Finite Group Databasebeta

← #10779129

Subgroups — #10779129

Order 768.

608 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81938C211yesmaximal, characteristic, complemented
2#80411C212yesmaximal
3C8×#237C211yesmaximal, characteristic
4C3×#1673C211yesmaximal, characteristic, complemented
5#71377C211yesmaximal, characteristic, complemented
6#71036C212yesmaximal, complemented
7#70853C211yesmaximal, characteristic, complemented
8#68589C211yesmaximal, characteristic, complemented
9#68570C211yesmaximal, characteristic, complemented
10#68466C212yesmaximal, complemented
11#68427C212yesmaximal, complemented
12#19273—31nomaximal, Sylow
13#5728C2212yes—
14C4×#237C2211yescharacteristic
15C3×#500C2211yescharacteristic
16C3×#461C2212yes—
17C3×C8×D4C2211yescharacteristic
18#5173—22no—
19#5164—22no—
20C8×#67—21no—
21C2×C8×Dic3—21no—
22#5090C2212yes—
23#5080C2212yes—
24D4×C3⋊C8C2211yescharacteristic
25#5008C2212yes—
26#4880—21no—
27#4880—21no—
28#4878—21no—
29#4878—21no—
30#4850C2212yes—
31#4835C2212yes—
32#4833C2211yescharacteristic
33#4830C2211yescharacteristic
34#4814—22no—
35#4789—22no—
36#67⋊C8—21no—
37#4718—21no—
38C4×D3×C8C2211yescharacteristic
39#4682C2211yescharacteristic
40C8×Dic6C2211yescharacteristic
41C3×#335C2211yescharacteristic
42C3×#332C2212yes—
43#4547C2212yes—
44#4481C2212yes—
45#4481C2212yes—
46#4480C2212yes—
47#4458C2211yescharacteristic
48#3160—31no—
49#3116—32no—
50C8×#41—31no—
51#1676—31no—
52#1675—31no—
53#1674—31no—
54#1673D311yescharacteristic, complemented
55#1672—32no—
56#1671—32no—
57#1568—31no—
58#1554—32no—
59C2×#237D411yescharacteristic
60C2×C3×SD16—22no—
61C22×C3×C8—21no—
62C3×C4⋊Q8C2312yes—
63C3×C4×D4C2311yescharacteristic
64#946—22no—
65Q8×Dic3—22no—
66C2×#216—22no—
67D4×Dic3D411yescharacteristic
68C4×#67—21no—
69C2×C4×Dic3—21no—
70C22×C3⋊C8—21no—
71C2×D3×C8D411yescharacteristic
72C2×D3×C8D411yescharacteristic
73#897—22no—
74#892—22no—
75#892D411yescharacteristic
76#890—22no—
77#887—22no—
78#887D411yescharacteristic
79#67⋊C4—21no—
80#877—21no—
81C42×D3C2311yescharacteristic
82#869C2312yes—
83C4×Dic6C2311yescharacteristic
84C3×#107—21no—
85C3×#107—21no—
86C3×C4⋊C8C2312yes—
87C3×C4⋊C8C2311yescharacteristic
88C3×#103—22no—
89C3×C22⋊C8—21no—
90C3×C4×C8C2311yescharacteristic
91#835—22no—
92#832—22no—
93#830—21no—
94(C2×D3)⋊C8—22no—
95(C2×D3)⋊C8—21no—
96#818—21no—
97#818—21no—
98#818—22no—
99#816—22no—
100#815—22no—
101Dic3⋊C8—21no—
102Dic3⋊C8—21no—
103C8×Dic3—21no—
104C8×Dic3—21no—
105C8×Dic3—21no—
106C8×Dic3D411yescharacteristic
107C8×Dic3D411yescharacteristic
108#808—22no—
109#808—22no—
110#808—22no—
111C4⋊(C3⋊C8)C2311yescharacteristic
112C4⋊(C3⋊C8)C2312yes—
113C4×C3⋊C8C2311yescharacteristic
114#533—32no—
115C4×#41—31no—
116#500—31no—
117#500C2×D311yescharacteristic
118#499—61no—
119#499—61no—
120#498—32no—
121#485—62no—
122#477—32no—
123#476—32no—
124#462—32no—
125#461C2×D312yes—
126#452—62no—
127C8×Q8—31no—
128C8×Q8—31no—
129#441—62no—
130#440—62no—
131C8×D4—31no—
132C8×D4—61no—
133C8×D4—61no—
134C8×D4C2×D311yescharacteristic
135#432—32no—
136#431—31no—
137#431—61no—
