Finite Group Databasebeta

← #10801792

Subgroups — #10801792

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×#4887C211yesmaximal, characteristic
2C2×#4716C211yesmaximal, characteristic, complemented
3C2×#4708C211yesmaximal, characteristic
4#73762C212yesmaximal
5C3×#1776C211yesmaximal, characteristic, complemented
6#70613C211yesmaximal, characteristic, complemented
7#70577C211yesmaximal, characteristic
8#70575C211yesmaximal, characteristic, complemented
9C4×#810C212yesmaximal, complemented
10#68884C212yesmaximal
11#68851C212yesmaximal, complemented
12#21530—31nomaximal, Sylow
13C22×Dic12C2211yescharacteristic
14C22×ΓL(1,𝔽25)C2211yescharacteristic
15C22×D3×C8C2211yescharacteristic
16C2×#897C2212yes—
17C2×#872C2211yescharacteristic
18C2×C4×Dic6C2212yes—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×#97C2211yescharacteristic
22C3×C4×C4⋊C4C2212yes—
23C2×#817C2211yescharacteristic
24C2×#816C2212yes—
25C2×#815C2211yescharacteristic
26C2×C8×Dic3C2211yescharacteristic
27#4987C2212yes—
28#4984C2212yes—
29#4973C2212yes—
30C2×#810C2212yes—
31C2×#810C2212yes—
32#4940—22no—
33#4928—22no—
34C2×C4×C3⋊C8C2211yescharacteristic
35#4887C418yescomplemented
36#4716—24no—
37#4708—24no—
38#4670—22no—
39#4653—22no—
40C3×#340C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×#467—31no—
44C2×#441—31no—
45C2×#405—31no—
46#2323—32no—
47#1971—31no—
48#1943—31no—
49#1933—32no—
50#1894—32no—
51#1807—31no—
52#1776D311yescharacteristic, complemented
53C4×#103—32no—
54C22×C4×D3C2311yescharacteristic
55C22×Dic6C2312yes—
56C22×C3×C8C2311yescharacteristic
57C22×C3×C8C2311yescharacteristic
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C42—22no—
61C2×C3×C42C2311yescharacteristic
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4D412yes—
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3—22no—
67C2×C4⋊Dic3C2312yes—
68C2×Dic3⋊C4—22no—
69C2×Dic3⋊C4D412yes—
70C2×C4×Dic3—22no—
71C2×C4×Dic3C2311yescharacteristic
72C22×C3⋊C8C2311yescharacteristic
73C22×C3⋊C8C2311yescharacteristic
74C2×Dic12—22no—
75C2×Dic12—22no—
76C2×Dic12C2×C414yes—
77C2×ΓL(1,𝔽25)—22no—
78C2×ΓL(1,𝔽25)—22no—
79C2×ΓL(1,𝔽25)—22no—
80C2×D3×C8—22no—
81C2×D3×C8—22no—
82C2×D3×C8C2×C414yes—
83#897—24no—
84#897C2×C418yes—
85#872—22no—
86#872—22no—
87C4×Dic6—24no—
88C4×Dic6—24no—
89C3×#107C2×C414yes—
90C3×#106—22no—
91C3×#97D414yes—
92C3×#95—22no—
93C3⋊#95—22no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97#817—22no—
98#816—24no—
99#815—22no—
100C8×Dic3—22no—
101#810—24no—
102#810C2×C418yes—
103C4×C3⋊C8—22no—
104C22×Q16—31no—
105C22×M16—31no—
106C23×C8—31no—
107C2×C22⋊Q8—32no—
108C2×C4×Q8—32no—
109C2×#117—31no—
110#467—38no—
111#441—64no—
112C2×#107C2×D311yescharacteristic
113C2×#106—31no—
114C2×#106C2×D311yescharacteristic
115C2×#103—32no—
116C2×#103—32no—
117C2×#103—32no—
118#405—64no—
119C2×#97—31no—
120C2×#97C2×D311yescharacteristic
121C2×C4×C8—31no—
122C2×C4×C8—31no—
123#391—32no—
124#389—32no—
125#389—62no—
126#388—62no—
127#388—62no—
128#387—62no—
129#380—32no—
130C4×C4⋊C4C2×D312yes—
131#340—32no—
132#340—32no—
133#340C2×D312yes—
134C23×D3C2×D411yescharacteristic
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4C2×D412yes—
138C22×C3×C4C2411yescharacteristic
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3C2×D412yes—
142C22×Dic3C2×D411yescharacteristic
143C2×C4×D3—21no—
144C2×C4×D3—21no—
145C2×C4×D3—21no—
146C2×C4×D3—21no—
147C2×C4×D3C22×C412yes—
148C2×C4×D3#4112yes—
149C2×Dic6—22no—
150C2×Dic6—22no—
151C2×Dic6—22no—
152C2×Dic6—22no—
153C2×Dic6—22no—
154C2×Dic6C22×C414yes—
155C2×C3×C8—21no—
156C2×C3×C8—21no—
157C2×C3×C8C22×C412yes—
158C2×C3×C8C2×D412yes—
159C2×C3×C8C2×D412yes—
160C2×C3×C8#4112yes—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4C22×C414yes—
165C3×C42—82no—
166C3×C42C2×D412yes—
167C3×C42#4112yes—
168(C2×D3)⋊C4—28no—
169(C2×D3)⋊C4—24no—
170(C2×D3)⋊C4—82no—
171C4⋊Dic3—28no—
172C4⋊Dic3—22no—
173C4⋊Dic3—22no—
174C4⋊Dic3—82no—
175Dic3⋊C4—24no—
176Dic3⋊C4—44no—
177C4×Dic3—21no—
178C4×Dic3—21no—
179C4×Dic3—44no—
180C2×C3⋊C8—21no—
181C2×C3⋊C8—21no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—21no—
184C2×C3⋊C8C22×C412yes—
