Finite Group Databasebeta

← #10801793

Subgroups — #10801793

Order 768.

954 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5188C211yesmaximal, characteristic
2C2×#5184C211yesmaximal, characteristic, complemented
3C2×#4708C211yesmaximal, characteristic
4#76611C212yesmaximal
5C3×#1971C211yesmaximal, characteristic, complemented
6#70577C211yesmaximal, characteristic
7#70553C211yesmaximal, characteristic, complemented
8#70271C212yesmaximal
9Dic3×#103C212yesmaximal, complemented
10#69819C211yesmaximal, characteristic, complemented
11#69674C212yesmaximal, complemented
12#21530—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×#946C2212yes—
15C2×Q8×Dic3C2212yes—
16C22×ΓL(1,𝔽25)C2211yescharacteristic
17C22×D3×C8C2211yescharacteristic
18C2×#872C2211yescharacteristic
19C2×C3×#106C2211yescharacteristic
20C2×C3×#103C2212yes—
21C2×C3×#97C2211yescharacteristic
22C3×#391C2212yes—
23C2×#835C2212yes—
24C2×#835C2212yes—
25#5188C418yescomplemented
26#5184—24no—
27C2×#818C2211yescharacteristic
28C2×#817C2211yescharacteristic
29C2×#815C2211yescharacteristic
30C2×C8×Dic3C2211yescharacteristic
31#4987C2212yes—
32Dic3×C4⋊C4C2212yes—
33#4928C2212yes—
34C2×C4×C3⋊C8C2211yescharacteristic
35#4708—24no—
36#4670—22no—
37#4669—22no—
38#4652—22no—
39#4647—22no—
40#4551C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×#467—31no—
44C2×#441—31no—
45C2×#405—31no—
46#2323—32no—
47#1971D311yescharacteristic, complemented
48#1943—31no—
49#1933—32no—
50#1894—32no—
51#1807—31no—
52#1776—31no—
53C4×#103—32no—
54C22×C3×Q8C2312yes—
55C22×C4×D3C2311yescharacteristic
56C2×C3×Q16—22no—
57C2×C3×Q16—22no—
58C2×C3×Q16C2×C414yes—
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C42C2311yescharacteristic
63#946—24no—
64#946C2×C418yes—
65Q8×Dic3—24no—
66Q8×Dic3—24no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4D412yes—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3C2312yes—
72C2×C4⋊Dic3C2312yes—
73C2×Dic3⋊C4—22no—
74C2×Dic3⋊C4—22no—
75C2×Dic3⋊C4D412yes—
76C2×C4×Dic3—22no—
77C2×C4×Dic3—22no—
78C2×C4×Dic3C2311yescharacteristic
79C22×C3⋊C8C2311yescharacteristic
80C22×C3⋊C8C2311yescharacteristic
81C2×ΓL(1,𝔽25)—22no—
82C2×ΓL(1,𝔽25)—22no—
83C2×ΓL(1,𝔽25)—22no—
84C2×D3×C8—22no—
85C2×D3×C8—22no—
86C2×D3×C8C2×C414yes—
87#872—22no—
88#872—22no—
89C3×#106—22no—
90C3×#103—24no—
91C3×#97—22no—
92C3×#95—22no—
93C3×#95—22no—
94#835—24no—
95#835C2×C418yes—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99#818C2×C414yes—
100#817—22no—
101#815D414yes—
102C8×Dic3—22no—
103C4×C3⋊C8—22no—
104C22×Q16C2×D311yescharacteristic
105C22×M16—31no—
106C23×C8—31no—
107C2×C22⋊Q8—32no—
108C2×C4×Q8—32no—
109C2×#117—31no—
110#467—38no—
111#441—64no—
112C2×#107—31no—
113C2×#106—31no—
114C2×#106C2×D311yescharacteristic
115C2×#103—32no—
116C2×#103—32no—
117C2×#103C2×D312yes—
118#405—64no—
119C2×#97—31no—
120C2×#97C2×D311yescharacteristic
121C2×C4×C8—31no—
122C2×C4×C8—31no—
123#391C2×D312yes—
124#389—32no—
125#389—62no—
126#388—62no—
127#388—62no—
128#387—62no—
129#380—32no—
130C4×C4⋊C4—32no—
131#340—32no—
132#340—32no—
133#340—32no—
134C23×D3C2×D411yescharacteristic