138#427—61no—
139#427—61no—
140C2×C4×C8—31no—
141C2×C4×C8—61no—
142#335—31no—
143#335C2×D311yescharacteristic
144#334—32no—
145#333—32no—
146#333—32no—
147#332—32no—
148#332C2×D312yes—
149C2×C3×Q8—22no—
150C2×C3×D4C2×D411yescharacteristic
151C22×C3×C4—21no—
152C2×#67—21no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155#237—22no—
156#237—22no—
157C2×C4×D3C2×D411yescharacteristic
158C2×C4×D3C2×D411yescharacteristic
159C2×C4×D3#4111yescharacteristic
160C2×Dic6—22no—
161C2×Dic6C2×D411yescharacteristic
162C3×SD16—44no—
163C2×C3×C8—21no—
164C2×C3×C8—22no—
165C2×C3×C8—42no—
166C2×C3×C8C2×D411yescharacteristic
167C2×C3×C8C2×D411yescharacteristic
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4C2×D411yescharacteristic
171C3×C22⋊C4—21no—
172C3×C42C2411yescharacteristic
173#217—21no—
174#216—44no—
175(C2×D3)⋊C4—21no—
176(C2×D3)⋊C4—42no—
177C4⋊Dic3—22no—
178C4⋊Dic3—22no—
179C4⋊Dic3—42no—
180C4⋊Dic3C2×D411yescharacteristic
181Dic3⋊C4—21no—
182Dic3⋊C4—21no—
183Dic3⋊C4—42no—
184C4×Dic3—21no—
185C4×Dic3—21no—
186C4×Dic3—21no—
187C4×Dic3—42no—
188C4×Dic3C2×D411yescharacteristic
189C4×Dic3C2×D411yescharacteristic
190C4×Dic3#4111yescharacteristic
191C2×C3⋊C8—22no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—42no—
194C2×C3⋊C8C2×D411yescharacteristic
195C2×C3⋊C8C2×D411yescharacteristic
196D3×C8—22no—
197D3×C8—22no—
198C2×#41—31no—
199C2×Q16—62no—
200C2×SD16—22no—
201C22×C8—21no—
202C22×C8—31no—
203C22×C8—31no—
204C22×C8—61no—
205C4⋊Q8—32no—
206C4⋊Q8C22×D312yes—
207#125—62no—
208C22⋊Q8—62no—
209C22⋊Q8—62no—
210C4×Q8—31no—
211C4×Q8—31no—
212C4×Q8—62no—
213C4×Q8—62no—
214C4×D4—31no—
215C4×D4—61no—
216C4×D4—61no—
217C4×D4C22×D311yescharacteristic
218#117—31no—
219#117—61no—
220#117—62no—
221C2×C42—31no—
222C2×C42—61no—
223#107—21no—
224#107—21no—
225#107—62no—
226#107—61no—
227#107—61no—
228#106—62no—
229#106—62no—
230#106—62no—
231C4⋊C8—31no—
232C4⋊C8—32no—
233C4⋊C8—61no—
234C4⋊C8—61no—
235C4⋊C8C22×D311yescharacteristic
236C4⋊C8C22×D312yes—
237#103—22no—
238#103—62no—
239#103—62no—
240#102—62no—
241C22⋊C8—21no—
242C22⋊C8—61no—
243C22⋊C8—62no—
244C22⋊C8—61no—
245#97—62no—
246C4×C8—31no—
247C4×C8—31no—
248C4×C8—31no—
249C4×C8—61no—
250C4×C8—61no—
251C4×C8—61no—
252C4×C8C22×D311yescharacteristic
253C23×C3—21no—
254C22×D3#12111yescharacteristic
255C3×Q8—42no—
256C3×Q8—42no—
257C3×D4—21no—
258C3×D4#13512yes—
259C2×C3×C4—21no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—42no—
263C2×C3×C4C22×D411yescharacteristic
264C2×C3×C4C22×D411yescharacteristic
265C2×C3×C4C2×#4111yescharacteristic
266#67—24no—
267C2×Dic3—24no—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3#12111yescharacteristic
273C2×Dic3#12111yescharacteristic
274C2×Dic3#12111yescharacteristic
275C4×D3—21no—
276C4×D3—21no—
277C4×D3—21no—
278C4×D3—21no—
279C4×D3—41no—
280Dic6—21no—
281Dic6—82no—
282Dic6#13512yes—
283C3×C8—21no—
284C3×C8—42no—
285C3×C8—41no—
286C3×C8#13712yes—
287C3⋊C8—21no—
288C3⋊C8—42no—
289C3⋊C8—41no—
290C3⋊C8#13712yes—
291#41—62no—
292#41—62no—
293C2×Q8—22no—
294C2×Q8—31no—
295C2×Q8—62no—
296C2×D4—61no—
297C2×D4D3×D411yescharacteristic
