Finite Group Databasebeta

← #10824627

Subgroups — #10824627

Order 768.

894 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×Q16C211yesmaximal, characteristic, complemented
2C2×#4833C211yesmaximal, characteristic
3#76479C212yesmaximal
4C3×#2056C211yesmaximal, characteristic, complemented
5#72071C211yesmaximal, characteristic
6#70462C211yesmaximal, characteristic, complemented
7#70274C212yesmaximal
8Dic3×#103C212yesmaximal, complemented
9#69682C212yesmaximal, complemented
10#68830C211yesmaximal, characteristic
11#68784C211yesmaximal, characteristic, complemented
12#23699—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×#946C2212yes—
15C2×Q8×Dic3C2212yes—
16C2×C42×D3C2211yescharacteristic
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×#103C2212yes—
19C3×#391C2212yes—
20C2×#835C2212yes—
21C2×#835C2212yes—
22Dic3×Q16—24no—
23C2×(C2×D3)⋊C8C2211yescharacteristic
24C2×(C2×D3)⋊C8C2211yescharacteristic
25C2×#818C2211yescharacteristic
26C2×C8×Dic3C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28#4987C2212yes—
29Dic3×C4⋊C4C2212yes—
30C2×#808C2212yes—
31#4928C2212yes—
32C2×C4⋊(C3⋊C8)C2211yescharacteristic
33#4833—24no—
34#4668—22no—
35#4666—22no—
36#4655—22no—
37#4649—22no—
38C3×#340C2211yescharacteristic
39#4551C2212yes—
40#4551C2211yescharacteristic
41#4529C2212yes—
42C2×C4×Q16—31no—
43C2×#431—31no—
44C4×C22⋊Q8—32no—
45#2086—31no—
46#2075—31no—
47#2066—32no—
48#2056D311yescharacteristic, complemented
49#2035—32no—
50#2001—31no—
51C4×#106—31no—
52C4×#103—32no—
53C22×C3×Q8C2312yes—
54C22×C4×D3D411yescharacteristic
55C22×C4×D3D411yescharacteristic
56C22×C4×D3C2311yescharacteristic
57C2×C3×Q16—24no—
58C2×C3×Q16—22no—
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C42C2311yescharacteristic
63#946—24no—
64#946—24no—
65Q8×Dic3—24no—
66Q8×Dic3—24no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—22no—
69C2×C4⋊Dic3—22no—
70C2×C4⋊Dic3C2312yes—
71C2×C4⋊Dic3C2312yes—
72C2×Dic3⋊C4—22no—
73C2×Dic3⋊C4—22no—
74C2×C4×Dic3—22no—
75C2×C4×Dic3—22no—
76C2×C4×Dic3D411yescharacteristic
77C2×C4×Dic3D411yescharacteristic
78C2×C4×Dic3C2311yescharacteristic
79C22×C3⋊C8C2311yescharacteristic
80C22×C3⋊C8C2311yescharacteristic
81C42×D3—22no—
82C42×D3—22no—
83C3×C4⋊C8—22no—
84C3×#103—24no—
85C3×#95—22no—
86C3×#95—22no—
87#835—24no—
88#835—24no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91C3⋊#95—22no—
92(C2×D3)⋊C8—22no—
93(C2×D3)⋊C8—22no—
94#818—22no—
95C8×Dic3—22no—
96C8×Dic3D414yes—
97#808—24no—
98C4⋊(C3⋊C8)—22no—
99C22×Q16C2×D311yescharacteristic
100C2×C22⋊Q8—32no—
101C2×C4×Q8—32no—
102C22×C42—31no—
103C4×Q16—64no—
104#431—64no—
105C2×#107—31no—
106C2×#106—32no—
107C2×C4⋊C8—31no—
108C2×C4⋊C8C2×D311yescharacteristic
109C2×#103—32no—
110C2×#103—32no—
111C2×#103C2×D312yes—
112C2×C22⋊C8—31no—
113C2×C22⋊C8—31no—
114C2×C4×C8—31no—
115C2×C4×C8—31no—
116#391C2×D312yes—
117#389—32no—
118#387—62no—
119#385—62no—
120#380—32no—
121C4×C4⋊C4—32no—
122C4×C4⋊C4—62no—
123C4×C22⋊C4—62no—
124#340—31no—
125#340—32no—
126#340—32no—
