Finite Group Databasebeta

← #10779113

Subgroups — #10779113

Order 768.

702 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#500C211yesmaximal, characteristic, complemented
2#80477C212yesmaximal
3C8×#239C211yesmaximal, characteristic
4C3×#1674C211yesmaximal, characteristic, complemented
5#71371C211yesmaximal, characteristic, complemented
6#71053C212yesmaximal, complemented
7#70944C211yesmaximal, characteristic, complemented
8#68578C211yesmaximal, characteristic, complemented
9#68569C211yesmaximal, characteristic, complemented
10#68443C212yesmaximal, complemented
11#68420C212yesmaximal, complemented
12#19271—31nomaximal, Sylow
13C2×#918—22no—
14D3×C4⋊Q8C2212yes—
15C4×#239C2211yescharacteristic
16C3×#500C2211yescharacteristic
17C3×#462C2212yes—
18C3×C8×Q8C2211yescharacteristic
19#5182—22no—
20#5083C2212yes—
21#5078C2212yes—
22#5031C2212yes—
23Q8×C3⋊C8C2211yescharacteristic
24D3×#107—21no—
25D3×#107—21no—
26#4874—21no—
27#4874—21no—
28#4841C2212yes—
29#4837C2212yes—
30#4834—21no—
31#4834C2211yescharacteristic
32#4833—21no—
33#4833C2211yescharacteristic
34#4818—22no—
35#4802—22no—
36C8×D12—21no—
37C8×D12C2211yescharacteristic
38C4×D3×C8—21no—
39C4×D3×C8C2211yescharacteristic
40#4682C2211yescharacteristic
41C3×#335C2211yescharacteristic
42C3×#332C2212yes—
43#4547C2212yes—
44#4486C2212yes—
45#4482C2212yes—
46#4479C2212yes—
47#4458C2211yescharacteristic
48C2×#500—31no—
49#3039—32no—
50C8×#41—31no—
51#1674—32no—
52#1674—31no—
53#1674D311yescharacteristic, complemented
54#1673—31no—
55#1673—31no—
56#1673—32no—
57#1567—31no—
58#1546—32no—
59C2×#239D411yescharacteristic
60C2×D3×Q8—22no—
61C2×C3×Q16—22no—
62C3×C4⋊Q8C2312yes—
63C3×C4×Q8C2311yescharacteristic
64Q8×Dic3D411yescharacteristic
65#944—22no—
66C2×#216—22no—
67C2×#215—22no—
68#918—48no—
69C2×#204—22no—
70C2×ΓL(1,𝔽25)—22no—
71C2×D3×C8—21no—
72C2×D3×C8—21no—
73C2×D3×C8D411yescharacteristic
74C2×D3×C8D411yescharacteristic
75#893—21no—
76#893D411yescharacteristic
77#892—21no—
78#892D411yescharacteristic
79D3×C4⋊C4—22no—
80D3×C4⋊C4—22no—
81#888—22no—
82#888—22no—
83C4×D12—21no—
84C4×D12C2311yescharacteristic
85C42×D3—21no—
86C42×D3C2311yescharacteristic
87#869C2312yes—
88C3×#107—21no—
89C3×#107—21no—
90C3×C4⋊C8—21no—
91C3×C4⋊C8C2312yes—
92C3×C4⋊C8C2311yescharacteristic
93C3×#103—22no—
94C3×C4×C8—21no—
95C3×C4×C8C2311yescharacteristic
96#835—22no—
97(C2×D3)⋊C8—21no—
98(C2×D3)⋊C8—21no—
99(C2×D3)⋊C8—21no—
100#818—21no—
101#818—21no—
102#816—22no—
103Dic3⋊C8—22no—
104C8×Dic3—21no—
105C8×Dic3D411yescharacteristic
106C8×Dic3D411yescharacteristic
107#810—22no—
108#807—22no—
109#807—22no—
110C4⋊(C3⋊C8)—21no—
111C4⋊(C3⋊C8)C2311yescharacteristic
112C4⋊(C3⋊C8)C2312yes—
113C4×C3⋊C8—21no—
114C4×C3⋊C8C2311yescharacteristic
115C2×#137—62no—
116C2×C4⋊Q8—32no—
117C4×#41—31no—
118#500—32no—
119#500—31no—
120#500—61no—
121#500—61no—
122#500C2×D311yescharacteristic
123#462—32no—
124#462—62no—
125#462C2×D312yes—
126#461—32no—
127#461—32no—
128#461—62no—
129C8×Q8—31no—
130C8×Q8C2×D311yescharacteristic
131C8×D4—31no—
132C8×D4—31no—
133C8×D4—61no—
134C8×D4—61no—
135#431—31no—
136#431—61no—
137C2×#107—61no—
138C2×#107—61no—
139#423—32no—
140#419—62no—
141C2×C4×C8—31no—
142C2×C4×C8—61no—
143#335—32no—
144#335—31no—
