Finite Group Databasebeta

← #10777939

Subgroups — #10777939

Order 768.

598 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81792C211yesmaximal, characteristic, complemented
2#80997C212yesmaximal
3#79938C211yesmaximal, characteristic
4C3×#1690C211yesmaximal, characteristic, complemented
5#71306C211yesmaximal, characteristic, complemented
6#71122C212yesmaximal, complemented
7#70964C211yesmaximal, characteristic, complemented
8#68542C211yesmaximal, characteristic, complemented
9#68527C211yesmaximal, characteristic, complemented
10#68452C212yesmaximal, complemented
11#68419C212yesmaximal, complemented
12#19161—31nomaximal, Sylow
13#5728C2212yes—
14C4×#239C2211yescharacteristic
15C3×#494C2211yescharacteristic
16C3×#475C2212yes—
17C3×#446C2211yescharacteristic
18#5188—21no—
19#5188—21no—
20Dic3×Q16—21no—
21Dic3×Q16—21no—
22#5087C2212yes—
23#5079C2212yes—
24#5031C2212yes—
25#5024C2211yescharacteristic
26#4868—22no—
27#4852—22no—
28#4841C2212yes—
29#4838C2211yescharacteristic
30#4836—21no—
31#4833—21no—
32#4833C2211yescharacteristic
33#4833C2212yes—
34#4808—22no—
35#4790—22no—
36#4707—21no—
37#4706—21no—
38#4688C2211yescharacteristic
39C4×D3×C8C2211yescharacteristic
40#4682C2211yescharacteristic
41C3×#334C2211yescharacteristic
42C3×#332C2212yes—
43#4547C2212yes—
44#4486C2212yes—
45#4482C2212yes—
46#4479C2212yes—
47#4457C2211yescharacteristic
48#3170—31no—
49#3107—32no—
50#2965—31no—
51#1700—32no—
52#1699—32no—
53#1694—31no—
54#1693—31no—
55#1690D311yescharacteristic, complemented
56#1689—31no—
57#1564—31no—
58#1544—32no—
59C2×#239D411yescharacteristic
60C2×C3×Q16—21no—
61C2×C3×Q16—21no—
62C3×C4⋊Q8C2312yes—
63C3×C4×Q8C2311yescharacteristic
64#946—22no—
65Q8×Dic3—22no—
66Q8×Dic3D411yescharacteristic
67C2×#216—22no—
68C2×#204—22no—
69C2×ΓL(1,𝔽25)—21no—
70C2×ΓL(1,𝔽25)—21no—
71C2×D3×C8D411yescharacteristic
72C2×D3×C8D411yescharacteristic
73#897—22no—
74#893—21no—
75#893D411yescharacteristic
76#892—21no—
77#892—22no—
78#892D411yescharacteristic
79#890—22no—
80#887—22no—
81C4×D12—21no—
82C4×D12C2311yescharacteristic
83C42×D3—21no—
84C42×D3C2311yescharacteristic
85#869C2312yes—
86C3×#106—22no—
87C3×C4⋊C8—21no—
88C3×C4⋊C8C2312yes—
89C3×C4⋊C8C2311yescharacteristic
90C3×#103—22no—
91C3×#97—21no—
92C3×C4×C8C2311yescharacteristic
93#835—22no—
94#835—22no—
95(C2×D3)⋊C8—22no—
96(C2×D3)⋊C8—21no—
97(C2×D3)⋊C8—21no—
98(C2×D3)⋊C8—21no—
99#818—21no—
100#818—21no—
101#816—22no—
102C8×Dic3—21no—
103C8×Dic3—22no—
104C8×Dic3D411yescharacteristic
105C8×Dic3D411yescharacteristic
106#810—22no—
107#809—22no—
108#807—22no—
109C4⋊(C3⋊C8)—21no—
110C4⋊(C3⋊C8)C2311yescharacteristic
111C4⋊(C3⋊C8)C2312yes—
112#803—21no—
113C4×C3⋊C8C2311yescharacteristic
114#533—32no—
115C4×#41—31no—
116#500—31no—
117#494C2×D311yescharacteristic
118#487—32no—
119#485—62no—
120#475C2×D312yes—
121#474—32no—
122#467—61no—
123#467—61no—
124#462—32no—
125#461—32no—
126#452—62no—
127#446—31no—
128#446C2×D311yescharacteristic
129C4×Q16—61no—
130C4×Q16—61no—
131C4×Q16—62no—
132C4×SD16—62no—
133#436—31no—
