Finite Group Databasebeta

← #10828668

Subgroups — #10828668

Order 768.

1010 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85746C211yesmaximal, characteristic
2C3×#2317C211yesmaximal, characteristic, complemented
3#82139C212yesmaximal, complemented
4#71591C212yesmaximal, complemented
5#71589C211yesmaximal, characteristic, complemented
6#69592C214yesmaximal, complemented
7#69563C212yesmaximal, complemented
8#69543C212yesmaximal, complemented
9#24081—31nomaximal, Sylow
10C2×C3×#117C2211yescharacteristic
11C2×C3×#117C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13#5791C2212yes—
14#5533—22no—
15#5532—22no—
16#5530—22no—
17#5529—22no—
18#5510C2211yescharacteristic
19#5504C2212yes—
20#5500C2211yescharacteristic
21#5494—22no—
22#5479—22no—
23C3×#389—22no—
24C3×#385—22no—
25C3×#384C2212yes—
26C3×#382C2212yes—
27#5207C2212yes—
28#4971—22no—
29#4970—22no—
30#4968—22no—
31#4967—22no—
32#4966C2214yes—
33#4964C2214yes—
34C2×#805C2214yes—
35#4530C2212yes—
36#4530C2212yes—
37#4523C2214yes—
38#3493—31no—
39#3092—32no—
40#2317D311yescharacteristic, complemented
41#2061—32no—
42#2040—32no—
43#1897—32no—
44#1859—34no—
45#1822—31no—
46C23×C3×C4—21no—
47C23×C3×C4C2311yescharacteristic
48C2×#235C2312yes—
49C3×#117—22no—
50C3×#117—22no—
51C3×#117—24no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C4⋊C4C2311yescharacteristic
59C2×C3×C4⋊C4C2311yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4D412yes—
62C2×C3×C42D412yes—
63C2×C3×C42C2312yes—
64#953—22no—
65#930—22no—
66#930D412yes—
67#930D412yes—
68#929—22no—
69C4×#67—24no—
70C4×#67—22no—
71C4×#67—22no—
72#926C2312yes—
73#924—22no—
74#924D412yes—
75#924D412yes—
76C2×#208C2314yes—
77#897—22no—
78#895—22no—
79#893—22no—
80#892—22no—
81D3×C4⋊C4—22no—
82#890—22no—
83#888—22no—
84#887—22no—
85#885—24no—
86#884—24no—
87#883—24no—
88#879—24no—
89#876—24no—
90#875—22no—
91C4×D12—22no—
92#872—24no—
93#870—22no—
94C4×Dic6—22no—
95C3×#95—24no—
96#830—22no—
97#830—22no—
98#810—24no—
99#809—24no—
100#808—24no—
101#807—24no—
102#805—28no—
103#556—62no—
104#555—62no—
105#553—62no—
106#552—62no—
107#544—31no—
108#538—62no—
109#537—31no—
110#529—62no—
111#525—32no—
112#520—32no—
113C2×#117C2×D311yescharacteristic
114C2×#117C2×D311yescharacteristic
115C22×C4⋊C4C2×D311yescharacteristic
116#484—62no—
117#483—62no—
118#481—62no—
119#480—62no—
120#428—34no—
121C2×#104—34no—
122#417—34no—
123#407—32no—
124#389—22no—
125#385—22no—
126#384C2×D312yes—
127#382C2×D312yes—
128#343—32no—
129#343—32no—
130#339—34no—
131C24×C3C2×D411yescharacteristic
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—42no—
137C22×C3×C4—82no—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2411yescharacteristic
144C2×#67—22no—
145C2×#67—22no—
146C2×#67—22no—
147#235—24no—
148#235—24no—
149C2×D12—22no—
150C2×D12C2×D412yes—
151C2×C4×D3—22no—
152C2×C4×D3—24no—
153C2×Dic6—22no—
154C2×Dic6C2×D412yes—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—41no—
161C3×C4⋊C4—84no—
162C3×C22⋊C4—24no—
163C3×C22⋊C4—24no—
164C3×C42—22no—
165C3×C42—22no—
166C3×C42—22no—
167C3×C42#4114yes—
168#217—22no—
169#217—22no—
170#217—22no—
171(C2×D3)⋊C4—24no—
172(C2×D3)⋊C4—24no—
173(C2×D3)⋊C4—22no—
174(C2×D3)⋊C4—22no—
175C4⋊Dic3—22no—
176C4⋊Dic3—22no—
177C4⋊Dic3C2×D412yes—
178C4⋊Dic3C2×D412yes—
179Dic3⋊C4—24no—
180Dic3⋊C4—24no—
181Dic3⋊C4—22no—
182Dic3⋊C4—22no—
183C4×Dic3—24no—
184C4×Dic3—22no—
185#208—24no—
186#208—24no—
187C2×C3⋊C8—24no—
188C2×#41—32no—
189C23×C4—21no—
190C23×C4C22×D311yescharacteristic
191C2×M16—34no—
192C4⋊Q8—62no—
193#126—64no—
194#126—64no—
195#125—62no—
196#125—62no—
197#124—62no—
198#124—64no—
199#123—64no—
200#123—62no—
201C22⋊Q8—32no—
202C22⋊Q8—32no—
203C22⋊Q8—62no—
204C22⋊Q8—62no—
205#121—32no—
206#121—32no—
207#121—64no—
208#121—62no—
209#121—62no—
210#120—62no—
211C4×Q8—62no—
212C4×Q8—62no—
213C4×D4—62no—
214C4×D4—62no—
215C4×D4—62no—
216C4×D4—62no—
217C4×D4—64no—
218#117—22no—
219#117—22no—
220#117—24no—
221#117—32no—
222#117—64no—
223#117—62no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—62no—
230C2×C4⋊C4C22×D312yes—
