Finite Group Databasebeta

← #10777635

Subgroups — #10777635

Order 768.

618 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C8×#41C211yesmaximal, characteristic
2#82507C212yesmaximal
3#78475C211yesmaximal, characteristic, complemented
4#70947C212yesmaximal, complemented
5#70945C211yesmaximal, characteristic, complemented
6#70942C211yesmaximal, characteristic, complemented
7#70855C212yesmaximal, complemented
8#70853C211yesmaximal, characteristic, complemented
9#70850C211yesmaximal, characteristic, complemented
10#70776C212yesmaximal, complemented
11C2×#4458C211yesmaximal, characteristic, complemented
12#19126—31nomaximal, Sylow
13C3×C4×#41C2211yescharacteristic
14#5470C2212yes—
15C3×C8×Q8C2211yescharacteristic
16C3×C8×Q8C2211yescharacteristic
17C3×C8×D4—21no—
18C3×C8×D4—21no—
19C3×C8×D4C2211yescharacteristic
20C3×C8×D4C2211yescharacteristic
21C3×#431—21no—
22C3×#431C2211yescharacteristic
23C2×C3×C4×C8—21no—
24C2×C3×C4×C8C2211yescharacteristic
25#5233—22no—
26#5116—21no—
27#5116—21no—
28#5022—22no—
29#5021C2212yes—
30#5020C2212yes—
31#5009—22no—
32#5008C2212yes—
33#5007C2212yes—
34#4998—22no—
35C2×C4⋊(C3⋊C8)C2212yes—
36#4699C2211yescharacteristic
37C4×Dic12—21no—
38C4×Dic12—21no—
39#4682C2211yescharacteristic
40#4681C2212yes—
41#4482C2212yes—
42#4482C2212yes—
43#4481C2212yes—
44#4481C2212yes—
45#4458C2211yescharacteristic
46#4458C2211yescharacteristic
47#4458C2212yes—
48#3170—31no—
49#3076—32no—
50C8×#41D311yescharacteristic
51#1684—32no—
52#1683—32no—
53#1676—31no—
54#1675—31no—
55#1670—31no—
56#1669—31no—
57C2×#335—31no—
58#1542—32no—
59C2×C3×#41D411yescharacteristic
60C22×C3×C8—21no—
61C22×C3×C8—21no—
62C22×C3×C8D411yescharacteristic
63C22×C3×C8D411yescharacteristic
64C3×C4×Q8D411yescharacteristic
65C3×C4×Q8C2311yescharacteristic
66C3×C4×D4—21no—
67C3×C4×D4—21no—
68C3×C4×D4D411yescharacteristic
69C3×C4×D4C2311yescharacteristic
70C3×#117—21no—
71C3×#117D411yescharacteristic
72C2×C3×C42—21no—
73C2×C3×C42C2311yescharacteristic
74#926—22no—
75#924—22no—
76#924—22no—
77C22×C3⋊C8—22no—
78C2×Dic12—21no—
79C2×Dic12—21no—
80#870—22no—
81#869C2312yes—
82#869C2312yes—
83C4×Dic6—22no—
84C4×Dic6—22no—
85C3×C4⋊C8—21no—
86C3×C4⋊C8—21no—
87C3×C4⋊C8C2311yescharacteristic
88C3×C4⋊C8C2311yescharacteristic
89C3×C22⋊C8—21no—
90C3×C22⋊C8—21no—
91C3×C22⋊C8—21no—
92C3×C4×C8—21no—
93C3×C4×C8—21no—
94C3×C4×C8—21no—
95C3×C4×C8D411yescharacteristic
96C3×C4×C8D411yescharacteristic
97C3×C4×C8C2311yescharacteristic
98C3×C4×C8C2311yescharacteristic
99#835—22no—
100#835—22no—
101#832—22no—
102#832—22no—
103#818—21no—
104#818—21no—
105#816—22no—
106#816—22no—
107#808—22no—
108#808—22no—
109#807—22no—
110#807—22no—
111C4⋊(C3⋊C8)—22no—
112C4⋊(C3⋊C8)C2312yes—
113C4⋊(C3⋊C8)C2312yes—
114#533—32no—
115C4×#41C2×D311yescharacteristic
116#500—31no—
117#494—31no—
118#487—32no—
119#479—62no—
120#478—62no—
121#477—32no—
122#476—32no—
123#475—32no—
124#474—32no—
125#467—61no—
126#467—61no—
127C8×Q8C2×D311yescharacteristic
128C8×Q8C2×D311yescharacteristic
129C4×Q16—61no—
130C4×Q16—61no—
131C8×D4—21no—
132C8×D4—21no—
133C8×D4C2×D311yescharacteristic
134C8×D4C2×D311yescharacteristic
135#431—21no—
136#431C2×D311yescharacteristic
137#427—62no—
138C2×C4⋊C8—32no—
139#416—62no—