134#436—31no—
135#435—61no—
136#435—61no—
137#431—31no—
138#431—32no—
139#431—61no—
140C4×M16—61no—
141C2×C4×C8—31no—
142#335—32no—
143#334—31no—
144#334C2×D311yescharacteristic
145#332—32no—
146#332—32no—
147#332—32no—
148#332C2×D312yes—
149C2×C3×Q8—22no—
150C2×C3×Q8C2×D411yescharacteristic
151#239—42no—
152C2×D12—21no—
153C2×D12C2×D411yescharacteristic
154C2×C4×D3—21no—
155C2×C4×D3—21no—
156C2×C4×D3—21no—
157C2×C4×D3C2×D411yescharacteristic
158C2×C4×D3C2×D411yescharacteristic
159C2×C4×D3#4111yescharacteristic
160C2×Dic6—22no—
161C3×Q16—42no—
162C3×Q16—42no—
163C2×C3×C8—22no—
164C2×C3×C8—21no—
165C2×C3×C8C2×D411yescharacteristic
166C2×C3×C8C2×D411yescharacteristic
167C3×C4⋊C4—21no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4C2×D411yescharacteristic
171C3×C42—21no—
172C3×C42C2411yescharacteristic
173#216—44no—
174(C2×D3)⋊C4—21no—
175(C2×D3)⋊C4—21no—
176(C2×D3)⋊C4—21no—
177(C2×D3)⋊C4—42no—
178C4⋊Dic3—21no—
179C4⋊Dic3—22no—
180C4⋊Dic3—22no—
181C4⋊Dic3—42no—
182C4⋊Dic3C2×D411yescharacteristic
183Dic3⋊C4—42no—
184C4×Dic3—21no—
185C4×Dic3—21no—
186C4×Dic3—42no—
187C4×Dic3C2×D411yescharacteristic
188C4×Dic3C2×D411yescharacteristic
189C4×Dic3#4111yescharacteristic
190C2×C3⋊C8—21no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8C2×D411yescharacteristic
193C2×C3⋊C8C2×D411yescharacteristic
194#204—44no—
195ΓL(1,𝔽25)—42no—
196ΓL(1,𝔽25)—42no—
197D3×C8—22no—
198D3×C8—41no—
199C2×#41—31no—
200C2×Q16—21no—
201C2×Q16—21no—
202C2×Q16—62no—
203C2×SD16—62no—
204C2×M16—61no—
205C2×M16—61no—
206C22×C8—31no—
207C22×C8—31no—
208C4⋊Q8—32no—
209C4⋊Q8C22×D312yes—
210#125—62no—
211C22⋊Q8—62no—
212C22⋊Q8—62no—
213C4×Q8—31no—
214C4×Q8—62no—
215C4×Q8—62no—
216C4×Q8C22×D311yescharacteristic
217C4×D4—31no—
218C4×D4—31no—
219C4×D4—61no—
220C4×D4—61no—
221#117—31no—
222#117—61no—
223#117—62no—
224C2×C42—31no—
225C2×C42—61no—
226#107—62no—
227#107—61no—
228#107—61no—
229#106—22no—
230C4⋊C8—21no—
231C4⋊C8—31no—
232C4⋊C8—32no—
233C4⋊C8—61no—
234C4⋊C8C22×D311yescharacteristic
235C4⋊C8C22×D312yes—
236#103—22no—
237#103—62no—
238#103—62no—
239#103—62no—
240#103—62no—
241#102—62no—
242C22⋊C8—61no—
243C22⋊C8—61no—
244C22⋊C8—62no—
245C22⋊C8—61no—
246#97—21no—
247#97—61no—
248C4×C8—31no—
249C4×C8—31no—
250C4×C8—31no—
251C4×C8—61no—
252C4×C8—62no—
253C4×C8C22×D311yescharacteristic
254C22×D3—21no—
255C22×D3#12111yescharacteristic
256C3×Q8—22no—
257C3×Q8—42no—
258C3×Q8—42no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—42no—
262C2×C3×C4C22×D411yescharacteristic
263C2×C3×C4C22×D411yescharacteristic
264C2×C3×C4C2×#4111yescharacteristic
265C2×Dic3—21no—
266C2×Dic3—42no—
267C2×Dic3#12111yescharacteristic
268C2×Dic3#12111yescharacteristic
269C2×Dic3#12111yescharacteristic
270D12—22no—
271D12—42no—
272C4×D3—21no—
273C4×D3—21no—
274C4×D3—22no—
275C4×D3—22no—
276C4×D3—42no—
277C4×D3—41no—
278C4×D3—41no—
279C4×D3—42no—
280Dic6—42no—
281Dic6—42no—
282C3×C8—21no—
283C3×C8—41no—
284C3×C8—42no—
285C3×C8#13512yes—
286C3⋊C8—21no—
287C3⋊C8—21no—
288C3⋊C8—41no—
289C3⋊C8—42no—
290#41—122no—
291C2×Q8—22no—
