Finite Group Databasebeta

← #10950860

Subgroups — #10950860

Order 768.

556 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81809C211yesmaximal, characteristic, complemented
2#81775C211yesmaximal, characteristic, complemented
3C3×#2250C211yesmaximal, characteristic, complemented
4#71487C212yesmaximal, complemented
5#71474C212yesmaximal, complemented
6#71469C212yesmaximal, complemented
7#71461C212yesmaximal, complemented
8#71319C211yesmaximal, characteristic, complemented
9#71289C211yesmaximal, characteristic, complemented
10D3×#346C211yesmaximal, characteristic, complemented
11#69069C211yesmaximal, characteristic, complemented
12#33746—31nomaximal, Sylow
13#5716—21no—
14#5680—21no—
15C2×C3×SD32C2212yes—
16C2×C3×D16C2212yes—
17C3×#495C2211yescharacteristic
18C3×#493C2211yescharacteristic
19#5188C2211yescharacteristic
20Dic3×Q16C2211yescharacteristic
21#5171—21no—
22Dic3×SD16—21no—
23#5157—21no—
24#5157C2211yescharacteristic
25#5157C2212yes—
26Dic3×D8—21no—
27Dic3×D8C2211yescharacteristic
28Dic3×D8C2212yes—
29#5070—21no—
30#5056—21no—
31C2×D3×C16C2212yes—
32C4×D3×C8C2211yescharacteristic
33#4683C2211yescharacteristic
34#4682C2211yescharacteristic
35C3×#346C2211yescharacteristic
36#4564C2212yes—
37#4562C2212yes—
38#4562C2212yes—
39#4562C2212yes—
40#4505C2212yes—
41#4504C2212yes—
42#4502C2212yes—
43#4461C2211yescharacteristic
44#3169—31no—
45#3168—31no—
46#2255—31no—
47#2250D311yescharacteristic, complemented
48#2244—31no—
49#2243—31no—
50#2217—32no—
51#2215—32no—
52#2177—32no—
53#2175—32no—
54C2×#346—31no—
55C2×#237—41no—
56C2×C3×Q16C2311yescharacteristic
57C2×C3×SD16—21no—
58C2×C3×D8—21no—
59C2×C3×D8C2311yescharacteristic
60C2×C3×D8C2312yes—
61C3×#127—21no—
62C3×#124—21no—
63#946—21no—
64Q8×Dic3—21no—
65#938—22no—
66#938—21no—
67#938—41no—
68#936—41no—
69D4×Dic3—22no—
70D4×Dic3—21no—
71D4×Dic3—41no—
72C2×D3×C8D411yescharacteristic
73C2×D3×C8C2311yescharacteristic
74#883—41no—
75#67⋊C4—41no—
76#879—41no—
77#877—41no—
78C42×D3D411yescharacteristic
79#870—21no—
80#869—21no—
81C3×SD32—24no—
82C3×D16—24no—
83C2×C3×C16C2312yes—
84C3×C4×C8C2311yescharacteristic
85#835—21no—
86#835—21no—
87#832—22no—
88#832—22no—
89#832—21no—
90#832—21no—
91#818—21no—
92#818C2312yes—
93#818C2311yescharacteristic
94#818C2311yescharacteristic
95#817—21no—
96C8×Dic3D411yescharacteristic
97C8×Dic3C2311yescharacteristic
98C2×C3⋊C16C2312yes—
99C4×C3⋊C8D411yescharacteristic
100D3×C16—24no—
101#532—61no—
102#529—61no—
103C2×SD32C2×D312yes—
104C2×D16C2×D312yes—
105C22×C16—32no—
106#500—31no—
107#499—31no—
108#495C2×D311yescharacteristic
109#493C2×D311yescharacteristic
110#488—61no—
111#486—61no—
112#467—31no—
113#466—31no—
114#466—32no—
115#466—61no—
116#465—61no—
117C4×Q16—31no—
118C4×SD16—61no—
119C4×D8—31no—
120C4×D8—32no—
121C4×D8—61no—
122C2×C4×C8—31no—
123#367—32no—
124#366—32no—
125#358—32no—
126#357—32no—
127#357—32no—