145#335C2×D311yescharacteristic
146#332—32no—
147#332—32no—
148#332—32no—
149#332C2×D312yes—
150C2×C3×Q8—22no—
151C2×C3×Q8C2×D411yescharacteristic
152#239—22no—
153#239D814yes—
154D3×Q8—44no—
155D3×Q8—44no—
156C2×D12—21no—
157C2×D12C2×D411yescharacteristic
158C2×C4×D3—21no—
159C2×C4×D3—21no—
160C2×C4×D3—21no—
161C2×C4×D3—42no—
162C2×C4×D3C2×D411yescharacteristic
163C2×C4×D3C2×D411yescharacteristic
164C2×C4×D3#4111yescharacteristic
165C2×Dic6—22no—
166C2×Dic6—42no—
167C3×Q16—44no—
168C2×C3×C8—21no—
169C2×C3×C8—22no—
170C2×C3×C8C2×D411yescharacteristic
171C2×C3×C8C2×D411yescharacteristic
172C3×C4⋊C4—21no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4C2×D411yescharacteristic
176C3×C42—21no—
177C3×C42C2411yescharacteristic
178#216—44no—
179#215—44no—
180(C2×D3)⋊C4—21no—
181(C2×D3)⋊C4—21no—
182(C2×D3)⋊C4—21no—
183C4⋊Dic3—21no—
184C4⋊Dic3—22no—
185C4⋊Dic3—22no—
186C4⋊Dic3C2×D411yescharacteristic
187Dic3⋊C4—42no—
188Dic3⋊C4—42no—
189C4×Dic3—21no—
190C4×Dic3—21no—
191C4×Dic3C2×D411yescharacteristic
192C4×Dic3C2×D411yescharacteristic
193C4×Dic3#4111yescharacteristic
194C2×C3⋊C8—21no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8C2×D411yescharacteristic
197C2×C3⋊C8C2×D411yescharacteristic
198#204—44no—
199ΓL(1,𝔽25)—44no—
200D3×C8—22no—
201D3×C8—22no—
202D3×C8—42no—
203D3×C8—42no—
204C2×#41—31no—
205C22×Q8—62no—
206#137—128no—
207C2×Q16—22no—
208C2×Q16—62no—
209C2×SD16—62no—
210C2×SD16—62no—
211C2×M16—62no—
212C22×C8—31no—
213C22×C8—31no—
214C22×C8—61no—
215C22×C8—61no—
216C4⋊Q8—32no—
217C4⋊Q8—62no—
218C4⋊Q8—62no—
219C4⋊Q8—62no—
220C4⋊Q8C22×D312yes—
221C4×Q8—31no—
222C4×Q8C22×D311yescharacteristic
223C4×D4—31no—
224C4×D4—31no—
225C4×D4—61no—
226C4×D4—61no—
227#117—31no—
228#117—61no—
229C2×C4⋊C4—62no—
230C2×C4⋊C4—62no—
231C2×C42—31no—
232C2×C42—61no—
233#107—21no—
234#107—21no—
235#107—62no—
236#107—62no—
237#107—61no—
238#107—61no—
239C4⋊C8—21no—
240C4⋊C8—31no—
241C4⋊C8—32no—
242C4⋊C8—61no—
243C4⋊C8—62no—
244C4⋊C8C22×D311yescharacteristic
245C4⋊C8C22×D312yes—
246#103—22no—
247#103—62no—
248#103—62no—
249#103—62no—
250C22⋊C8—61no—
251C22⋊C8—61no—
252C22⋊C8—61no—
253C4×C8—21no—
254C4×C8—31no—
255C4×C8—31no—
256C4×C8—31no—
257C4×C8—61no—
258C4×C8—61no—
259C4×C8C22×D311yescharacteristic
260C22×D3—21no—
261C22×D3#12111yescharacteristic
262C3×Q8—21no—
263C3×Q8—42no—
264C3×Q8—42no—
265C3×Q8C2×D812yes—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—42no—
269C2×C3×C4C22×D411yescharacteristic
270C2×C3×C4C22×D411yescharacteristic
271C2×C3×C4C2×#4111yescharacteristic
272C2×Dic3—21no—
273C2×Dic3—42no—
274C2×Dic3#12111yescharacteristic
275C2×Dic3#12111yescharacteristic
276C2×Dic3#12111yescharacteristic
277D12—24no—
278D12—21no—
279D12C2×D812yes—
280C4×D3—21no—
281C4×D3—21no—
282C4×D3—24no—
283C4×D3—22no—
284C4×D3—41no—
285C4×D3—41no—
286C4×D3—42no—
287C4×D3—84no—
288C4×D3C2×D812yes—
289C4×D3#13512yes—
290Dic6—42no—
291Dic6—42no—
292Dic6—84no—
293C3×C8—21no—
294C3×C8—42no—
295C3×C8—41no—
296C3×C8#13712yescomplemented
297C3⋊C8—21no—
298C3⋊C8—42no—
299C3⋊C8—41no—
300C3⋊C8#13712yescomplemented
301#41—34no—
302#41—62no—
303C2×Q8—22no—
304C2×Q8—62no—
305C2×Q8—122no—
306C2×Q8—124no—
307C2×Q8—124no—
308C2×Q8D3×D411yescharacteristic
309C2×D4—31no—
310C2×D4—61no—
