Finite Group Databasebeta

← #10824623

Subgroups — #10824623

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
2Dic3×C22⋊Q8C212yesmaximal
3C2×#4718C211yesmaximal, characteristic
4C3×#2001C211yesmaximal, characteristic
5#72068C211yesmaximal, characteristic, complemented
6#71694C212yesmaximal
7#70261C212yesmaximal, complemented
8Dic3×#103C212yesmaximal, complemented
9#69938C211yesmaximal, characteristic
10#68838C211yesmaximal, characteristic, complemented
11#68784C211yesmaximal, characteristic, complemented
12#23699—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×C3×C22⋊Q8C2212yes—
15C2×Q8×Dic3C2212yes—
16C22×C4×Dic3C2211yescharacteristic
17C2×C3×#103C2212yes—
18C2×C3×C22⋊C8C2211yescharacteristic
19#5229C2212yes—
20C2×#835C2212yes—
21C2×#835C2212yes—
22#5212C2212yes—
23C2×#830C2211yescharacteristic
24Dic3×Q16—24no—
25C2×#818C2211yescharacteristic
26C2×Dic3⋊C8C2211yescharacteristic
27C2×Dic3⋊C8C2211yescharacteristic
28C2×C8×Dic3C2211yescharacteristic
29C2×C8×Dic3C2211yescharacteristic
30#4984C2212yes—
31#4980—22no—
32Dic3×C4⋊C4—22no—
33Dic3×C4⋊C4C2212yes—
34C2×#808C2212yes—
35#4950—22no—
36Dic3×C22⋊C4—22no—
37#4718—24no—
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#2056—31no—
49#2035—32no—
50#2001D311yescharacteristic
51C4×#106—31no—
52C4×#103—32no—
53C22×C3×Q8C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×Dic3D411yescharacteristic
56C23×Dic3D411yescharacteristic
57C2×C3×Q16—24no—
58C2×C3×Q16—22no—
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×C22⋊Q8—24no—
62C3×C22⋊Q8—24no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C22⋊C4—22no—
66C2×#217—22no—
67Q8×Dic3—24no—
68Q8×Dic3—24no—
69C2×C4⋊Dic3—22no—
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3C2312yes—
72C2×C4⋊Dic3C2312yes—
73C2×C4×Dic3—22no—
74C2×C4×Dic3—22no—
75C2×C4×Dic3—22no—
76C2×C4×Dic3—22no—
77C2×C4×Dic3D411yescharacteristic
78C2×C4×Dic3D411yescharacteristic
79C2×C4×Dic3C2311yescharacteristic
80C2×C4×Dic3C2311yescharacteristic
81C22×C3⋊C8C2311yescharacteristic
82C22×C3⋊C8C2311yescharacteristic
83C3×#103—24no—
84C3×C22⋊C8—22no—
85#835—24no—
86#835—24no—
87C3⋊#95—22no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91C3⋊#95—22no—
92#830—22no—
93#818—22no—
94Dic3⋊C8—22no—
95Dic3⋊C8—22no—
96C8×Dic3—22no—
97C8×Dic3D414yes—
98#808—24no—
99C22×Q16C2×D311yescharacteristic
100C2×C22⋊Q8C2×D312yes—
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⋊C8—31no—
109C2×#103—32no—
110C2×#103—32no—
111C2×#103C2×D312yes—
112C2×C22⋊C8—31no—
113C2×C22⋊C8C2×D311yescharacteristic
114C2×C4×C8—31no—
115C2×C4×C8—31no—
116#391—32no—
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
128C24×C3C2×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×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2411yescharacteristic
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3—24no—
148C22×Dic3—22no—
149C22×Dic3—22no—
150C22×Dic3C2×D411yescharacteristic
151C22×Dic3C2×D411yescharacteristic
152C22×Dic3C2×D411yescharacteristic
153C22×Dic3C2×D411yescharacteristic
154C3×Q16—28no—
155C2×C3×C8—22no—
156C2×C3×C8—21no—
157C2×C3×C8—22no—
158C2×C3×C8C2×D412yes—
159C3×C4⋊C4—28no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4#4114yes—
162C3×C22⋊C4—28no—
163#217—44no—
164C4⋊Dic3—22no—
165C4⋊Dic3—22no—
166C4⋊Dic3—22no—
167C4⋊Dic3—22no—
168C4⋊Dic3—44no—
169C4⋊Dic3—44no—
170C4×Dic3—21no—
171C4×Dic3—21no—
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×C4C22×D311yescharacteristic
186C2×Q16—24no—
187C2×Q16—22no—
188C22×C8—31no—
189C22×C8—31no—
190C22×C8C22×D311yescharacteristic
191C22×C8C22×D311yescharacteristic
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C4×Q8—64no—
195C4×Q8—64no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—32no—
198C2×C4⋊C4—32no—
199C2×C4⋊C4—62no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4C22×D312yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—62no—
204C2×C42—31no—
205C2×C42—31no—
206C2×C42—31no—
207C2×C42—31no—
208C2×C42—62no—
