Finite Group Databasebeta

← #10848605

Subgroups — #10848605

Order 768.

1188 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86190C211yesmaximal, characteristic
2C3×#2858C211yesmaximal, characteristic, complemented
3#83209C211yesmaximal, characteristic, complemented
4#83198C211yesmaximal, characteristic, complemented
5#71766C211yesmaximal, characteristic, complemented
6#71731C214yesmaximal, complemented
7#71723C214yesmaximal, complemented
8#69396C211yesmaximal, characteristic, complemented
9C4×#805C211yesmaximal, characteristic, complemented
10#25590—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×#127C2211yescharacteristic
13C2×C3×#124C2214yes—
14C2×#942C2212yes—
15#5806—24no—
16#5805C2212yes—
17#5798C2212yes—
18#5719—22no—
19#5718—22no—
20#5716—22no—
21#5713—22no—
22C4×#235C2211yescharacteristic
23#5470C2211yescharacteristic
24C3×#390—24no—
25C3×C43C2211yescharacteristic
26#5225C2212yes—
27#5225C2214yes—
28C2×#833C2214yes—
29#5090—22no—
30#5083—22no—
31#5082—22no—
32#5079—22no—
33C2×#805C2212yes—
34#4924C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4523C2212yes—
37#4523C2214yes—
38#3508—31no—
39#3139—31no—
40#3039—31no—
41#2858D311yescharacteristic, complemented
42#2004—34no—
43#1994—31no—
44#1965—31no—
45#1885—34no—
46C4×#104—31no—
47C22×C3×Q8C2312yes—
48C22×C3×D4D414yes—
49C22×C3×D4C2312yes—
50C2×#237—24no—
51C2×#235C2311yescharacteristic
52C3×C4⋊Q8—21no—
53C3×C4⋊Q8—21no—
54C3×C4⋊Q8—22no—
55C3×#127—22no—
56C3×#127—21no—
57C3×#127—21no—
58C3×#124—28no—
59C3×#124—44no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C22⋊C4—28no—
62C2×C3×C42—22no—
63C2×C3×C42C2312yes—
64C2×C3×C42C2311yescharacteristic
65#953—24no—
66C2×#217—24no—
67C2×#216—22no—
68#942—44no—
69C2×#215—22no—
70#940—22no—
71#939—28no—
72#938—24no—
73#938—24no—
74#936—24no—
75#936—22no—
76#935—28no—
77D4×Dic3—24no—
78D4×Dic3—24no—
79D4×Dic3—24no—
80C4×#67—22no—
81#926C2312yes—
82#926C2311yescharacteristic
83#924—22no—
84C2×#208C2312yes—
85C2×#208C2312yes—
86C4×D12D412yes—
87#872—21no—
88C42×D3—21no—
89#870D412yes—
90#869D412yes—
91C4×Dic6—22no—
92C4×Dic6D412yes—
93#835—28no—
94#835—24no—
95#833—44no—
96#805—24no—
97C4⋊(C3⋊C8)—22no—
98#803—21no—
99C4×C3⋊C8—21no—
100C2×#137—32no—
101#559—62no—
102#550—62no—
103#546—64no—
104#540—62no—
105#535—32no—
106#533—31no—
107#532—62no—
108C2×C4⋊Q8C2×D311yescharacteristic
109C2×#127C2×D311yescharacteristic
110C2×#124C2×D314yes—
111#518—32no—
112C4×#41—31no—
113#491—62no—
114#487—62no—
115#462—62no—
116#461—62no—
117#423—31no—
118C2×#104—32no—
119#419—34no—
120#419—32no—
121C2×#100—34no—
122C4×M16—31no—
123#390—24no—
124C43C2×D311yescharacteristic
125#339—32no—
126#339—34no—
127C24×C3C2×D414yes—
128C2×C3×Q8—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8—24no—
131C2×C3×Q8—24no—
132C2×C3×D4—24no—
