Finite Group Databasebeta

← #10828024

Subgroups — #10828024

Order 768.

1214 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3486C211yesmaximal, characteristic
2#84062C211yesmaximal, characteristic, complemented
3#82915C212yesmaximal, complemented
4#72143C212yesmaximal, complemented
5#72133C211yesmaximal, characteristic, complemented
6#71863C214yesmaximal, complemented
7#71712C212yesmaximal, complemented
8#71658C212yesmaximal, complemented
9#24023—31nomaximal, Sylow
10C2×C3×2−1+4C2211yescharacteristic
11C23×C3×Q8C2211yescharacteristic
12C3×D4×Q8—24no—
13C3×#548—24no—
14C3×#536—22no—
15C3×#536C2212yes—
16C3×#525—22no—
17C3×#525C2212yes—
18C3×#519C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C2×#837C2214yes—
22#5232C2214yes—
23#5229—22no—
24#5229C2212yes—
25#5229C2212yes—
26C2×#836C2212yes—
27C2×#836C2212yes—
28#5211—22no—
29#5207C2212yes—
30#5045—24no—
31#5035—24no—
32#4995C2214yes—
33#4532C2214yes—
34#3486D311yescharacteristic
35#3093—32no—
36#2310—31no—
37#2038—34no—
38#1897—32no—
39#1857—32no—
40#1793—32no—
41#1784—31no—
42C3×2−1+4—24no—
43C3×2−1+4—42no—
44C2×C3×#41—22no—
45C2×C3×#41—22no—
46C2×C3×#41C2312yes—
47C22×C3×Q8—22no—
48C22×C3×Q8—24no—
49C22×C3×Q8—24no—
50C22×C3×Q8—42no—
51C22×C3×Q8D412yes—
52C22×C3×Q8C2311yescharacteristic
53C22×C3×Q8C2311yescharacteristic
54C23×C3×C4D412yes—
55C23×C3×C4C2311yescharacteristic
56C3×C4⋊Q8—24no—
57C3×C4⋊Q8—22no—
58C3×#124—24no—
59C3×#124—22no—
60C3×#123—24no—
61C3×#123—22no—
62C3×C22⋊Q8—24no—
63C3×C22⋊Q8—28no—
64C3×C22⋊Q8—24no—
65C3×C22⋊Q8D414yes—
66C3×#121—24no—
67C3×#121D414yes—
68C3×#120—22no—
69C3×C4×Q8—24no—
70C3×C4×D4—28no—
71C3×C4×D4—24no—
72C3×#117—22no—
73C3×#117C2312yes—
74C2×#217—21no—
75C2×#217—21no—
76#926—24no—
77#926—42no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3C2312yes—
80C2×C4×Dic3—21no—
81C2×C4×Dic3C2312yes—
82C2×#208C2314yes—
83#837—44no—
84#836—24no—
85#836—42no—
86#835—24no—
87#832—24no—
88C3⋊#95—24no—
89C3⋊#95—24no—
90#830—24no—
91#808—24no—
92#807—24no—
93C2×2−1+4C2×D311yescharacteristic
94C23×Q8C2×D311yescharacteristic
95D4×Q8—24no—
96#548—24no—
97#536—22no—
98#536C2×D312yes—
99#525—22no—
100#525C2×D312yes—
101#519C2×D311yescharacteristic
102C2×#117—31no—
103C2×#117—31no—
104#485—64no—
105#482—64no—
106#428—34no—
107C2×#104—34no—
108#417—34no—
109C2×#101—32no—
110C2×#101—32no—
111#407—32no—
112#391—32no—
113#391—32no—
114#391—62no—
115#381—62no—
116#339—34no—
117C24×C3C2×D411yescharacteristic
118C3×#41—24no—
119C3×#41—24no—
120C3×#41—44no—
121C3×#41—48no—
122C3×#41—44no—
123C2×C3×Q8—22no—
124C2×C3×Q8—24no—
125C2×C3×Q8—24no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—42no—
132C2×C3×Q8—416no—
133C2×C3×Q8—48no—
134C2×C3×Q8C2×D412yes—
135C2×C3×Q8#4114yes—
136C2×C3×D4—22no—
137C2×C3×D4—24no—
138C2×C3×D4—22no—
139C2×C3×D4C2×D412yes—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—24no—
143C22×C3×C4—24no—
144C22×C3×C4—24no—
145C22×C3×C4—22no—
146C22×C3×C4—42no—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D414yes—
150C22×C3×C4C2×D412yes—
151C22×C3×C4C2411yescharacteristic
152C22×Dic3—22no—
153C22×Dic3—22no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—28no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4C2×D414yes—
158C3×C22⋊C4—28no—
159C3×C22⋊C4—24no—
160C3×C22⋊C4—22no—
161C3×C22⋊C4—24no—
162C3×C42—24no—
163C3×C42—22no—
164#217—42no—
165#217—42no—
166C4⋊Dic3—24no—
167C4⋊Dic3—42no—
168C4×Dic3—24no—
169C4×Dic3—42no—
170#208—24no—
171#208—24no—
172C2×C3⋊C8—24no—
1732−1+4—24no—
1742−1+4—42no—
175C2×#41—22no—
176C2×#41—22no—
177C2×#41C22×D312yes—
178C22×Q8—24no—
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—42no—
182C22×Q8#6712yes—
183C22×Q8C22×D311yescharacteristic
184C22×Q8C22×D311yescharacteristic
185C23×C4#6712yes—
186C23×C4C22×D311yescharacteristic
187C2×M16—34no—
188C4⋊Q8—24no—
189C4⋊Q8—22no—
190#124—24no—
191#124—22no—
192#123—24no—
193#123—22no—
194C22⋊Q8—24no—
195C22⋊Q8—28no—
196C22⋊Q8—24no—
197C22⋊Q8#6714yes—
