Finite Group Databasebeta

← #10876286

Subgroups — #10876286

Order 768.

728 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2922C211yesmaximal, characteristic, complemented
2#82536C211yesmaximal, characteristic
3#71644C211yesmaximal, characteristic, complemented
4#70549C212yesmaximal, complemented
5#70548C212yesmaximal, complemented
6#70010C211yesmaximal, characteristic
7#70009C211yesmaximal, characteristic
8#70002C211yesmaximal, characteristic, complemented
9#70001C211yesmaximal, characteristic, complemented
10#69874C212yesmaximal, complemented
11#69793C212yesmaximal, complemented
12#27598—31nomaximal, Sylow
13#5964C2211yescharacteristic
14C3×C4×#41C2211yescharacteristic
15#5819C2211yescharacteristic
16C3×#433—22no—
17C3×#431—22no—
18C3×#424C2211yescharacteristic, complemented
19C3×#424C2211yescharacteristic, complemented
20C3×#408C2211yescharacteristic
21C3×#408C2211yescharacteristic
22C3×C4×M16C2211yescharacteristic, complemented
23C2×C3×C4×C8C2211yescharacteristic, complemented
24#5240C2211yescharacteristic
25#5237C2211yescharacteristic
26#5139C2212yes—
27#5132C2212yes—
28#5124C2212yescomplemented
29#5113C2212yes—
30#5110C2212yes—
31#5102C2212yescomplemented
32#5005—22no—
33#5005C2212yes—
34C2×#805—22no—
35C2×#805C2212yes—
36#4559C2212yescomplemented
37#4552C2212yescomplemented
38#4531C2212yescomplemented
39#4523C2212yescomplemented
40#2922D311yescharacteristic, complemented
41#2901—31no—
42#2098—32no—
43#2080—32no—
44#1937—32no—
45#1932—32no—
46#1868—31no—
47#1858—31no—
48#1827—31no—
49#1794—31no—
50#1791—31no—
51C2×C3×#41C2311yescharacteristic
52#1010—22no—
53#1009—22no—
54D3×#41—42no—
55#1007—21no—
56#1004—21no—
57#1004—22no—
58C2×#235—22no—
59C2×#235C2312yes—
60C2×C3×M16C2311yescharacteristic
61C2×C3×M16C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×C4×Q8—21no—
65C3×C4×D4—22no—
66C3×C4×D4—21no—
67C3×#117D412yes—
68C3×#117C2311yescharacteristic
69C2×C3×C42D412yes—
70C2×C3×C42C2311yescharacteristic
71#948—42no—
72#926C2312yes—
73C2×#208—22no—
74C2×#208C2312yes—
75C2×#208C2312yes—
76C3×C4⋊C8—22no—
77C3×C4⋊C8—22no—
78C3×C22⋊C8—22no—
79C3×C22⋊C8—22no—
80C3×#97—21no—
81C3×C4×C8—21no—
82C3×C4×C8—21no—
83C3×C4×C8—21no—
84#836—21no—
85#834—22no—
86#833—21no—
87#825—42no—
88#824—22no—
89#823—22no—
90#822—42no—
91#821—22no—
92(C2×D3)⋊C8—42no—
93#816—22no—
94Dic3⋊C8—42no—
95#805—22no—
96#805—24no—
97#805—22no—
98#585—31no—
99#568—31no—
100C4×#41C2×D311yescharacteristic
101#433—22no—
102#431—22no—
103#430—32no—
104#424—32no—
105#424C2×D311yescharacteristic, complemented
106#424C2×D311yescharacteristic, complemented
107#421—32no—
108#421—32no—
109#421—62no—
110C2×#104—32no—
111C2×#104—62no—
112#416—32no—
113#413—31no—
114#413—32no—
115#410—31no—
116#408—32no—
117#408C2×D311yescharacteristic
118#408C2×D311yescharacteristic
119C4×M16C2×D311yescharacteristic, complemented
120C2×C4×C8C2×D311yescharacteristic, complemented
121#344—32no—
122#341—32no—
123#339—32no—
124H(3,ℤ/4)—32no—
125C3×#41—22no—