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8C22×C414yes—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2411yescharacteristic
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3C2×D411yescharacteristic
148C22×Dic3C2×D412yes—
149C2×C4×D3—21no—
150C2×C4×D3—21no—
151C2×C4×D3—21no—
152C2×C4×D3—21no—
153C2×C4×D3C22×C412yes—
154C2×C4×D3#4112yes—
155C3×Q16—28no—
156C2×C3×C8—21no—
157C2×C3×C8—21no—
158C2×C3×C8C22×C412yes—
159C2×C3×C8C2×D412yes—
160C2×C3×C8#4112yes—
161C2×C3×C8#4112yes—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C42—21no—
165C3×C42—21no—
166(C2×D3)⋊C4—28no—
167(C2×D3)⋊C4—24no—
168(C2×D3)⋊C4—82no—
169C4⋊Dic3—22no—
170C4⋊Dic3—22no—
171C4⋊Dic3—22no—
172C4⋊Dic3—44no—
173C4⋊Dic3C22×C414yes—
174Dic3⋊C4—28no—
175Dic3⋊C4—24no—
176Dic3⋊C4—82no—
177C4×Dic3—44no—
178C4×Dic3—82no—
179C4×Dic3C2×D412yes—
180C4×Dic3#4112yes—
181C2×C3⋊C8—21no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—21no—
184C2×C3⋊C8—21no—
185C2×C3⋊C8C22×C412yes—
186C2×C3⋊C8C2×D412yes—
187ΓL(1,𝔽25)—28no—
188D3×C8—28no—
189C22×Q8C22×D312yes—
190C23×C4—31no—
191C2×Q16—22no—
192C2×Q16—22no—
193C2×Q16C4×D314yes—
194C2×M16—62no—
195C2×M16—62no—
196C2×M16—62no—
197C22×C8—31no—
198C22×C8—31no—
199C22×C8—34no—
200C22×C8—62no—
201C22×C8—62no—
202C22×C8C22×D311yescharacteristic
203C22×C8C22×D311yescharacteristic
204C22⋊Q8—38no—
205C22⋊Q8—64no—
206C4×Q8—64no—
207C4×Q8—64no—
208#117—62no—
209#117—62no—
210C2×C4⋊C4—32no—
211C2×C4⋊C4—32no—
212C2×C4⋊C4—32no—
213C2×C4⋊C4—62no—
214C2×C4⋊C4—62no—
215C2×C4⋊C4—62no—
216C2×C4⋊C4C22×D312yes—
217C2×C22⋊C4—32no—
218C2×C22⋊C4—62no—
219C2×C22⋊C4—62no—
220C2×C42—31no—
221C2×C42—62no—
222C2×C42—62no—
223C2×C42C22×D311yescharacteristic
224#107—34no—
225#106—22no—
226#106—62no—
227#103—24no—
228#103—38no—
229#103—64no—
230#97—22no—
231#97—34no—
232C4×C8—62no—
233C4×C8—62no—
234#95—22no—
235#95—22no—
236#95—62no—
237#95—62no—
238#95—62no—
239C23×C3C22×D411yescharacteristic
240C22×D3—21no—
241C22×D3—21no—
242C22×D3—22no—
243C22×D3—22no—
244C22×D3C4×D412yes—
245C3×Q8—24no—
246C3×Q8—24no—
247C3×Q8—42no—
248C3×Q8—42no—
249C2×C3×C4—24no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4C23×C411yescharacteristic
259C2×C3×C4C22×D411yescharacteristic
260C2×C3×C4C2×#4111yescharacteristic
261C2×C3×C4C2×#4111yescharacteristic
262C2×C3×C4C2×#4111yescharacteristic
263C2×C3×C4C2×#4111yescharacteristic
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—22no—
267C2×Dic3—22no—
268C2×Dic3—24no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3C4×D412yes—
275C2×Dic3#12114yes—
276C4×D3—22no—
277C4×D3—22no—
278C4×D3—22no—
279C4×D3—22no—
280C3×C8—22no—
281C3×C8C4×D414yes—
282C3⋊C8—22no—
283C3⋊C8—22no—
284C24—31no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8C2×C4×D314yes—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—31no—
294C22×C4—32no—
295C22×C4—32no—
296C22×C4—32no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4—62no—
302C22×C4—62no—
303C22×C4D3×D412yes—
304C22×C4C23×D311yescharacteristic
305Q16—28no—
306M16—68no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—32no—
310C2×C8—32no—