127#340C2×D311yescharacteristic
128C23×D3C2×D411yescharacteristic
129C2×C3×Q8—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—22no—
133C2×C3×Q8—22no—
134C2×C3×Q8#4114yes—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2411yescharacteristic
140C22×Dic3—22no—
141C22×Dic3—22no—
142C22×Dic3C2×D411yescharacteristic
143C22×Dic3C2×D411yescharacteristic
144C22×Dic3C2×D411yescharacteristic
145C2×C4×D3—22no—
146C2×C4×D3—22no—
147C2×C4×D3—22no—
148C2×C4×D3—21no—
149C2×C4×D3—21no—
150C2×C4×D3—22no—
151C2×C4×D3—22no—
152C2×C4×D3—22no—
153C2×C4×D3—24no—
154C3×Q16—28no—
155C2×C3×C8—22no—
156C2×C3×C8—21no—
157C2×C3×C8—22no—
158C2×C3×C8C2×D412yes—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C42—21no—
162C3×C42—21no—
163(C2×D3)⋊C4—28no—
164(C2×D3)⋊C4—44no—
165C4⋊Dic3—22no—
166C4⋊Dic3—22no—
167C4⋊Dic3—22no—
168C4⋊Dic3—44no—
169C4⋊Dic3#4114yes—
170Dic3⋊C4—28no—
171Dic3⋊C4—44no—
172C4×Dic3—22no—
173C4×Dic3—22no—
174C4×Dic3—44no—
175C4×Dic3—44no—
176C4×Dic3C2×D412yes—
177C4×Dic3#4112yes—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—21no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8C2×D412yes—
182C22×Q8C22×D312yes—
183C23×C4—31no—
184C23×C4—31no—
185C23×C4—31no—
186C2×Q16—24no—
187C2×Q16—22no—
188C22×C8—31no—
189C22×C8—31no—
190C22×C8C22×D311yescharacteristic
191C22×C8C22×D311yescharacteristic
192C22⋊Q8—64no—
193C22⋊Q8—64no—
194C4×Q8—64no—
195C4×Q8—64no—
196C2×C4⋊C4—32no—
197C2×C4⋊C4—32no—
198C2×C4⋊C4—62no—
199C2×C4⋊C4—62no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4C22×D312yes—
202C2×C22⋊C4—62no—
203C2×C22⋊C4—62no—
204C2×C42—31no—
205C2×C42—31no—
206C2×C42—31no—
207C2×C42—62no—
208C2×C42—62no—
209C2×C42—62no—
210C2×C42—62no—
211C2×C42C22×D311yescharacteristic
212#107—62no—
213#106—64no—
214C4⋊C8—22no—
215C4⋊C8—62no—
216#103—24no—
217#103—64no—
218#103—64no—
219C22⋊C8—62no—
220C22⋊C8—62no—
221C4×C8—34no—
222C4×C8—62no—
223#95—22no—
224#95—22no—
225#95—62no—
226#95—62no—
227#95—62no—
228C23×C3C22×D411yescharacteristic
229C22×D3—21no—
230C22×D3—22no—
231C22×D3—21no—
232C22×D3—22no—
233C22×D3#13512yes—
234C3×Q8—24no—
235C3×Q8—24no—
236C3×Q8—42no—
237C3×Q8—42no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4#13512yes—
249C2×C3×C4C22×D411yescharacteristic
250C2×C3×C4C2×#4112yes—
251C2×C3×C4C2×#4111yescharacteristic
252C2×C3×C4C2×#4112yes—
253C2×Dic3—21no—
254C2×Dic3—21no—
255C2×Dic3—21no—
256C2×Dic3—22no—
257C2×Dic3—24no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3#12112yes—
264C2×Dic3#12112yes—
265C2×Dic3#13714yes—
266C4×D3—28no—
267C4×D3—41no—
268C4×D3—42no—
269C4×D3—41no—
270C4×D3—44no—
271C3×C8—22no—
272C3×C8—41no—
273C3⋊C8—22no—
274C3⋊C8—41no—
275C24—31no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8#23714yes—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—62no—
288C22×C4—62no—
289C22×C4—64no—
290C22×C4—62no—
291C22×C4—62no—
292C22×C4—62no—
293C22×C4—61no—
294C22×C4—62no—
295C22×C4—61no—
296C22×C4—62no—
297C22×C4—62no—
298C22×C4D3×D411yescharacteristic
299C22×C4D3×D411yescharacteristic
