#4112yes—
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2411yescharacteristic
130C2×#67—21no—
131C2×#67—22no—
132C22×Dic3—21no—
133#235—22no—
134#235—22no—
135C2×D12C2×Q811yescharacteristic
136C2×C4×D3—21no—
137C2×Dic6C2×Q811yescharacteristic
138C3×M16—22no—
139C3×M16—22no—
140C3×M16—22no—
141C3×M16—42no—
142C2×C3×C8—21no—
143C2×C3×C8—21no—
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C2×C3×C8—42no—
147C3×C4⋊C4—22no—
148C3×C22⋊C4—22no—
149C3×C42—22no—
150#217—22no—
151#217—22no—
152(C2×D3)⋊C4—22no—
153C4⋊Dic3—22no—
154Dic3⋊C4—22no—
155C4×Dic3—21no—
156C4×Dic3—22no—
157C4×Dic3C2×Q811yescharacteristic
158C4×Dic3C2×Q811yescharacteristic
159#208—22no—
160#208—24no—
161#208—22no—
162#208—22no—
163#208—42no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—42no—
168C2×C3⋊C8C2×D412yes—
169C2×C3⋊C8C2×D411yescharacteristic
170C2×C3⋊C8C2×D411yescharacteristic
171C2×C3⋊C8#4112yes—
172C2×C3⋊C8#4112yes—
173ΓL(1,𝔽25)—41no—
174ΓL(1,𝔽25)—41no—
175D3×C8—42no—
176C2×#41—31no—
177C2×#41C22×D311yescharacteristic
178#131—42no—
179#131—62no—
180#131—62no—
181#131—62no—
182#131—62no—
183#131—122no—
184C2×M16—22no—
185C2×M16—32no—
186C2×M16—31no—
187C2×M16—32no—
188C2×M16—61no—
189C2×M16—62no—
190C2×M16C22×D311yescharacteristic
191C2×M16C22×D312yes—
192C22×C8—31no—
193C22×C8—32no—
194C22×C8—61no—
195C22×C8C22×D311yescharacteristic
196C4⋊Q8—31no—
197#127—31no—
198#124—31no—
199#124—31no—
200#121—62no—
201C4×D4—62no—
202#117—32no—
203#117C22×D312yes—
204C2×C42—31no—
205#108—62no—
206#108—62no—
207#108—64no—
208#107—62no—
209#106—62no—
210C4⋊C8—42no—
211C4⋊C8—62no—
212C4⋊C8—122no—
213#104—122no—
214#104—122no—
215#99—22no—
216#99—62no—
217C22⋊C8—22no—
218C22⋊C8—62no—
219#97—61no—
220#97—61no—
221C23×C3C22⋊Q812yes—
222C23×C3C2×#4111yescharacteristic
223C22×D3—21no—
224C3×Q8—22no—
225C3×D4—24no—
226C3×D4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—42no—
230C2×C3×C4C22⋊Q812yes—
231C2×C3×C4C22×D411yescharacteristic
232C2×C3×C4C22×Q811yescharacteristic
233C2×C3×C4C2×#4112yes—
234C2×C3×C4H(5,𝔽2)12yes—
235#67—42no—
236#67—82no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—42no—
241C2×Dic3—81no—
242D12—22no—
243C4×D3—41no—
244C4×D3—41no—
245Dic6—22no—
246C3×C8—41no—
247C3×C8—41no—
248C3×C8—42no—
249C3⋊C8—22no—
250C3⋊C8—42no—
251C3⋊C8#12312yes—
252C3⋊C8#12311yescharacteristic
253C3⋊C8#12311yescharacteristic
254#41—22no—
255#41—22no—
256#41—62no—
257#41—62no—
258C2×Q8—31no—
259C2×Q8D3×Q811yescharacteristic
260C2×D4—31no—
261C2×D4—61no—
262C2×D4—62no—
263C2×D4#23512yes—
264C2×D4D3×Q811yescharacteristic
265C22×C4—61no—
266C22×C4—61no—
267C22×C4D3×D412yes—
268C22×C4C23×D311yescharacteristic
269M16—22no—
270M16—22no—
271M16—22no—
272M16—42no—
273M16—64no—
274M16—62no—
275M16—62no—
276M16—62no—
277M16—121no—
278M16—121no—
279M16—122no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—31no—
285C2×C8—31no—
286C2×C8—32no—
287C2×C8—32no—
288C2×C8—32no—
289C2×C8—42no—
290C2×C8—62no—
291C2×C8—62no—
292C2×C8—62no—
293C2×C8—122no—
294C2×C8—122no—
295C4⋊C4—22no—
296C4⋊C4—62no—
297C4⋊C4—62no—
298C22⋊C4—22no—
299C22⋊C4—62no—
300C22⋊C4—62no—
301C22⋊C4—62no—
302C42—22no—
303C42—31no—
304C42—31no—
305C42—61no—
306C42—62no—
307C22×C3—22no—
308C22×C3—41no—
309C22×C3—42no—
310C22×C3C2×C22⋊Q811yescharacteristic
311C22×C3#55312yes—
312C2×D3—42no—
313C2×D3—41no—
314C3×C4—22no—
315C3×C4—82no—
316C3×C4C2×#12311yescharacteristic
317C3×C4#53511yescharacteristic
318C3×C4#55412yes—
319Dic3—41no—
320Dic3—41no—
321Dic3—41no—
322Dic3—82no—
323C23—61no—
324C23#89612yes—
325C23C2×#23511yescharacteristic
326Q8—22no—
327Q8—62no—
328D4—24no—
329D4—22no—
330D4—62no—
331D4—122no—
332D4—242no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—122no—
342C2×C4—241no—
343C2×C4#89612yes—
344C2×C4C2×D3×D411yescharacteristic
345C2×C4#100412yes—
346C2×C4C2×D3×Q811yescharacteristic
347C2×C4D3×#4112yes—
348C8—31no—
349C8—31no—
350C8—32no—
351C8—41no—
352C8—41no—
353C8—42no—
354C8—62no—
355C8—122no—
356C2×C3—21no—
357C2×C3—22no—
358C2×C3—42no—
359C2×C3#248711yescharacteristic
360D3—81no—
361C22—22no—
362C22—41no—
363C22—42no—
364C22—122no—
365C22—121no—
366C22C2×#89611yescharacteristic
367C22#554012yes—
368C4—22no—
369C4—82no—
370C4—121no—
371C4—121no—
372C4—121no—
373C4—242no—
374C4#559211yescharacteristic
375C4#563612yes—
376C4D3×#12311yescharacteristic
377C3#2769411yescharacteristic, complemented, Sylow
378C2—22no—
379C2—21no—
380C2—42no—
381C2—241no—
382C2#7445711yescharacteristic
subgroupG/H
centerC4#5592
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1868C2
socleC2×C3#2487
prime radical#5832C22
hypercenter#1868D3
hypocenterC3#27694
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1868

Complements

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

NH ≅ G/Ntype
1. #83810C2semidirect
3. C3×#1868C2semidirect
5. #70549C2semidirect
6. #70538C2semidirect
8. #70041C2semidirect
9. #70002C2semidirect
10. #69032C2semidirect
11. #69024C2semidirect
48. #1868D3semidirect
377. C3#27694semidirect

Maximal subgroups

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

subgroupindex
#838102
#786582
C3×#18682
#720212
#705492
#705382
#700452
#700412
#700022
#690322
#690242
#276943