292C2×Q8—62no—
293C2×Q8D3×D411yescharacteristic
294C2×D4—31no—
295C2×D4—61no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4—61no—
302Q16—42no—
303Q16—42no—
304Q16—124no—
305SD16—124no—
306M16—122no—
307M16—122no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—31no—
311C2×C8—31no—
312C2×C8—62no—
313C2×C8—61no—
314C2×C8—62no—
315C2×C8—121no—
316C2×C8D3×D411yescharacteristic
317C2×C8D3×D411yescharacteristic
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—31no—
322C4⋊C4—61no—
323C4⋊C4—62no—
324C4⋊C4—62no—
325C4⋊C4—122no—
326C4⋊C4—122no—
327C4⋊C4D3×D411yescharacteristic
328C22⋊C4—61no—
329C22⋊C4—61no—
330C22⋊C4—61no—
331C22⋊C4—122no—
332C42—21no—
333C42—31no—
334C42—31no—
335C42—31no—
336C42—61no—
337C42—61no—
338C42—122no—
339C42C23×D311yescharacteristic
340C22×C3C2×#12111yescharacteristic
341C2×D3—21no—
342C2×D3—21no—
343C2×D3—21no—
344C2×D3—41no—
345C2×D3—41no—
346C2×D3—41no—
347C3×C4—21no—
348C3×C4—41no—
349C3×C4—41no—
350C3×C4—82no—
351C3×C4C2×#13511yescharacteristic
352C3×C4#57511yescharacteristic
353C3×C4#57512yes—
354Dic3—21no—
355Dic3—21no—
356Dic3—41no—
357Dic3—82no—
358Dic3#47312yes—
359C23—31no—
360C23—61no—
361Q8—22no—
362Q8—42no—
363Q8—42no—
364Q8—122no—
365Q8—122no—
366D4—62no—
367D4—122no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—31no—
373C2×C4—42no—
374C2×C4—62no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—62no—
378C2×C4—61no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—121no—
382C2×C4—121no—
383C2×C4—122no—
384C2×C4C2×D3×D411yescharacteristic
385C2×C4C2×D3×D411yescharacteristic
386C2×C4D3×#4111yescharacteristic
387C8—21no—
388C8—41no—
389C8—42no—
390C8—61no—
391C8—61no—
392C8—121no—
393C8—122no—
394C8#91612yes—
395C2×C3#303511yescharacteristic
396C2×C3#305411yescharacteristic
397C2×C3C2×#47311yescharacteristic
398D3—41no—
399D3—81no—
400C22—61no—
401C22—61no—
402C22—61no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22D3×#12111yescharacteristic
407C4—21no—
408C4—32no—
409C4—41no—
410C4—41no—
411C4—61no—
412C4—61no—
413C4—82no—
414C4—121no—
415C4—242no—
416C4C2×#91611yescharacteristic
417C4#577311yescharacteristic
418C4#577712yes—
419C3#1916111yescharacteristic, complemented, Sylow
420C2—121no—
421C2—241no—
422C2#8038211yescharacteristic
423C2#8090311yescharacteristic
424C2D3×#47311yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1690C2
socleC22×C3C2×#121
prime radicalC2×#239D4
hypercenter#1690D3
hypocenterC3#19161
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1690

Complements

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

NH ≅ G/Ntype
1. #81792C2semidirect
4. C3×#1690C2semidirect
5. #71306C2semidirect
6. #71122C2semidirect
7. #70964C2semidirect
8. #68542C2semidirect
9. #68527C2semidirect
10. #68452C2semidirect
11. #68419C2semidirect
55. #1690D3semidirect
419. C3#19161semidirect

Maximal subgroups

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

subgroupindex
#817922
#809972
#799382
C3×#16902
#713062
#711222
#709642
#685422
#685272
#684522
#684192
#191613