300C22×C4C23×D311yescharacteristic
301Q16—28no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—22no—
305C2×C8—32no—
306C2×C8—62no—
307C2×C8—61no—
308C2×C8—62no—
309C2×C8D3×D412yes—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—34no—
313C4⋊C4—68no—
314C4⋊C4—62no—
315C4⋊C4—62no—
316C4⋊C4—62no—
317C4⋊C4—124no—
318C4⋊C4—124no—
319C22⋊C4—68no—
320C22⋊C4—124no—
321C42—21no—
322C42—21no—
323C42—32no—
324C42—32no—
325C42—62no—
326C42—62no—
327C42—124no—
328C42—124no—
329C22×C3C2×#12111yescharacteristic
330C22×C3#52512yes—
331C22×C3C2×#13511yescharacteristic
332C22×C3C2×#13511yescharacteristic
333C22×C3C2×#13712yes—
334C2×D3—24no—
335C2×D3—22no—
336C2×D3—42no—
337C2×D3—41no—
338C2×D3—41no—
339C3×C4—22no—
340C3×C4—44no—
341C3×C4—41no—
342C3×C4—82no—
343C3×C4#52511yescharacteristic
344C3×C4#52912yes—
345C3×C4#53211yescharacteristic
346Dic3—22no—
347Dic3—44no—
348Dic3—41no—
349Dic3—82no—
350Dic3#47014yes—
351C23—32no—
352C23—61no—
353C23—62no—
354C23—61no—
355C23—62no—
356C23C2×D3×D411yescharacteristic
357Q8—24no—
358Q8—24no—
359Q8—42no—
360Q8—42no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—24no—
366C2×C4—34no—
367C2×C4—32no—
368C2×C4—32no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—68no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—64no—
379C2×C4—62no—
380C2×C4—124no—
381C2×C4—122no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4—121no—
385C2×C4—121no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4#91912yes—
390C2×C4C2×D3×D411yescharacteristic
391C2×C4C2×#23712yes—
392C2×C4D3×#4111yescharacteristic
393C2×C4D3×#4112yes—
394C8—22no—
395C8—41no—
396C8—62no—
397C8—121no—
398C2×C3#239211yescharacteristic
399C2×C3#305011yescharacteristic
400C2×C3C2×#47011yescharacteristic
401C2×C3#310712yes—
402C2×C3#310812yes—
403D3—42no—
404C22—62no—
405C22—64no—
406C22—121no—
407C22—122no—
408C22—121no—
409C22D3×#12111yescharacteristic
410C22#560612yes—
411C22C2×#91911yescharacteristic
412C22D3×#13511yescharacteristic
413C22D3×#13712yes—
414C4—22no—
415C4—34no—
416C4—41no—
417C4—44no—
418C4—62no—
419C4—82no—
420C4—124no—
421C4—121no—
422C4—242no—
423C4#558611yescharacteristic
424C4#564912yes—
425C4#566811yescharacteristic
426C3#2369911yescharacteristic, complemented, Sylow
427C2—122no—
428C2#7689111yescharacteristic
429C2#8045712yes—
430C2#8056912yes—
431C2#8075811yescharacteristic
432C2D3×#47011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2056C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#2056D3
hypocenterC3#23699
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2056

Complements

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

NH ≅ G/Ntype
1. C2×Dic3×Q16C2semidirect
4. C3×#2056C2semidirect
6. #70462C2semidirect
8. Dic3×#103C2semidirect
9. #69682C2semidirect
11. #68784C2semidirect
48. #2056D3semidirect
426. C3#23699semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×Q162
C2×#48332
#764792
C3×#20562
#720712
#704622
#702742
Dic3×#1032
#696822
#688302
#687842
#236993