185C2×C3⋊C8#4112yes—
186Dic12—28no—
187ΓL(1,𝔽25)—28no—
188D3×C8—28no—
189C22×Q8—32no—
190C23×C4—31no—
191C2×Q16—34no—
192C2×Q16—62no—
193C2×Q16—62no—
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—62no—
213C2×C4⋊C4—62no—
214C2×C4⋊C4—62no—
215C2×C4⋊C4C22×D312yes—
216C2×C4⋊C4C22×D312yes—
217C2×C22⋊C4—32no—
218C2×C22⋊C4—62no—
219C2×C22⋊C4—62no—
220C2×C42—22no—
221C2×C42—31no—
222C2×C42—62no—
223C2×C42C22×D311yescharacteristic
224#107C4×D314yes—
225#106—22no—
226#106—62no—
227#103—38no—
228#103—64no—
229#103—64no—
230#97—62no—
231#97D1214yes—
232C4×C8—62no—
233C4×C8—62no—
234#95—22no—
235#95—62no—
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—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—24no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4#12114yes—
254C2×C3×C4C23×C411yescharacteristic
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C2×#4111yescharacteristic
257C2×C3×C4C2×#4111yescharacteristic
258C2×C3×C4C2×#4111yescharacteristic
259C2×C3×C4C2×#4111yescharacteristic
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—24no—
263C2×Dic3—22no—
264C2×Dic3—22no—
265C2×Dic3—22no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3C4×D412yes—
272C4×D3—22no—
273C4×D3—22no—
274C4×D3—22no—
275C4×D3—22no—
276Dic6—24no—
277Dic6—24no—
278Dic6—42no—
279Dic6—42no—
280C3×C8—22no—
281C3×C8C4×D414yes—
282C3⋊C8—22no—
283C3⋊C8—22no—
284C24—31no—
285C2×Q8—34no—
286C2×Q8—62no—
287C2×Q8—62no—
288C2×Q8—62no—
289C2×Q8—62no—
290C2×Q8—62no—
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—68no—
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×C8C2×D1212yes—
318C2×C8C2×D1212yes—
319C2×C8#23512yes—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—68no—
324C4⋊C4—64no—
325C4⋊C4—62no—
326C4⋊C4—62no—
327C4⋊C4—124no—
328C4⋊C4—242no—
329C4⋊C4C2×C4×D314yes—
330C22⋊C4—68no—
331C22⋊C4—64no—
332C22⋊C4—242no—
333C42—61no—
334C42—61no—
335C42—82no—
336C42—124no—
337C42C2×D1212yes—
338C42#23912yes—
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—24no—
350C3×C4—44no—
351C3×C4—82no—
352C3×C4C2×C4×D411yescharacteristic
353C3×C4C4×#4111yescharacteristic
354C3×C4#52511yescharacteristic
355C3×C4#52911yescharacteristic
356Dic3—22no—
357Dic3—42no—
358Dic3—44no—
359Dic3—82no—
360C23—32no—
361C23—61no—
362C23—62no—
363C23—62no—
364C23—61no—
365C23C2×D3×D411yescharacteristic
366Q8—64no—
367Q8—64no—
368Q8—122no—
369Q8—122no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—32no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—62no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—64no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4#89514yes—
395C2×C4C22×C4×D311yescharacteristic
396C2×C4C22×D1211yescharacteristic
397C2×C4C2×#23511yescharacteristic
398C2×C4C2×#23911yescharacteristic
399C2×C4D3×#4111yescharacteristic
400C2×C4D3×#4111yescharacteristic
401C8—22no—
402C8—62no—
403C8—62no—
404C8C4×D1214yes—
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—
416C22C2×#89511yescharacteristic
417C22C4×D3×D411yescharacteristic
418C22#554811yescharacteristic
419C22#577512yes—
420C22#577912yes—
421C4—24no—
422C4—44no—
423C4—62no—
424C4—82no—
425C4—124no—
426C4—122no—
427C4—242no—
428C4C2×C4×D1211yescharacteristic
429C4#552411yescharacteristic
430C4C4×#23911yescharacteristic
431C4#558011yescharacteristic
432C3#2153011yescharacteristic, complemented, Sylow
433C2—122no—
434C2C4×#89511yescharacteristic
435C2#7921412yes—
436C2#7976912yes—
437C2#7983111yescharacteristic
438C2#7986811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1776C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
cosocleC22×C3×C4C24
hypercenter#1776D3
hypocenterC3#21530
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
2. C2×#4716C2semidirect
5. C3×#1776C2semidirect
6. #70613C2semidirect
8. #70575C2semidirect
9. C4×#810C2semidirect
11. #68851C2semidirect
35. #4887C4semidirect
52. #1776D3semidirect
432. C3#21530semidirect

Maximal subgroups

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

subgroupindex
C2×#48872
C2×#47162
C2×#47082
#737622
C3×#17762
#706132
#705772
#705752
C4×#8102
#688842
#688512
#215303