133C2×C3×D4—24no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—24no—
137C2×C3×D4—24no—
138C2×C3×D4—48no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2411yescharacteristic
143C2×#67—21no—
144C2×#67—28no—
145C22×Dic3—28no—
146#237—88no—
147#235—22no—
148#235—22no—
149C2×D12C2×D411yescharacteristic
150C2×C4×D3—22no—
151C2×C4×D3—21no—
152C2×Dic6—22no—
153C2×Dic6C2×D411yescharacteristic
154C3×C4⋊C4—42no—
155C3×C4⋊C4—42no—
156C3×C22⋊C4—416no—
157C3×C42—22no—
158C3×C42—44no—
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D412yes—
162C3×C42#4114yescomplemented
163#217—28no—
164#217—28no—
165#217—22no—
166#217—21no—
167#216—44no—
168#215—44no—
169(C2×D3)⋊C4—22no—
170C4⋊Dic3—22no—
171C4⋊Dic3C2×D412yes—
172C4⋊Dic3C2×D414yes—
173Dic3⋊C4—24no—
174Dic3⋊C4—22no—
175C4×Dic3—22no—
176C4×Dic3—22no—
177C4×Dic3—21no—
178#208—22no—
179#208—22no—
180#208—24no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—22no—
183C2×#41—31no—
184C2×#41—64no—
185C22×Q8C22×D312yes—
186C22×D4#6714yes—
187C22×D4C22×D312yes—
188#137—124no—
189C2×Q16—62no—
190C2×SD16—62no—
191C2×M16—32no—
192C2×M16—32no—
193C4⋊Q8—22no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—32no—
197#127—21no—
198#127—21no—
199#127—22no—
200#127—62no—
201#125—32no—
202#124—28no—
203#124—44no—
204#124—62no—
205#124—64no—
206#123—68no—
207C22⋊Q8—62no—
208#121—64no—
209#121—64no—
210#121—68no—
211#120—64no—
212C4×Q8—32no—
213C4×Q8—62no—
214C4×D4—32no—
215C4×D4—64no—
216C4×D4—64no—
217C4×D4—64no—
218C4×D4—62no—
219#117—31no—
220#117—32no—
221#117—61no—
222C2×C4⋊C4—22no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—64no—
225C2×C42—22no—
226C2×C42—61no—
227C2×C42C22×D312yes—
228C2×C42C22×D311yescharacteristic
229C4⋊C8—62no—
230#104—64no—
231#103—64no—
232#103—68no—
233#100—124no—
234#97—61no—
235C4×C8—61no—
236C23×C3—24no—
237C23×C3—28no—
238C23×C3—48no—
239C23×C3C22×D411yescharacteristic
240C22×D3—21no—
241C3×Q8—42no—
242C3×Q8—42no—
243C3×Q8—44no—
244C3×D4—42no—
245C3×D4—42no—
246C3×D4—44no—
247C3×D4—48no—
248C3×D4—84no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—44no—
252C2×C3×C4—42no—
253C2×C3×C4—44no—
254C2×C3×C4#12012yes—
255C2×C3×C4#12012yes—
256C2×C3×C4C22×D411yescharacteristic
257C2×C3×C4C22×D411yescharacteristic
258C2×C3×C4C2×#4112yes—
259C2×C3×C4H(5,𝔽2)12yes—
260#67—42no—
261#67—88no—
262C2×Dic3—21no—
263C2×Dic3—22no—
264C2×Dic3—24no—
265C2×Dic3—88no—
266D12—21no—
267D12—21no—
268C4×D3—42no—
269C4×D3—44no—
270Dic6—21no—
271Dic6—21no—
272Dic6—44no—
273C3⋊C8—42no—
274C3⋊C8—42no—
275C24C2×#6714yes—
276#41—62no—
277#41—62no—
278#41—248no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—31no—
284C2×Q8—62no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—31no—
292C2×D4—48no—
293C2×D4—61no—
294C2×D4—68no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—68no—
