Finite Group Databasebeta

← #10696440

Subgroups — #10696440

Order 768.

618 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83240C211yesmaximal, characteristic
2C2×#5080C211yesmaximal, characteristic, complemented
3C2×#4846C211yesmaximal, characteristic
4C2×C3×#332C211yesmaximal, characteristic, complemented
5#70456C211yesmaximal, characteristic, complemented
6C2×#4483C211yesmaximal, characteristic
7#67704C211yesmaximal, characteristic, complemented
8#10074—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#947—21no—
11C2×C4×D12C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15#5229—21no—
16C2×#821—21no—
17C2×#818—21no—
18C2×#816—22no—
19#5080—22no—
20#5080—22no—
21#4992—22no—
22#4987—22no—
23C2×#808—22no—
24C4×C4⋊Dic3C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26#4846—22no—
27#4846—22no—
28C3×#332—22no—
29#4483—22no—
30C2×#498—31no—
31C2×#474—31no—
32#2547—31no—
33#1981—31no—
34C2×#332D311yescharacteristic, complemented
35C2×D4⋊C8—31no—
36#1279—31no—
37C22×C3×Q8—21no—
38C22×D12D411yescharacteristic
39C22×C4×D3—21no—
40C22×Dic6—21no—
41C22×C3×C8—21no—
42C22×C3×C8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4D412yes—
49C2×C3×C42C2311yescharacteristic
50#947—42no—
51#947—42no—
52C2×(C2×D3)⋊C4—24no—
53C2×(C2×D3)⋊C4—21no—
54C2×C4⋊Dic3—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3D412yes—
57C2×C4⋊Dic3D411yescharacteristic
58C2×C4×Dic3—21no—
59C22×C3⋊C8—21no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12—21no—
63C4×D12—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67#869D412yes—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8D412yes—
72C3⋊#95—24no—
73C3⋊#95—21no—
74#821—24no—
75#818—24no—
76#816—44no—
77#808—28no—
78C4×C3⋊C8—21no—
79C4×C3⋊C8D412yes—
80C2×C4⋊Q8—31no—
81C2×C4⋊Q8C2×D311yescharacteristic
82C2×#124—61no—
83C2×C4×D4—31no—
84#498—62no—
85#498—62no—
86#474—62no—
87#474—62no—
88C2×#107—61no—
89C2×#106—62no—
90C2×C4⋊C8C2×D311yescharacteristic
91C2×C4⋊C8C2×D311yescharacteristic
92C2×#103—62no—
93C2×#102—61no—
94C2×C4×C8—31no—
95#397—62no—
96#391—61no—
97#380—62no—
98C4×C4⋊C4—31no—
99#332—22no—
100D4⋊C8—62no—
101C23×D3—21no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×Q8—41no—
107C2×C3×Q8—41no—
108C22×C3×C4—22no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×Dic3—21no—
113C22×Dic3—22no—
114C22×Dic3—41no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12SD1612yes—
121C2×C4×D3—41no—
122C2×C4×D3—41no—
123C2×C4×D3—42no—
124C2×C4×D3—42no—
125C2×Dic6—22no—
126C2×Dic6—42no—
127C2×Dic6—42no—
128C2×Dic6—41no—
129C2×C3×C8—22no—
130C2×C3×C8—41no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8—42no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42#4111yescharacteristic
142(C2×D3)⋊C4—48no—
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145C4⋊Dic3—22no—
146C4⋊Dic3—21no—
147C4⋊Dic3—22no—
148C4⋊Dic3—21no—
149C4⋊Dic3—22no—
150C4⋊Dic3—22no—
151C4⋊Dic3—41no—
152C4⋊Dic3—41no—
153C4×Dic3—41no—
154C4×Dic3—41no—
155C2×C3⋊C8—24no—
156C2×C3⋊C8—22no—
157C22×Q8—21no—
158C22×Q8—61no—
159C22×D4—31no—
160C23×C4—61no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—61no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—32no—
169C4⋊Q8—61no—
170C4⋊Q8—61no—
171C4⋊Q8—61no—
172#124—122no—
173#124—122no—
174C4×D4—61no—
175C4×D4—61no—
176C4×D4—61no—
177C4×D4—61no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—31no—
180C2×C4⋊C4—32no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4D1212yes—
184C2×C22⋊C4—64no—
185C2×C22⋊C4—61no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188#107—64no—
189#106—68no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8#6712yes—
194#103—124no—
195#102—64no—
196C4×C8—32no—
197C4×C8—61no—
198#95—64no—
199#95—61no—
200C23×C3#12011yescharacteristic
201C22×D3—22no—
202C22×D3—41no—
203C22×D3—42no—
204C22×D3—41no—
205C22×D3—42no—
206C3×Q8—81no—
207C3×Q8—81no—
208C3×Q8—81no—
209C3×Q8—81no—
210C2×C3×C4—22no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12711yescharacteristic
220C2×C3×C4C2×SD1611yescharacteristic