311C22×C4—31no—
312C22×C4—31no—
313C22×C4—31no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—122no—
318Q16—44no—
319Q16—124no—
320SD16—124no—
321SD16—124no—
322M16—124no—
323C2×C8—21no—
324C2×C8—22no—
325C2×C8—31no—
326C2×C8—31no—
327C2×C8—62no—
328C2×C8—62no—
329C2×C8—61no—
330C2×C8—62no—
331C2×C8—122no—
332C2×C8—122no—
333C2×C8D3×D411yescharacteristic
334C2×C8D3×D411yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—31no—
339C4⋊C4—61no—
340C4⋊C4—62no—
341C4⋊C4—62no—
342C4⋊C4—122no—
343C4⋊C4—122no—
344C4⋊C4D3×D411yescharacteristic
345C22⋊C4—61no—
346C22⋊C4—61no—
347C22⋊C4—61no—
348C42—21no—
349C42—31no—
350C42—31no—
351C42—31no—
352C42—61no—
353C42—61no—
354C42C23×D311yescharacteristic
355C22×C3C2×#12111yescharacteristic
356C2×D3—21no—
357C2×D3—21no—
358C2×D3—22no—
359C2×D3—41no—
360C2×D3—41no—
361C2×D3#46612yes—
362C3×C4—22no—
363C3×C4—21no—
364C3×C4—41no—
365C3×C4—82no—
366C3×C4C22×D811yescharacteristic
367C3×C4C2×#13511yescharacteristic
368C3×C4C2×#13711yescharacteristic
369C3×C4C2×#13711yescharacteristic
370Dic3—21no—
371Dic3—41no—
372Dic3—82no—
373Dic3#46612yes—
374Dic3#47312yes—
375C23—31no—
376C23—61no—
377Q8—21no—
378Q8—42no—
379Q8—42no—
380Q8—122no—
381Q8—122no—
382Q8—244no—
383Q8D3×D812yes—
384D4—32no—
385D4—64no—
386D4—61no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—31no—
390C2×C4—31no—
391C2×C4—32no—
392C2×C4—32no—
393C2×C4—31no—
394C2×C4—42no—
395C2×C4—62no—
396C2×C4—64no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—122no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—122no—
404C2×C4—244no—
405C2×C4C2×D3×D411yescharacteristic
406C2×C4C2×D3×D411yescharacteristic
407C2×C4D3×#4111yescharacteristic
408C8—21no—
409C8—32no—
410C8—42no—
411C8—41no—
412C8—61no—
413C8—122no—
414C8—121no—
415C8#91812yescomplemented
416C2×C3#303911yescharacteristic
417C2×C3C2×#46611yescharacteristic
418C2×C3C2×#47311yescharacteristic
419D3—22no—
420D3—42no—
421C22—32no—
422C22—61no—
423C22—61no—
424C22—62no—
425C22—121no—
426C22—121no—
427C22D3×#12111yescharacteristic
428C4—21no—
429C4—22no—
430C4—32no—
431C4—32no—
432C4—41no—
433C4—61no—
434C4—82no—
435C4—121no—
436C4—242no—
437C4C2×D3×D811yescharacteristic
438C4C2×#91811yescharacteristic
439C4D3×#13511yescharacteristic
440C4D3×#13711yescharacteristic
441C3#1927111yescharacteristic, complemented, Sylow
442C2—62no—
443C2—122no—
444C2#8047711yescharacteristic
445C2D3×#46611yescharacteristic
446C2D3×#47311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1674C2
socleC22×C3C2×#121
prime radicalC2×#239D4
hypercenter#1674D3
hypocenterC3#19271
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1674

Complements

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

NH ≅ G/Ntype
1. D3×#500C2semidirect
4. C3×#1674C2semidirect
5. #71371C2semidirect
6. #71053C2semidirect
7. #70944C2semidirect
8. #68578C2semidirect
9. #68569C2semidirect
10. #68443C2semidirect
11. #68420C2semidirect
53. #1674D3semidirect
296. C3×C8#137semidirect
300. C3⋊C8#137semidirect
415. C8#918semidirect
441. C3#19271semidirect

Maximal subgroups

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

subgroupindex
D3×#5002
#804772
C8×#2392
C3×#16742
#713712
#710532
#709442
#685782
#685692
#684432
#684202
#192713