298C22×C4—62no—
299C22×C4—61no—
300C22×C4C2×#6712yes—
301C22×C4C23×D311yescharacteristic
302Q16—124no—
303SD16—124no—
304M16—62no—
305M16—62no—
306M16—64no—
307C2×C8—62no—
308C2×C8—62no—
309C4⋊C4—32no—
310C4⋊C4—34no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—62no—
314C4⋊C4—62no—
315C4⋊C4—64no—
316C22⋊C4—416no—
317C22⋊C4—68no—
318C22⋊C4—68no—
319C22⋊C4—61no—
320C22⋊C4—62no—
321C22⋊C4—62no—
322C42—22no—
323C42—44no—
324C42—62no—
325C42—61no—
326C42—62no—
327C42D3×D411yescharacteristic
328C42D3×D411yescharacteristic
329C42D3×D412yes—
330C42#23714yescomplemented
331C22×C3—22no—
332C22×C3—44no—
333C22×C3—48no—
334C22×C3—88no—
335C22×C3C2×#12011yescharacteristic
336C22×C3#54612yes—
337C2×D3—41no—
338C2×D3—42no—
339C3×C4—22no—
340C3×C4—44no—
341C3×C4—82no—
342C3×C4#45514yes—
343C3×C4#53211yescharacteristic
344C3×C4#53511yescharacteristic
345C3×C4#55012yes—
346Dic3—41no—
347Dic3—44no—
348Dic3—42no—
349C23—24no—
350C23—28no—
351C23—48no—
352C23—61no—
353C23C22×#6711yescharacteristic
354Q8—42no—
355Q8—42no—
356Q8—44no—
357Q8—61no—
358Q8—61no—
359Q8—124no—
360D4—42no—
361D4—42no—
362D4—44no—
363D4—48no—
364D4—61no—
365D4—61no—
366D4—84no—
367D4—122no—
368D4—248no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—61no—
375C2×C4—62no—
376C2×C4—64no—
377C2×C4—122no—
378C2×C4—124no—
379C2×C4—248no—
380C2×C4#93712yes—
381C2×C4#93712yes—
382C2×C4C2×D3×D411yescharacteristic
383C2×C4C2×D3×D411yescharacteristic
384C2×C4C2×#23712yes—
385C2×C4#100412yes—
386C8—122no—
387C8—122no—
388C2×C3—22no—
389C2×C3—84no—
390C2×C3#243111yescharacteristic
391C2×C3C2×#45512yes—
392D3—81no—
393C22—22no—
394C22—48no—
395C22—44no—
396C22—88no—
397C22—122no—
398C22—121no—
399C22C2×#93711yescharacteristic
400C22#580612yes—
401C4—22no—
402C4—44no—
403C4—82no—
404C4—122no—
405C4—124no—
406C4—121no—
407C4#506114yes—
408C4#571511yescharacteristic
409C4#571611yescharacteristic
410C4#571812yes—
411C3#2559011yescharacteristic, complemented, Sylow
412C2—22no—
413C2—84no—
414C2—241no—
415C2C2×#506112yes—
416C2#8318711yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2858C2
socleC22×C3C2×#120
prime radical#86190C2
hypercenter#2858D3
hypocenterC3#25590
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2858

Complements

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

NH ≅ G/Ntype
2. C3×#2858C2semidirect
3. #83209C2semidirect
4. #83198C2semidirect
5. #71766C2semidirect
6. #71731C2semidirect
7. #71723C2semidirect
8. #69396C2semidirect
9. C4×#805C2semidirect
41. #2858D3semidirect
162. C3×C42#41semidirect
330. C42#237semidirect
411. C3#25590semidirect

Maximal subgroups

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

subgroupindex
#861902
C3×#28582
#832092
#831982
#717662
#717312
#717232
#693962
C4×#8052
#255903