140C2×C4×C8—21no—
141C2×C4×C8C2×D311yescharacteristic
142#335—31no—
143#335—31no—
144#335—32no—
145#333—32no—
146#333—32no—
147#332—32no—
148#332—32no—
149C3×#41—22no—
150C3×#41Q1614yescomplemented
151C2×C3×Q8C2×D411yescharacteristic
152C2×C3×D4—21no—
153C2×C3×D4C2×D411yescharacteristic
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4#4111yescharacteristic
160C2×Dic6—22no—
161C2×Dic6—22no—
162C2×C3×C8—22no—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8—41no—
166C2×C3×C8—42no—
167C2×C3×C8—42no—
168C2×C3×C8C2×D411yescharacteristic
169C2×C3×C8C2×D411yescharacteristic
170C2×C3×C8C2×D411yescharacteristic
171C2×C3×C8C2×D411yescharacteristic
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4C2×D411yescharacteristic
175C3×C4⋊C4C2×D411yescharacteristic
176C3×C22⋊C4—21no—
177C3×C22⋊C4—21no—
178C3×C22⋊C4—21no—
179C3×C42—21no—
180C3×C42—21no—
181C3×C42—21no—
182C3×C42C2×D411yescharacteristic
183C3×C42C2×D411yescharacteristic
184C3×C42#4111yescharacteristic
185C3×C42C2411yescharacteristic
186#217—42no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191Dic3⋊C4—42no—
192Dic3⋊C4—42no—
193C4×Dic3—42no—
194C2×C3⋊C8—22no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—44no—
197Dic12—42no—
198Dic12—42no—
199C2×#41D1211yescharacteristic
200C2×Q16—61no—
201C2×Q16—61no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—62no—
205C22×C8#6711yescharacteristic
206C22×C8#6711yescharacteristic
207C4⋊Q8—32no—
208C4⋊Q8—32no—
209#125—62no—
210C22⋊Q8—62no—
211C22⋊Q8—62no—
212C4×Q8—62no—
213C4×Q8—62no—
214C4×Q8D1211yescharacteristic
215C4×Q8C22×D311yescharacteristic
216C4×D4—21no—
217C4×D4—21no—
218C4×D4D1211yescharacteristic
219C4×D4C22×D311yescharacteristic
220#117—21no—
221#117—62no—
222#117D1211yescharacteristic
223C2×C42—21no—
224C2×C42C22×D311yescharacteristic
225#107—61no—
226#107—61no—
227#107—62no—
228#107—62no—
229#106—62no—
230#106—62no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—32no—
234C4⋊C8—32no—
235C4⋊C8—62no—
236C4⋊C8C22×D311yescharacteristic
237C4⋊C8C22×D311yescharacteristic
238#103—62no—
239#103—62no—
240#103—62no—
241#103—62no—
242#102—62no—
243#102—62no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8#6711yescharacteristic
251C4×C8#6711yescharacteristic
252C4×C8C22×D311yescharacteristic
253C4×C8C22×D311yescharacteristic
254C23×C3—21no—
255C23×C3#12111yescharacteristic
256C3×Q8—21no—
257C3×Q8C2×Q1612yes—
258C3×D4—24no—
259C3×D4—21no—
260C3×D4C2×Q1612yes—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—24no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4#12111yescharacteristic
272C2×C3×C4#12111yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4C2×Q1612yes—
275C2×C3×C4#13512yes—
276C2×C3×C4C22×D411yescharacteristic
277C2×C3×C4C22×D411yescharacteristic
278C2×C3×C4C2×#4111yescharacteristic
279C2×Dic3—42no—
280C2×Dic3—42no—
281Dic6—42no—
282Dic6—42no—
283Dic6—82no—
284C3×C8—21no—
285C3×C8—21no—
286C3×C8—21no—
287C3×C8—41no—
288C3×C8—41no—
289C3×C8#13512yes—
290C3⋊C8—42no—
291C3⋊C8—42no—
292#41—22no—
293#41Dic1214yescomplemented
294C2×Q8—62no—
295C2×Q8—62no—
296C2×Q8C2×D1211yescharacteristic
297C2×D4—21no—
298C2×D4C2×D1211yescharacteristic
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4C2×D1211yescharacteristic