298C22×C4—21no—
299C22×C4—31no—
300C22×C4—31no—
301C22×C4—31no—
302C22×C4—61no—
303C22×C4—61no—
304Q16—124no—
305SD16—44no—
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—31no—
309C2×C8—31no—
310C2×C8—42no—
311C2×C8—62no—
312C2×C8—62no—
313C2×C8—61no—
314C2×C8—62no—
315C2×C8—122no—
316C2×C8D3×D411yescharacteristic
317C2×C8D3×D411yescharacteristic
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—31no—
321C4⋊C4—61no—
322C4⋊C4—61no—
323C4⋊C4—62no—
324C4⋊C4—62no—
325C4⋊C4—122no—
326C4⋊C4—122no—
327C4⋊C4D3×D411yescharacteristic
328C22⋊C4—21no—
329C22⋊C4—61no—
330C22⋊C4—61no—
331C22⋊C4—122no—
332C42—31no—
333C42—31no—
334C42—31no—
335C42—61no—
336C42—61no—
337C42—61no—
338C42—122no—
339C42C23×D311yescharacteristic
340C22×C3—22no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3C2×#12111yescharacteristic
344C2×D3—21no—
345C2×D3—21no—
346C2×D3—21no—
347C3×C4—21no—
348C3×C4—41no—
349C3×C4—82no—
350C3×C4C2×#13511yescharacteristic
351C3×C4C2×#13511yescharacteristic
352C3×C4C2×#13711yescharacteristic
353C3×C4C2×#13711yescharacteristic
354Dic3—21no—
355Dic3—22no—
356Dic3—21no—
357Dic3—21no—
358Dic3—41no—
359Dic3—82no—
360C23—21no—
361C23—31no—
362Q8—32no—
363Q8—42no—
364Q8—42no—
365Q8—61no—
366Q8—242no—
367D4—21no—
368D4—64no—
369D4#91212yes—
370C2×C4—21no—
371C2×C4—31no—
372C2×C4—31no—
373C2×C4—31no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—64no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—121no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4C2×D3×D411yescharacteristic
388C2×C4C2×D3×D411yescharacteristic
389C2×C4D3×#4111yescharacteristic
390C8—21no—
391C8—32no—
392C8—42no—
393C8—41no—
394C8—61no—
395C8—122no—
396C8—121no—
397C8#91512yes—
398C2×C3—42no—
399C2×C3#303911yescharacteristic
400C2×C3#305011yescharacteristic
401C2×C3#306711yescharacteristic
402D3—41no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—61no—
407C22—61no—
408C22—61no—
409C22D3×#12111yescharacteristic
410C4—21no—
411C4—41no—
412C4—61no—
413C4—61no—
414C4—61no—
415C4—62no—
416C4—82no—
417C4—121no—
418C4—242no—
419C4C2×#91211yescharacteristic
420C4C2×#91511yescharacteristic
421C4D3×#13511yescharacteristic
422C4D3×#13711yescharacteristic
423C3#1927311yescharacteristic, complemented, Sylow
424C2—42no—
425C2—121no—
426C2#8042911yescharacteristic
427C2#8071411yescharacteristic
428C2#8094311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1673C2
socleC22×C3C2×#121
prime radicalC2×#237D4
hypercenter#1673D3
hypocenterC3#19273
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1673

Complements

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

NH ≅ G/Ntype
1. #81938C2semidirect
4. C3×#1673C2semidirect
5. #71377C2semidirect
6. #71036C2semidirect
7. #70853C2semidirect
8. #68589C2semidirect
9. #68570C2semidirect
10. #68466C2semidirect
11. #68427C2semidirect
54. #1673D3semidirect
423. C3#19273semidirect

Maximal subgroups

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

subgroupindex
#819382
#804112
C8×#2372
C3×#16732
#713772
#710362
#708532
#685892
#685702
#684662
#684272
#192733