126C3×#41—22no—
127C2×C3×Q8#4111yescharacteristic
128C2×C3×D4#4112yes—
129C2×C3×D4#4111yescharacteristic
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2411yescharacteristic
134C2×#67—22no—
135C2×#67—42no—
136#239—42no—
137D3×Q8—42no—
138#237—42no—
139#237—44no—
140D3×D4—42no—
141D3×D4—44no—
142#235—24no—
143#235—22no—
144#235—22no—
145#235—22no—
146#235—22no—
147#235—42no—
148#235—42no—
149C2×D12—22no—
150C2×D12#4112yes—
151C2×C4×D3—22no—
152C2×C4×D3—42no—
153C2×Dic6—22no—
154C2×Dic6#4112yes—
155C3×M16—22no—
156C3×M16—22no—
157C2×C3×C8—21no—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—21no—
165C3×C22⋊C4—22no—
166C3×C22⋊C4—21no—
167C3×C42—22no—
168C3×C42—21no—
169C3×C42C2×D411yescharacteristic
170C3×C42C2×D411yescharacteristic
171C3×C42C2×D412yes—
172#217—22no—
173C4⋊Dic3—22no—
174C4×Dic3—22no—
175#208—22no—
176#208—22no—
177#208—22no—
178#208—22no—
179#208—42no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—42no—
1832−1+4—62no—
1842−1+4—61no—
185H(5,𝔽2)—62no—
186H(5,𝔽2)—62no—
187H(5,𝔽2)—61no—
188C2×#41—32no—
189C2×#41—62no—
190C2×#41—122no—
191C2×#41C22×D311yescharacteristic
192#131—122no—
193C2×M16—32no—
194C2×M16—32no—
195C2×M16—62no—
196C2×M16C22×D311yescharacteristic
197C2×M16C22×D311yescharacteristic
198C22×C8C22×D311yescharacteristic
199C22×C8C22×D311yescharacteristic
200C4×Q8—21no—
201C4×D4—22no—
202C4×D4—21no—
203#117—32no—
204#117#6712yes—
205#117C22×D311yescharacteristic
206C2×C42#6712yes—
207C2×C42C22×D311yescharacteristic
208#108—122no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—122no—
212#104—62no—
213#104—62no—
214#104—64no—
215#104—122no—
216#103—62no—
217#102—62no—
218#101—62no—
219#101—61no—
220#100—62no—
221#100—61no—
222#99—62no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—122no—
226#97—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C23×C3#12312yes—
231C23×C3C2×#4111yescharacteristic
232C22×D3—22no—
233C22×D3—42no—
234C3×Q8—22no—
235C3×D4—24no—
236C3×D4—22no—
237C2×C3×C4—21no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—21no—
242C2×C3×C4—42no—
243C2×C3×C4—44no—
244C2×C3×C4#12312yes—
245C2×C3×C4C22×D411yescharacteristic
246C2×C3×C4C2×#4111yescharacteristic
247C2×C3×C4C2×#4112yes—
248C2×C3×C4H(5,𝔽2)12yes—
249#67—44no—
250#67—42no—
251#67—42no—
252C2×Dic3—22no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255D12—22no—
256D12—22no—
257D12—42no—
258C4×D3—44no—
259C4×D3—42no—
260C4×D3—42no—
261Dic6—22no—
262Dic6—22no—
263Dic6—42no—
264C3×C8—41no—
265C3×C8—41no—
266C3×C8—41no—
267C3×C8—41no—
268C3⋊C8—42no—
269C3⋊C8—42no—
270#41—22no—
271#41—22no—
272#41—64no—
273#41—62no—
274#41—62no—
275#41—62no—
276#41—62no—
277#41—122no—
278#41—124no—
279#41—122no—
280#41—122no—
281#41—122no—
282C2×Q8—32no—
283C2×Q8—62no—
284C2×Q8—122no—
285C2×Q8#23511yescharacteristic
286C2×D4—32no—
287C2×D4—62no—
288C2×D4—62no—
289C2×D4—122no—
290C2×D4—124no—
291C2×D4—122no—
292C2×D4#23512yes—
293C2×D4#23511yescharacteristic