303C22×C4#23511yescharacteristic
304C22×C4C2×#6711yescharacteristic
305Q16—122no—
306Q16—122no—
307C2×C8—22no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—62no—
314C2×C8—62no—
315C2×C8—124no—
316C2×C8C2×#6711yescharacteristic
317C2×C8C2×#6711yescharacteristic
318C2×C8C2×#6711yescharacteristic
319C2×C8C2×#6711yescharacteristic
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—62no—
323C4⋊C4—62no—
324C4⋊C4—62no—
325C4⋊C4—62no—
326C4⋊C4—122no—
327C4⋊C4—122no—
328C4⋊C4C2×D1211yescharacteristic
329C4⋊C4C2×D1211yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—122no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—122no—
338C42C2×D1211yescharacteristic
339C42#23511yescharacteristic
340C42C2×#6711yescharacteristic
341C42C23×D311yescharacteristic
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—22no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3#46712yes—
348C22×C3C2×#12111yescharacteristic
349C3×C4—21no—
350C3×C4—22no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—41no—
354C3×C4—41no—
355C3×C4#46712yes—
356C3×C4C22×Q1611yescharacteristic
357C3×C4C2×#13511yescharacteristic
358C3×C4C2×#13511yescharacteristic
359C3×C4#57511yescharacteristic
360Dic3—82no—
361Dic3—82no—
362C23—21no—
363C23#93011yescharacteristic
364Q8—21no—
365Q8—122no—
366Q8—122no—
367Q8—242no—
368Q8C2×Dic1212yes—
369D4—24no—
370D4—21no—
371D4C2×Dic1212yes—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—122no—
383C2×C4—122no—
384C2×C4#90412yes—
385C2×C4C2×Dic1212yes—
386C2×C4#93011yescharacteristic
387C2×C4#93011yescharacteristic
388C2×C4#93011yescharacteristic
389C2×C4C22×D1211yescharacteristic
390C2×C4C2×#23511yescharacteristic
391C2×C4C22×#6711yescharacteristic
392C8—21no—
393C8—21no—
394C8—21no—
395C8—41no—
396C8—41no—
397C8—122no—
398C8—122no—
399C8#95012yes—
400C2×C3—22no—
401C2×C3—42no—
402C2×C3#303511yescharacteristic
403C2×C3C2×#46711yescharacteristic
404C2×C3#306711yescharacteristic
405C22—21no—
406C22—21no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22#511612yes—
411C22C2×#93011yescharacteristic
412C4—22no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—41no—
417C4—41no—
418C4—242no—
419C4—242no—
420C4#511612yes—
421C4C2×#90411yescharacteristic
422C4C22×Dic1211yescharacteristic
423C4C2×#95011yescharacteristic
424C4#582211yescharacteristic
425C3#1912611yescharacteristic, complemented, Sylow
426C2—22no—
427C2—42no—
428C2#8252511yescharacteristic
429C2C2×#511611yescharacteristic
430C2#8380511yescharacteristic
subgroupG/H
centerC2×C4#930
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×C8×#41C2
socleC22×C3C2×#121
prime radicalC2×C3×#41D4
hypercenterC8×#41D3
hypocenterC3#19126
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
3. #78475C2semidirect
4. #70947C2semidirect
5. #70945C2semidirect
6. #70942C2semidirect
7. #70855C2semidirect
8. #70853C2semidirect
9. #70850C2semidirect
10. #70776C2semidirect
11. C2×#4458C2semidirect
150. C3×#41Q16semidirect
293. #41Dic12semidirect
425. C3#19126semidirect

Maximal subgroups

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

subgroupindex
C3×C8×#412
#825072
#784752
#709472
#709452
#709422
#708552
#708532
#708502
#707762
C2×#44582
#191263