128#357—32no—
129#346—32no—
130#346—31no—
131#346C2×D311yescharacteristic
132C2×C3×Q8—21no—
133C2×C3×D4—21no—
134C2×C3×D4—22no—
135C2×C3×D4—41no—
136C2×C3×D4—41no—
137C2×#67—41no—
138C2×#67—42no—
139C22×Dic3—41no—
140C22×Dic3—42no—
141#237—84no—
142C2×C4×D3D811yescharacteristic
143C2×C4×D3D811yescharacteristic
144C2×C4×D3C2×D411yescharacteristic
145C2×Dic6—41no—
146C3×Q16—22no—
147C3×SD16—42no—
148C3×D8—22no—
149C3×D8—22no—
150C3×D8—22no—
151C3×D8—42no—
152C2×C3×C8C2×D411yescharacteristic
153C2×C3×C8C2411yescharacteristic
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C42C2×D411yescharacteristic
157#217—41no—
158#217—41no—
159#217—42no—
160#217—42no—
161(C2×D3)⋊C4—41no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—22no—
165C4⋊Dic3—41no—
166C4⋊Dic3—41no—
167Dic3⋊C4—41no—
168Dic3⋊C4—41no—
169Dic3⋊C4—41no—
170C4×Dic3—41no—
171C4×Dic3D811yescharacteristic
172C4×Dic3D811yescharacteristic
173C4×Dic3C2×D411yescharacteristic
174C2×C3⋊C8C2×D411yescharacteristic
175C2×C3⋊C8C2×D411yescharacteristic
176D3×C8—22no—
177D3×C8—22no—
178C3×C16#4114yescomplemented
179C3⋊C16—22no—
180C2×#41—121no—
181C2×Q16C22×D311yescharacteristic
182C2×SD16—21no—
183C2×D8—21no—
184C2×D8C22×D312yes—
185C2×D8C22×D311yescharacteristic
186C22×C8—31no—
187C22×C8—31no—
188C4⋊Q8—61no—
189#127—21no—
190#126—121no—
191#125—61no—
192#124—21no—
193#124—121no—
194#123—121no—
195C22⋊Q8—61no—
196#121—61no—
197#121—62no—
198#121—121no—
199C4×Q8—61no—
200C4×D4—61no—
201C4×D4—62no—
202C4×D4—121no—
203C4×D4—121no—
204#117—121no—
205C2×C42—31no—
206SD32—24no—
207D16—24no—
208C2×C16—32no—
209C2×C16—64no—
210C2×C16C22×D312yes—
211#107—31no—
212#107—31no—
213#107—32no—
214#107—61no—
215#106—61no—
216#103—61no—
217#103—61no—
218#102—61no—
219#102—62no—
220#102—62no—
221#102—61no—
222C4×C8—31no—
223C4×C8—31no—
224C4×C8—31no—
225C4×C8C22×D311yescharacteristic
226C23×C3—41no—
227C23×C3—42no—
228C22×D3C2×D811yescharacteristic
229C3×Q8—41no—
230C3×Q8—41no—
231C3×D4—41no—
232C3×D4—42no—
233C3×D4—42no—
234C3×D4—41no—
235C3×D4—82no—
236C3×D4—82no—
237C2×C3×C4—41no—
238C2×C3×C4C2×D811yescharacteristic
239C2×C3×C4C2×D811yescharacteristic
240C2×C3×C4C22×D411yescharacteristic
241#67—84no—
242#67—82no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—42no—
246C2×Dic3—84no—
247C2×Dic3—82no—
248C2×Dic3C2×D811yescharacteristic
249C2×Dic3C2×D811yescharacteristic
250C2×Dic3C2×D811yescharacteristic
251C4×D3—22no—
252C4×D3—21no—
253C4×D3—21no—
254C4×D3—41no—
255Dic6—82no—
256C3×C8#13512yes—
257C3×C8C2×#4112yes—
258C3⋊C8—21no—
259C3⋊C8—21no—
260#41—244no—
261C2×Q8—21no—
262C2×Q8—121no—
263C2×D4—21no—
264C2×D4—22no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—121no—
268C2×D4—122no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—121no—
273C22×C4—122no—
274Q16—22no—
275SD16—42no—
276D8—22no—
277D8—22no—
278D8—22no—
279D8—42no—
280C2×C8—31no—