209C2×C42—62no—
210C2×C42—62no—
211C2×C42—62no—
212#107—62no—
213#106—64no—
214C4⋊C8—62no—
215C4⋊C8—62no—
216#103—24no—
217#103—64no—
218#103—64no—
219C22⋊C8—22no—
220C22⋊C8—62no—
221C4×C8—34no—
222C4×C8—62no—
223#95—62no—
224#95—62no—
225#95—62no—
226#95—62no—
227#95—62no—
228C23×C3—21no—
229C23×C3—22no—
230C23×C3—21no—
231C23×C3—22no—
232C23×C3#13512yes—
233C23×C3C22×D411yescharacteristic
234C3×Q8—24no—
235C3×Q8—24no—
236C3×Q8—42no—
237C3×Q8—42no—
238C2×C3×C4—21no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4C22×D411yescharacteristic
250C2×C3×C4C2×#4112yes—
251C2×C3×C4C2×#4112yes—
252C2×C3×C4C2×#4111yescharacteristic
253C2×Dic3—28no—
254C2×Dic3—21no—
255C2×Dic3—21no—
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—21no—
259C2×Dic3—22no—
260C2×Dic3—44no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3#12112yes—
268C2×Dic3#12112yes—
269C2×Dic3#13512yes—
270C2×Dic3#13714yes—
271C3×C8—22no—
272C3×C8—41no—
273C3⋊C8—22no—
274C3⋊C8—41no—
275C24D3×D411yescharacteristic
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—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—31no—
289C22×C4—31no—
290C22×C4—31no—
291C22×C4—31no—
292C22×C4—62no—
293C22×C4—62no—
294C22×C4—62no—
295C22×C4—64no—
296C22×C4—62no—
297C22×C4—62no—
298C22×C4—62no—
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—28no—
311C4⋊C4—22no—
312C4⋊C4—62no—
313C4⋊C4—62no—
314C4⋊C4—62no—
315C4⋊C4—62no—
316C4⋊C4—124no—
317C4⋊C4—124no—
318C4⋊C4#23714yes—
319C22⋊C4—28no—
320C22⋊C4—124no—
321C42—32no—
322C42—32no—
323C42—62no—
324C42—62no—
325C42—61no—
326C42—61no—
327C42—124no—
328C42—124no—
329C22×C3—24no—
330C22×C3—22no—
331C22×C3—42no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3C2×#12111yescharacteristic
335C22×C3#52512yes—
336C22×C3C2×#13511yescharacteristic
337C22×C3C2×#13511yescharacteristic
338C22×C3C2×#13712yes—
339C3×C4—44no—
340C3×C4—41no—
341C3×C4—44no—
342C3×C4#52511yescharacteristic
343C3×C4#52912yes—
344C3×C4#53211yescharacteristic
345Dic3—22no—
346Dic3—22no—
347Dic3—41no—
348Dic3—82no—
349Dic3—82no—
350Dic3#47014yes—
351C23—21no—
352C23—22no—
353C23—22no—
354C23—21no—
355C23#91912yes—
356C23C2×D3×D411yescharacteristic
357Q8—24no—
358Q8—24no—
359Q8—42no—
360Q8—42no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—34no—
366C2×C4—32no—
367C2×C4—32no—
368C2×C4—32no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—68no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—62no—
382C2×C4—61no—
383C2×C4—124no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4C2×D3×D411yescharacteristic
391C2×C4C2×#23712yes—
392C2×C4C2×#23712yes—
393C2×C4D3×#4111yescharacteristic
394C8—22no—
395C8—41no—
396C8—62no—
397C8—121no—
398C2×C3—42no—
399C2×C3#239211yescharacteristic
400C2×C3#305011yescharacteristic
401C2×C3C2×#47011yescharacteristic
402C2×C3#310712yes—
403C2×C3#310812yes—
404C22—24no—
405C22—22no—
406C22—41no—
407C22—42no—
408C22—41no—
409C22#558612yes—
410C22D3×#12111yescharacteristic
411C22C2×#91911yescharacteristic
412C22D3×#13511yescharacteristic
413C22D3×#13712yes—
414C4—34no—
415C4—44no—
416C4—41no—
417C4—44no—
418C4—62no—
419C4—62no—
420C4—121no—
421C4—242no—
422C4—242no—
423C4#558611yescharacteristic
424C4#560012yes—
425C4#571611yescharacteristic
426C3#2369911yescharacteristic, complemented, Sylow
427C2—42no—
428C2#7666611yescharacteristic
429C2#8017312yes—
430C2#8021312yes—
431C2#8075811yescharacteristic
432C2D3×#47011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2001C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2001D3
hypocenterC3#23699
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

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

NH ≅ G/Ntype
1. C2×Dic3×Q16C2semidirect
5. #72068C2semidirect
7. #70261C2semidirect
8. Dic3×#103C2semidirect
10. #68838C2semidirect
11. #68784C2semidirect
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
Dic3×C22⋊Q82
C2×#47182
C3×#20012
#720682
#716942
#702612
Dic3×#1032
#699382
#688382
#687842
#236993