198#121—24no—
199#121#6714yes—
200#120—22no—
201C4×Q8—24no—
202C4×D4—28no—
203C4×D4—24no—
204#117—22no—
205#117—64no—
206#117—122no—
207#117C22×D312yes—
208C2×C4⋊C4—32no—
209C2×C4⋊C4—61no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C42—32no—
213C2×C42—61no—
214#107—64no—
215#106—64no—
216#104—124no—
217#103—64no—
218#102—64no—
219#101—64no—
220#101—122no—
221C22⋊C8—64no—
222#95—64no—
223#95—64no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—22no—
227C23×C3—42no—
228C23×C3#12012yes—
229C23×C3C22×D411yescharacteristic
230C3×Q8—28no—
231C3×Q8—24no—
232C3×Q8—48no—
233C3×Q8—44no—
234C3×Q8—44no—
235C3×D4—24no—
236C3×D4—44no—
237C3×D4—48no—
238C3×D4—42no—
239C3×D4—42no—
240C2×C3×C4—22no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—22no—
244C2×C3×C4—28no—
245C2×C3×C4—24no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—48no—
251C2×C3×C4—48no—
252C2×C3×C4—44no—
253C2×C3×C4#12012yes—
254C2×C3×C4#12314yes—
255C2×C3×C4C22×D412yes—
256C2×C3×C4C2×#4112yes—
257C2×C3×C4H(5,𝔽2)12yes—
258C2×Dic3—44no—
259C2×Dic3—44no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C3⋊C8—44no—
263C24C2×#6711yescharacteristic
264#41—24no—
265#41—24no—
266#41—44no—
267#41—48no—
268#41—44no—
269C2×Q8—22no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—42no—
278C2×Q8—416no—
279C2×Q8—48no—
280C2×Q8#23714yes—
281C2×Q8C2×#6712yes—
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4C2×#6712yes—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—42no—
293C22×C4—62no—
294C22×C4—62no—
295C22×C4C2×#6712yes—
296C22×C4C2×#6712yes—
297C22×C4C2×#6711yescharacteristic
298C22×C4C2×#6714yes—
299C22×C4C23×D311yescharacteristic
300M16—64no—
301M16—64no—
302C2×C8—64no—
303C4⋊C4—24no—
304C4⋊C4—28no—
305C4⋊C4—24no—
306C4⋊C4—64no—
307C4⋊C4—122no—
308C4⋊C4C2×#6714yes—
309C22⋊C4—28no—
310C22⋊C4—24no—
311C22⋊C4—22no—
312C22⋊C4—24no—
313C22⋊C4—122no—
314C22⋊C4—122no—
315C42—24no—
316C42—22no—
317C42—64no—
318C42—122no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—42no—
326C22×C3—44no—
327C22×C3C2×#12011yescharacteristic
328C22×C3C2×#12311yescharacteristic
329C22×C3#53411yescharacteristic
330C3×C4—22no—
331C3×C4—24no—
332C3×C4—42no—
333C3×C4—42no—
334C3×C4—44no—
335C3×C4#54614yes—
336Dic3—82no—
337Dic3—82no—
338C23—21no—
339C23—21no—
340C23—22no—
341C23—42no—
342C23#95312yes—
343C23C22×#6711yescharacteristic
344Q8—28no—
345Q8—24no—
346Q8—48no—
347Q8—44no—
348Q8—44no—
349D4—24no—
350D4—44no—
351D4—48no—
352D4—42no—
353D4—42no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—28no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—48no—
365C2×C4—48no—
366C2×C4—44no—
367C2×C4—124no—
368C2×C4—124no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4#93514yes—
372C2×C4#95312yes—
373C2×C4C2×#23712yes—
374C2×C4#100412yes—
375C2×C4C22×#6712yes—
376C8—124no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—42no—
380C2×C3—82no—
381C2×C3#240511yescharacteristic
382C2×C3#302012yes—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—44no—
391C22C2×#93511yescharacteristic
392C22#580411yescharacteristic
393C22C2×#95311yescharacteristic
394C4—22no—
395C4—24no—
396C4—42no—
397C4—42no—
398C4—44no—
399C4—242no—
400C4—242no—
401C4#580614yes—
402C3#2402311yescharacteristic, complemented, Sylow
403C2—21no—
404C2—21no—
405C2—42no—
406C2—82no—
407C2#8404711yescharacteristic
408C2#8411612yes—
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3486C2
socleC22×C3C2×#120
hypercenter#3486D3
hypocenterC3#24023
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3486

Complements

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

NH ≅ G/Ntype
2. #84062C2semidirect
3. #82915C2semidirect
4. #72143C2semidirect
5. #72133C2semidirect
6. #71863C2semidirect
7. #71712C2semidirect
8. #71658C2semidirect
402. C3#24023semidirect

Maximal subgroups

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

subgroupindex
C3×#34862
#840622
#829152
#721432
#721332
#718632
#717122
#716582
#240233