281C2×C8—31no—
282C2×C8—62no—
283C2×C8—62no—
284C2×C8D3×D411yescharacteristic
285C2×C8C23×D311yescharacteristic
286C4⋊C4—62no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—121no—
290C4⋊C4—121no—
291C4⋊C4—121no—
292C4⋊C4—121no—
293C4⋊C4—121no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—121no—
297C22⋊C4—122no—
298C22⋊C4—121no—
299C22⋊C4—122no—
300C22⋊C4—121no—
301C42—31no—
302C42—31no—
303C42—31no—
304C42—121no—
305C42D3×D411yescharacteristic
306C16—62no—
307C16#23714yescomplemented
308C22×C3—82no—
309C22×C3—81no—
310C22×C3—81no—
311C22×C3—82no—
312C22×C3—82no—
313C22×C3—81no—
314C22×C3C22×D811yescharacteristic
315C2×D3—21no—
316C2×D3—21no—
317C2×D3#50912yes—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—81no—
321C3×C4#53211yescharacteristic
322C3×C4C2×#13511yescharacteristic
323Dic3—21no—
324Dic3—21no—
325Dic3—21no—
326Dic3—82no—
327Dic3—81no—
328Dic3—81no—
329C23—31no—
330C23—41no—
331C23—42no—
332Q8—41no—
333Q8—41no—
334Q8—242no—
335D4—41no—
336D4—42no—
337D4—42no—
338D4—41no—
339D4—82no—
340D4—82no—
341D4—244no—
342D4—242no—
343C2×C4—31no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—41no—
347C2×C4—62no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—121no—
353C2×C4—122no—
354C2×C4—244no—
355C2×C4—242no—
356C2×C4D3×D811yescharacteristic
357C2×C4D3×D811yescharacteristic
358C2×C4C2×D3×D411yescharacteristic
359C8—61no—
360C8—61no—
361C8#91212yes—
362C8C2×#23712yes—
363C2×C3—162no—
364C2×C3—161no—
365C2×C3#316911yescharacteristic
366C2×C3C2×#50911yescharacteristic
367C2×C3#341411yescharacteristic
368D3—22no—
369C22—32no—
370C22—61no—
371C22—61no—
372C22—82no—
373C22—81no—
374C22—82no—
375C22—81no—
376C22—81no—
377C22—82no—
378C22C2×D3×D811yescharacteristic
379C4—21no—
380C4—21no—
381C4—61no—
382C4—61no—
383C4—61no—
384C4—81no—
385C4—242no—
386C4—241no—
387C4—241no—
388C4#571611yescharacteristic
389C4C2×#91211yescharacteristic
390C3#3374611yescharacteristic, complemented, Sylow
391C2—62no—
392C2—162no—
393C2—161no—
394C2#8177511yescharacteristic
395C2D3×#50911yescharacteristic
396C2#8203511yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2250C2
socleC22×C3C22×D8
hypercenter#2250D3
hypocenterC3#33746
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2250

Complements

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

NH ≅ G/Ntype
1. #81809C2semidirect
2. #81775C2semidirect
3. C3×#2250C2semidirect
4. #71487C2semidirect
5. #71474C2semidirect
6. #71469C2semidirect
7. #71461C2semidirect
8. #71319C2semidirect
9. #71289C2semidirect
10. D3×#346C2semidirect
11. #69069C2semidirect
47. #2250D3semidirect
178. C3×C16#41semidirect
307. C16#237semidirect
390. C3#33746semidirect

Maximal subgroups

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

subgroupindex
#818092
#817752
C3×#22502
#714872
#714742
#714692
#714612
#713192
#712892
D3×#3462
#690692
#337463