231C2×C4⋊C4C22×D311yescharacteristic
232C2×C4⋊C4C22×D311yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4#6712yes—
235C2×C42#6712yes—
236C2×C42C22×D312yes—
237#107—64no—
238#106—64no—
239#104—68no—
240#103—64no—
241#102—64no—
242C22⋊C8—62no—
243C22⋊C8—62no—
244#95—24no—
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3#12012yes—
250C23×C3C22×D411yescharacteristic
251C22×D3—22no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—24no—
255C2×C3×C4—24no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—24no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—44no—
264C2×C3×C4—42no—
265C2×C3×C4—84no—
266C2×C3×C4#12011yescharacteristic
267C2×C3×C4#12011yescharacteristic
268C2×C3×C4C22×D412yes—
269C2×C3×C4C2×#4112yes—
270C2×C3×C42−1+412yes—
271#67—42no—
272#67—42no—
273#67—82no—
274#67—82no—
275C2×Dic3—22no—
276C2×Dic3—22no—
277C2×Dic3—22no—
278D12—24no—
279D12—82no—
280C4×D3—44no—
281C4×D3—84no—
282Dic6—24no—
283Dic6—82no—
284C3⋊C8—44no—
285C24C2×#6711yescharacteristic
286#41—64no—
287#41—64no—
288C2×Q8—32no—
289C2×Q8—62no—
290C2×D4—32no—
291C2×D4—62no—
292C2×D4—62no—
293C2×D4—62no—
294C2×D4—62no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—64no—
301C22×C4—62no—
302C22×C4—82no—
303C22×C4D3×D412yes—
304C22×C4D3×D412yes—
305C22×C4C2×#6711yescharacteristic
306C22×C4C2×#6712yes—
307C22×C4C2×#6712yes—
308C22×C4C23×D311yescharacteristic
309M16—64no—
310M16—64no—
311C2×C8—64no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—24no—
316C4⋊C4—32no—
317C4⋊C4—32no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—64no—
321C4⋊C4—64no—
322C4⋊C4—62no—
323C4⋊C4—62no—
324C4⋊C4—62no—
325C4⋊C4—62no—
326C4⋊C4—84no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—64no—
330C22⋊C4—64no—
331C22⋊C4—62no—
332C22⋊C4—62no—
333C22⋊C4—62no—
334C22⋊C4—62no—
335C22⋊C4—62no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—64no—
340C42—62no—
341C42#23514yes—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—42no—
348C22×C3—42no—
349C22×C3C2×#12011yescharacteristic
350C22×C3#52511yescharacteristic
351C22×C3#53611yescharacteristic
352C2×D3—42no—
353C2×D3—42no—
354C2×D3—42no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—44no—
358C3×C4—42no—
359C3×C4—42no—
360C3×C4—82no—
361C3×C4#54714yes—
362Dic3—42no—
363Dic3—42no—
364Dic3—42no—
365C23—21no—
366C23—21no—
367C23—41no—
368C23—41no—
369C23—62no—
370C23#93712yes—
371C23C22×#6711yescharacteristic
372Q8—64no—
373Q8—242no—
374D4—64no—
375D4—122no—
376D4—122no—
377D4—242no—
378D4—242no—
379D4—242no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—62no—
394C2×C4—62no—
395C2×C4—62no—
396C2×C4—84no—
397C2×C4—124no—
398C2×C4—244no—
399C2×C4#93711yescharacteristic
400C2×C4#93711yescharacteristic
401C2×C4C2×#23512yes—
402C2×C4C2×D3×D412yes—
403C2×C4#100712yes—
404C8—124no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—42no—
408C2×C3#240611yescharacteristic
409C2×C3#302111yescharacteristic
410C2×C3#302211yescharacteristic
411D3—82no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—42no—
418C22—42no—
419C22—122no—
420C22—122no—
421C22—122no—
422C22#579111yescharacteristic
423C22C2×#93711yescharacteristic
424C22#581611yescharacteristic
425C4—21no—
426C4—21no—
427C4—44no—
428C4—42no—
429C4—42no—
430C4—82no—
431C4—122no—
432C4—122no—
433C4—122no—
434C4#550714yes—
435C3#2408111yescharacteristic, complemented, Sylow
436C2—21no—
437C2—21no—
438C2—42no—
439C2—242no—
440C2#8219211yescharacteristic
441C2#8339511yescharacteristic
442C2#8341611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2317C2
socleC22×C3C2×#120
prime radical#85746C2
hypercenter#2317D3
hypocenterC3#24081
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. C3×#2317C2semidirect
3. #82139C2semidirect
4. #71591C2semidirect
5. #71589C2semidirect
6. #69592C2semidirect
7. #69563C2semidirect
8. #69543C2semidirect
40. #2317D3semidirect
435. C3#24081semidirect

Maximal subgroups

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

subgroupindex
#857462
C3×#23172
#821392
#715912
#715892
#695922
#695632
#695432
#240813