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—62no—
297C22×C4—122no—
298C22×C4C2×#6712yes—
299C22×C4C23×D311yescharacteristic
300M16—22no—
301M16—22no—
302M16—62no—
303M16—62no—
304M16—62no—
305M16—62no—
306M16—122no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—62no—
314C2×C8—62no—
315C2×C8—122no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—62no—
319C22⋊C4—22no—
320C22⋊C4—21no—
321C22⋊C4—62no—
322C42—22no—
323C42—21no—
324C42—62no—
325C42C2×#6711yescharacteristic
326C42C2×#6711yescharacteristic
327C42C2×#6712yes—
328C22×C3—22no—
329C22×C3—41no—
330C22×C3—42no—
331C22×C3C2×#12311yescharacteristic
332C22×C3#55112yes—
333C2×D3—42no—
334C2×D3—82no—
335C2×D3—162no—
336C3×C4—22no—
337C3×C4—22no—
338C3×C4—42no—
339C3×C4—41no—
340C3×C4#52511yescharacteristic
341C3×C4#53511yescharacteristic
342C3×C4#55112yes—
343Dic3—42no—
344Dic3—82no—
345C23—62no—
346C23—122no—
347C23#92912yes—
348C23C2×#23511yescharacteristic
349Q8—22no—
350Q8—62no—
351Q8—62no—
352Q8—122no—
353D4—24no—
354D4—22no—
355D4—62no—
356D4—62no—
357D4—122no—
358D4—124no—
359D4—122no—
360D4—122no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—62no—
369C2×C4—124no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4#92912yes—
375C2×C4C2×#23511yescharacteristic
376C2×C4D3×#4112yes—
377C2×C4#100912yes—
378C2×C4C22×#6711yescharacteristic
379C8—41no—
380C8—41no—
381C8—41no—
382C8—41no—
383C8—122no—
384C8—122no—
385C2×C3—22no—
386C2×C3—21no—
387C2×C3—42no—
388C2×C3#248211yescharacteristic
389D3—82no—
390C22—22no—
391C22—41no—
392C22—42no—
393C22—122no—
394C22—242no—
395C22—482no—
396C22#552712yes—
397C22C2×#92911yescharacteristic
398C4—22no—
399C4—22no—
400C4—42no—
401C4—41no—
402C4—122no—
403C4—242no—
404C4#552712yes—
405C4#582811yescharacteristic
406C4#583011yescharacteristic
407C3#2759811yescharacteristic, complemented, Sylow
408C2—21no—
409C2—22no—
410C2—42no—
411C2—242no—
412C2#8228211yescharacteristic
subgroupG/H
centerC4#5830
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2922C2
socleC2×C3#2482
prime radical#5964C22
hypercenter#2922D3
hypocenterC3#27598
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2922

Complements

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

NH ≅ G/Ntype
1. C3×#2922C2semidirect
3. #71644C2semidirect
4. #70549C2semidirect
5. #70548C2semidirect
8. #70002C2semidirect
9. #70001C2semidirect
10. #69874C2semidirect
11. #69793C2semidirect
18. C3×#424C22semidirect
19. C3×#424C22semidirect
22. C3×C4×M16C22semidirect
23. C2×C3×C4×C8C22semidirect
28. #5124C22semidirect
31. #5102C22semidirect
36. #4559C22semidirect
37. #4552C22semidirect
38. #4531C22semidirect
39. #4523C22semidirect
40. #2922D3semidirect
105. #424C2×D3semidirect
106. #424C2×D3semidirect
119. C4×M16C2×D3semidirect
120. C2×C4×C8C2×D3semidirect
407. C3#27598semidirect

Maximal subgroups

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

subgroupindex
C3×#29222
#825362
#716442
#705492
#705482
#700102
#700092
#700022
#700012
#698742
#697932
#275983