Finite Group Databasebeta

← #10849652

Subgroups — #10849652

Order 768.

1072 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3508C211yesmaximal, characteristic
2#83342C211yesmaximal, characteristic, complemented
3#83180C211yesmaximal, characteristic, complemented
4#82509C211yesmaximal, characteristic, complemented
5#71852C214yesmaximal, complemented
6#71840C211yesmaximal, characteristic, complemented
7#71766C211yesmaximal, characteristic, complemented
8#71647C214yesmaximal, complemented
9#71643C211yesmaximal, characteristic, complemented
10#25690—31nomaximal, Sylow
11C3×#559—22no—
12C3×#550—22no—
13C3×#546—24no—
14C3×#540—22no—
15C3×#535C2212yes—
16C3×#533C2211yescharacteristic
17C3×#532—22no—
18C2×C3×#127C2211yescharacteristic
19C3×#518C2212yes—
20C3×C4×#41C2211yescharacteristic
21C2×#870C2211yescharacteristic
22C2×#837C2212yes—
23#5232C2212yes—
24#5222—24no—
25#5221C2214yes—
26C2×#833C2214yes—
27#5081—22no—
28#5065—22no—
29#5022—22no—
30#5009—22no—
31#4995C2212yes—
32#4995C2214yes—
33#4932C2211yescharacteristic
34#4924C2211yescharacteristic
35C4×#208C2211yescharacteristic
36#4532C2214yes—
37#4532C2212yes—
38#3508D311yescharacteristic
39#3166—31no—
40#3080—31no—
41#2641—31no—
42#2037—34no—
43#1995—31no—
44#1965—31no—
45#1856—34no—
46#1759—31no—
47C2×C3×#41—24no—
48C2×C3×#41C2311yescharacteristic
49C22×C3×D4D414yes—
50C22×C3×D4C2312yes—
51C3×C4⋊Q8D412yes—
52C3×#127—22no—
53C3×#127—22no—
54C3×#127—21no—
55C3×#127—21no—
56C3×#125D412yes—
57C3×#124—24no—
58C3×#124—22no—
59C3×#123—28no—
60C3×C22⋊Q8—22no—
61C3×#121—24no—
62C3×#121—28no—
63C3×#121—24no—
64C3×#120—24no—
65C3×C4×Q8—22no—
66C3×C4×Q8D412yes—
67C3×C4×D4—24no—
68C3×C4×D4—22no—
69C3×C4×D4—24no—
70C3×C4×D4—24no—
71C3×C4×D4D412yes—
72C3×#117—21no—
73C3×#117C2311yescharacteristic
74C3×#117C2312yes—
75C2×C3×C22⋊C4—24no—
76C2×C3×C42—21no—
77C2×C3×C42C2311yescharacteristic
78C2×#217—28no—
79C2×C4⋊Dic3C2312yes—
80C2×Dic3⋊C4—22no—
81C2×C4×Dic3C2312yes—
82C2×#208C2312yes—
83C2×#208C2312yes—
84#870—22no—
85#870—21no—
86#870—21no—
87#837—24no—
88#835—22no—
89#833—28no—
90#832—22no—
91C3⋊#95—22no—
92#808—22no—
93#808—24no—
94#807—22no—
95#807—24no—
96C4⋊(C3⋊C8)—22no—
97#803—21no—
98C4×C3⋊C8—21no—
99#559—22no—
100#550—22no—
101#546—24no—
102#540—22no—
103#535C2×D312yes—
104#533C2×D311yescharacteristic
105#532—22no—
106C2×#127C2×D311yescharacteristic
107C2×#125—31no—
108#518C2×D312yes—
109C4×#41C2×D311yescharacteristic
110#501—62no—
111#499—62no—
112#479—62no—
113#478—62no—
114#428—32no—
115#428—34no—
116#423—31no—
117C2×#104—32no—
118#417—32no—
119C2×#100—34no—
120C4×M16—31no—
121#394—64no—
122#390—34no—
123#376—31no—
124#339—32no—
125#339—34no—
126C24×C3C2×D414yes—
127C3×#41—22no—
128C3×#41—22no—
129C3×#41—88no—
130C2×C3×Q8—22no—
131C2×C3×Q8C2×D411yescharacteristic
132C2×C3×D4—21no—
133C2×C3×D4—28no—
134C2×C3×D4—24no—
135C2×C3×D4—24no—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4—84no—
141C2×C3×D4C2×D411yescharacteristic
142C22×C3×C4—28no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2411yescharacteristic
147C22×Dic3—22no—
148C22×Dic3—22no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4C2×D412yes—
153C3×C4⋊C4C2×D414yes—
154C3×C22⋊C4—28no—
155C3×C22⋊C4—28no—
156C3×C22⋊C4—21no—
157C3×C22⋊C4—22no—
158C3×C22⋊C4—22no—
159C3×C42—22no—
160C3×C42—21no—
161C3×C42—22no—
162C3×C42C2×D411yescharacteristic
163C3×C42C2×D411yescharacteristic
164C3×C42C2×D412yes—
165#217—88no—
166C4⋊Dic3—22no—
167C4⋊Dic3—22no—
168Dic3⋊C4—42no—
169Dic3⋊C4—42no—
170C4×Dic3—22no—
171C4×Dic3—22no—
172#208—22no—
173#208—24no—
174#208#4114yescomplemented
175C2×C3⋊C8—22no—
176C2×C3⋊C8—22no—
177C2×#41—24no—
178C2×#41C22×D311yescharacteristic
179C22×D4#6714yes—
180C22×D4C22×D312yes—
181C2×M16—32no—
182C2×M16—32no—
183C4⋊Q8#6712yes—
184#127—22no—
185#127—21no—
186#127—21no—
187#127—22no—
188#125—62no—
189#125—61no—
190#125—61no—
191#125#6712yes—
192#124—24no—
193#124—22no—
194#123—28no—
195C22⋊Q8—22no—
196#121—24no—
197#121—28no—
198#121—24no—
199#120—24no—
200C4×Q8—22no—
201C4×Q8#6712yes—
202C4×D4—24no—