221C2×C3×C4#13511yescharacteristic
222C2×C3×C4#13512yes—
223C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C4AGL(1,ℤ/8)12yes—
225C2×C3×C4#13711yescharacteristic
226C2×Dic3—42no—
227C2×Dic3—41no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C2×Dic3—42no—
232C2×Dic3—82no—
233D12—22no—
234D12—22no—
235D12—41no—
236D12—41no—
237C4×D3—82no—
238C4×D3—81no—
239C4×D3—81no—
240Dic6—44no—
241Dic6—81no—
242Dic6—81no—
243C3×C8—42no—
244C3×C8—81no—
245C3⋊C8—24no—
246C24—61no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—62no—
254C2×Q8—121no—
255C2×Q8—122no—
256C2×Q8—122no—
257C2×D4—32no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—61no—
263C22×C4—22no—
264C22×C4—61no—
265C22×C4—62no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4—122no—
269C22×C4—122no—
270C22×C4—121no—
271C22×C4C2×D1211yescharacteristic
272C22×C4D3×D411yescharacteristic
273C22×C4D3×D411yescharacteristic
274C2×C8—22no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—41no—
279C2×C8—64no—
280C2×C8—62no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—62no—
286C4⋊C4—61no—
287C4⋊C4—62no—
288C4⋊C4—62no—
289C4⋊C4—62no—
290C4⋊C4—61no—
291C4⋊C4—121no—
292C4⋊C4—121no—
293C22⋊C4—128no—
294C22⋊C4—121no—
295C22⋊C4—121no—
296C42—121no—
297C42—121no—
298C42#23511yescharacteristic
299C42D3×D411yescharacteristic
300C42D3×D411yescharacteristic
301C42C2×#6711yescharacteristic
302C22×C3#39211yescharacteristic
303C22×C3#44912yes—
304C22×C3#45011yescharacteristic
305C22×C3#45211yescharacteristic
306C22×C3#45612yes—
307C2×D3—44no—
308C2×D3—42no—
309C2×D3—82no—
310C2×D3—81no—
311C2×D3—81no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—82no—
315C3×C4#44811yescharacteristic
316C3×C4#44911yescharacteristic
317C3×C4#46312yes—
318C3×C4#46511yescharacteristic
319C3×C4#46811yescharacteristic
320C3×C4#49511yescharacteristic
321C3×C4#49611yescharacteristic
322Dic3—82no—
323Dic3—82no—
324Dic3—81no—
325C23—62no—
326C23—121no—
327C23—122no—
328C23—121no—
329C23—122no—
330C23#88211yescharacteristic
331Q8—81no—
332Q8—81no—
333Q8—81no—
334Q8—81no—
335Q8—124no—
336Q8—241no—
337Q8—241no—
338D4—62no—
339D4—62no—
340D4—121no—
341D4—121no—
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—121no—
347C2×C4—122no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—242no—
353C2×C4—241no—
354C2×C4—241no—
355C2×C4—242no—
356C2×C4#88211yescharacteristic
357C2×C4#88411yescharacteristic
358C2×C4#88411yescharacteristic
359C2×C4#91111yescharacteristic
360C2×C4#91211yescharacteristic
361C2×C4D3×SD1611yescharacteristic
362C2×C4#91412yes—
363C2×C4#91511yescharacteristic
364C2×C4#91912yes—
365C2×C4#93011yescharacteristic
366C2×C4#93711yescharacteristic
367C2×C4#94011yescharacteristic
368C8—42no—
369C8—64no—
370C8—81no—
371C2×C3#165212yes—
372C2×C3#165412yes—
373C2×C3#200011yescharacteristic
374C2×C3#200511yescharacteristic
375C2×C3#201211yescharacteristic
376D3—82no—
377C22—122no—
378C22—124no—
379C22—242no—
380C22—241no—
381C22—241no—
382C22#475212yes—
383C22#477611yescharacteristic
384C22#478311yescharacteristic
385C22#481012yes—
386C22#496011yescharacteristic
387C4—21no—
388C4—21no—
389C4—82no—
390C4—242no—
391C4—241no—
392C4—242no—
393C4#477811yescharacteristic
394C4#478011yescharacteristic
395C4#481912yes—
396C4#515811yescharacteristic
397C4#517011yescharacteristic
398C4#517611yescharacteristic
399C4#519111yescharacteristic
400C3#1007411yescharacteristic, complemented, Sylow
401C2—242no—
402C2#6843912yes—
403C2#6846212yes—
404C2#7011911yescharacteristic
405C2#7020511yescharacteristic
406C2#7027711yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#332C2
socleC23×C3#120
prime radicalC22×D12D4
hypercenterC2×#332D3
hypocenterC3#10074
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. C2×#5080C2semidirect
4. C2×C3×#332C2semidirect
5. #70456C2semidirect
7. #67704C2semidirect
34. C2×#332D3semidirect
400. C3#10074semidirect

Maximal subgroups

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

subgroupindex
#832402
C2×#50802
C2×#48462
C2×C3×#3322
#704562
C2×#44832
#677042
#100743