311C2×C8—61no—
312C2×C8—61no—
313C2×C8—61no—
314C2×C8—61no—
315C2×C8—68no—
316C2×C8C2×C4×D312yes—
317C2×C8D3×D412yes—
318C2×C8#23712yes—
319C2×C8#23712yes—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—34no—
323C4⋊C4—64no—
324C4⋊C4—68no—
325C4⋊C4—62no—
326C4⋊C4—62no—
327C4⋊C4—62no—
328C4⋊C4—124no—
329C4⋊C4—242no—
330C22⋊C4—64no—
331C22⋊C4—68no—
332C22⋊C4—242no—
333C42—21no—
334C42—21no—
335C42—32no—
336C42—32no—
337C42—124no—
338C42—242no—
339C22×C3C2×C4×D411yescharacteristic
340C22×C3C2×#12111yescharacteristic
341C22×C3#53211yescharacteristic
342C22×C3#57712yes—
343C22×C3#57812yes—
344C2×D3—22no—
345C2×D3—22no—
346C2×D3—22no—
347C2×D3—42no—
348C2×D3—42no—
349C3×C4—42no—
350C3×C4—44no—
351C3×C4—82no—
352C3×C4C2×C4×D411yescharacteristic
353C3×C4C4×#4111yescharacteristic
354C3×C4#52511yescharacteristic
355C3×C4#52911yescharacteristic
356Dic3—24no—
357Dic3—22no—
358Dic3—44no—
359Dic3—82no—
360C23—32no—
361C23—61no—
362C23—62no—
363C23—62no—
364C23—61no—
365C23C2×D3×D411yescharacteristic
366Q8—24no—
367Q8—24no—
368Q8—42no—
369Q8—42no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—32no—
375C2×C4—34no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—61no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—61no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—64no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4C22×C4×D311yescharacteristic
396C2×C4C2×D3×D411yescharacteristic
397C2×C4C2×#23711yescharacteristic
398C2×C4C2×#23711yescharacteristic
399C2×C4D3×#4111yescharacteristic
400C2×C4D3×#4111yescharacteristic
401C8—22no—
402C8—62no—
403C8—62no—
404C8#67⋊C414yes—
405C2×C3C4×#12111yescharacteristic
406C2×C3#295011yescharacteristic
407C2×C3#295211yescharacteristic
408C2×C3#306312yes—
409C2×C3#317812yes—
410D3—42no—
411C22—62no—
412C22—62no—
413C22—62no—
414C22—122no—
415C22—122no—
416C22C4×D3×D411yescharacteristic
417C22D3×#12111yescharacteristic
418C22#571611yescharacteristic
419C22#577812yes—
420C22#577912yes—
421C4—44no—
422C4—42no—
423C4—62no—
424C4—64no—
425C4—82no—
426C4—124no—
427C4—242no—
428C4C2×#67⋊C411yescharacteristic
429C4C4×#23711yescharacteristic
430C4#560611yescharacteristic
431C4#568011yescharacteristic
432C3#2153011yescharacteristic, complemented, Sylow
433C2—122no—
434C2#7629111yescharacteristic
435C2#7983111yescharacteristic
436C2#7983211yescharacteristic
437C2#8090712yes—
438C2#8186212yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1971C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
cosocleC22×C3×C4C24
hypercenter#1971D3
hypocenterC3#21530
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1971

Complements

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

NH ≅ G/Ntype
2. C2×#5184C2semidirect
5. C3×#1971C2semidirect
7. #70553C2semidirect
9. Dic3×#103C2semidirect
10. #69819C2semidirect
11. #69674C2semidirect
25. #5188C4semidirect
47. #1971D3semidirect
432. C3#21530semidirect

Maximal subgroups

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

subgroupindex
C2×#51882
C2×#51842
C2×#47082
#766112
C3×#19712
#705772
#705532
#702712
Dic3×#1032
#698192
#696742
#215303