203C4×D4—22no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4#6712yes—
207#117—21no—
208#117C22×D311yescharacteristic
209#117C22×D312yes—
210C2×C4⋊C4—32no—
211C2×C4⋊C4—62no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—68no—
214C2×C42—21no—
215C2×C42—32no—
216C2×C42C22×D311yescharacteristic
217#107—62no—
218#107—64no—
219#106—62no—
220#106—64no—
221C4⋊C8—62no—
222#104—64no—
223#103—62no—
224#102—62no—
225#100—68no—
226#97—61no—
227C4×C8—61no—
228#95—62no—
229C23×C3—24no—
230C23×C3—21no—
231C23×C3—28no—
232C23×C3—48no—
233C23×C3C22×D411yescharacteristic
234C3×Q8—21no—
235C3×Q8—21no—
236C3×Q8—44no—
237C3×D4—21no—
238C3×D4—21no—
239C3×D4—48no—
240C3×D4—42no—
241C3×D4—42no—
242C3×D4—42no—
243C3×D4—44no—
244C3×D4—84no—
245C3×D4—88no—
246C2×C3×C4—21no—
247C2×C3×C4—22no—
248C2×C3×C4—24no—
249C2×C3×C4—42no—
250C2×C3×C4—44no—
251C2×C3×C4—42no—
252C2×C3×C4—88no—
253C2×C3×C4#12012yes—
254C2×C3×C4#12012yes—
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C22×D411yescharacteristic
257C2×C3×C4C2×#4112yes—
258C2×C3×C42−1+412yes—
259C2×Dic3—44no—
260C2×Dic3—44no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C3⋊C8—24no—
264C3⋊C8—42no—
265C24C2×#6714yes—
266#41—22no—
267#41—22no—
268#41—88no—
269C2×Q8—22no—
270C2×Q8C2×#6711yescharacteristic
271C2×D4—21no—
272C2×D4—28no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—84no—
280C2×D4C2×#6711yescharacteristic
281C22×C4—28no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—62no—
285C22×C4—62no—
286C22×C4C2×#6712yes—
287C22×C4C23×D311yescharacteristic
288M16—34no—
289M16—62no—
290M16—64no—
291C2×C8—62no—
292C2×C8—62no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—62no—
297C4⋊C4—62no—
298C4⋊C4—122no—
299C4⋊C4—122no—
300C4⋊C4C2×#6712yes—
301C4⋊C4C2×#6714yes—
302C22⋊C4—28no—
303C22⋊C4—28no—
304C22⋊C4—21no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—248no—
308C42—22no—
309C42—21no—
310C42—22no—
311C42—62no—
312C42—62no—
313C42C2×#6711yescharacteristic
314C42C2×#6711yescharacteristic
315C42C2×#6712yes—
316C22×C3—22no—
317C22×C3—41no—
318C22×C3—44no—
319C22×C3—42no—
320C22×C3—48no—
321C22×C3—88no—
322C22×C3C2×#12011yescharacteristic
323C22×C3#54712yes—
324C3×C4—22no—
325C3×C4—22no—
326C3×C4—41no—
327C3×C4—42no—
328C3×C4—44no—
329C3×C4#53211yescharacteristic
330C3×C4#53611yescharacteristic
331C3×C4#54812yes—
332Dic3—82no—
333Dic3—82no—
334C23—24no—
335C23—21no—
336C23—28no—
337C23—48no—
338C23C22×#6711yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—44no—
342D4—21no—
343D4—21no—
344D4—48no—
345D4—42no—
346D4—42no—
347D4—44no—
348D4—42no—
349D4—84no—
350D4—88no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—88no—
358C2×C4—124no—
359C2×C4—124no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4#95312yes—
363C2×C4#95312yes—
364C2×C4D3×#4112yes—
365C2×C4#101012yes—
366C2×C4C22×#6711yescharacteristic
367C2×C4C22×#6711yescharacteristic
368C8—64no—
369C8—122no—
370C2×C3—22no—
371C2×C3—81no—
372C2×C3—84no—
373C2×C3#243211yescharacteristic
374C2×C3#301912yes—
375C22—22no—
376C22—41no—
377C22—44no—
378C22—42no—
379C22—48no—
380C22—88no—
381C22#582412yes—
382C22C2×#95311yescharacteristic
383C4—22no—
384C4—22no—
385C4—41no—
386C4—42no—
387C4—44no—
388C4—242no—
389C4—242no—
390C4#582511yescharacteristic
391C4#583112yes—
392C4#583211yescharacteristic
393C3#2569011yescharacteristic, complemented, Sylow
394C2—22no—
395C2—84no—
396C2—81no—
397C2#8409311yescharacteristic
398C2#8410412yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3508C2
socleC22×C3C2×#120
hypercenter#3508D3
hypocenterC3#25690
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3508

Complements

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

NH ≅ G/Ntype
2. #83342C2semidirect
3. #83180C2semidirect
4. #82509C2semidirect
5. #71852C2semidirect
6. #71840C2semidirect
7. #71766C2semidirect
8. #71647C2semidirect
9. #71643C2semidirect
174. #208#41semidirect
393. C3#25690semidirect

Maximal subgroups

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

subgroupindex
C3×#35082
#833422
#831802
#825092
#718522
#718402
#717662
#716472
#716432
#256903