Finite Group Databasebeta

← #10777938

Subgroups — #10777938

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
1C3×#3170C211yesmaximal, characteristic, complemented
2#83352C212yesmaximal
3#78447C211yesmaximal, characteristic
4#71829C212yesmaximal, complemented
5#71379C211yesmaximal, characteristic, complemented
6#71373C211yesmaximal, characteristic, complemented
7#71309C211yesmaximal, characteristic, complemented
8#71306C211yesmaximal, characteristic, complemented
9#71242C212yesmaximal, complemented
10#71234C212yesmaximal, complemented
11#69729C211yesmaximal, characteristic, complemented
12#19161—31nomaximal, Sylow
13C3×#533C2212yes—
14C4×#235C2211yescharacteristic
15C3×#500C2211yescharacteristic
16C3×#494C2211yescharacteristic
17C3×#487C2212yes—
18C3×#467—21no—
19C3×#467—21no—
20C3×C4×Q16—21no—
21C3×C4×Q16—21no—
22C2×C3×C4×C8C2211yescharacteristic
23#5111—21no—
24#5111—21no—
25#5094C2212yes—
26#5090C2212yes—
27#5087C2212yes—
28#5083C2212yes—
29#5047—22no—
30#5045—22no—
31C4×#216—22no—
32C4×#215—22no—
33#4925C2212yes—
34#4925C2211yescharacteristic
35#4688C2211yescharacteristic
36#4688C2211yescharacteristic
37C4×ΓL(1,𝔽25)—21no—
38#4685—21no—
39#4679C2211yescharacteristic
40#4679C2211yescharacteristic
41#4547C2212yes—
42#4547C2212yes—
43#4547C2212yes—
44#4547C2212yes—
45#4458C2212yes—
46#4457C2211yescharacteristic
47#4457C2211yescharacteristic
48#3170D311yescharacteristic, complemented
49#3107—32no—
50#2965—31no—
51#1700—32no—
52#1699—32no—
53#1694—31no—
54#1693—31no—
55#1690—31no—
56#1689—31no—
57#1564—31no—
58#1544—32no—
59C2×#235D411yescharacteristic
60C2×C3×Q16—21no—
61C2×C3×Q16—21no—
62C22×C3×C8D411yescharacteristic
63C22×C3×C8D411yescharacteristic
64C3×C4⋊Q8C2312yes—
65C3×C4⋊Q8C2312yes—
66C3×#125—22no—
67C3×C22⋊Q8—22no—
68C3×C22⋊Q8—22no—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×#117—22no—
72C2×C3×C42C2311yescharacteristic
73C2×#216—22no—
74C2×#215—22no—
75C4×#67—21no—
76C4×#67—21no—
77#926D411yescharacteristic
78C2×ΓL(1,𝔽25)—21no—
79C2×ΓL(1,𝔽25)—21no—
80C4×D12D411yescharacteristic
81C4×D12C2311yescharacteristic
82#872—21no—
83C42×D3—21no—
84C4×Dic6D411yescharacteristic
85C4×Dic6C2311yescharacteristic
86C3×#107—21no—
87C3×#107—21no—
88C3×#103—22no—
89C3×#103—22no—
90C3×C4×C8D411yescharacteristic
91C3×C4×C8D411yescharacteristic
92C3×C4×C8C2311yescharacteristic
93C3×C4×C8C2311yescharacteristic
94#835—22no—
95#835—22no—
96#830—22no—
97#830—21no—
98(C2×D3)⋊C8—21no—
99(C2×D3)⋊C8—21no—
100#815—21no—
101#815—21no—
102Dic3⋊C8—21no—
103Dic3⋊C8—21no—
104#810—22no—
105#809—22no—
106#808—22no—
107#807—22no—
108C4⋊(C3⋊C8)C2311yescharacteristic
109C4⋊(C3⋊C8)C2311yescharacteristic
110C4⋊(C3⋊C8)C2312yes—
111C4⋊(C3⋊C8)C2312yes—
112C4×C3⋊C8—21no—
113C4×C3⋊C8—22no—
114#533C2×D312yes—
115C4×#41—31no—
116#500C2×D311yescharacteristic
117#494C2×D311yescharacteristic
118#487C2×D312yes—
119#485—62no—
120#475—32no—
121#474—32no—
122#467—21no—
123#467—21no—
124#462—32no—
125#461—32no—
126#452—62no—
127#446—31no—
128#446—31no—
129C4×Q16—21no—
130C4×Q16—21no—
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×C8C2×D311yescharacteristic
142#335—32no—
143#334—31no—
144#334—31no—
145#332—32no—
146#332—32no—
147#332—32no—
148#332—32no—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4#4111yescharacteristic
154C2×#67—21no—
155#235—42no—
156C2×D12C2×D411yescharacteristic
157C2×C4×D3—21no—
158C2×C4×D3—21no—
159C2×C4×D3—21no—
160C2×Dic6C2×D411yescharacteristic
161C3×Q16—42no—
162C3×Q16—42no—
163C2×C3×C8—22no—
164C2×C3×C8—41no—
165C2×C3×C8C2×D411yescharacteristic
166C2×C3×C8C2×D411yescharacteristic
167C2×C3×C8C2×D411yescharacteristic
168C2×C3×C8C2×D411yescharacteristic
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—42no—
175C3×C22⋊C4—42no—
176C3×C42—42no—
177C3×C42C2×D411yescharacteristic
178C3×C42C2×D411yescharacteristic
179C3×C42#4111yescharacteristic
180C3×C42C2411yescharacteristic
181#217—21no—
182#216—44no—
183#215—44no—
184(C2×D3)⋊C4—21no—
185(C2×D3)⋊C4—21no—
186C4⋊Dic3C2×D411yescharacteristic
187C4⋊Dic3C2×D411yescharacteristic
188Dic3⋊C4—21no—
189Dic3⋊C4—21no—
190C4×Dic3—21no—
191C4×Dic3—21no—
192C4×Dic3—21no—
193C2×C3⋊C8—22no—
194C2×C3⋊C8—22no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197ΓL(1,𝔽25)—42no—
198ΓL(1,𝔽25)—42no—
199C2×#41—31no—
200C2×Q16—21no—
201C2×Q16—21no—
202C2×Q16—62no—
203C2×SD16—62no—
204C2×M16—61no—
205C2×M16—61no—
206C22×C8#6711yescharacteristic
207C22×C8#6711yescharacteristic
208C4⋊Q8C22×D312yes—
209C4⋊Q8C22×D312yes—
210#125—22no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—31no—
216C4×Q8—31no—
217C4×D4—31no—
218C4×D4—31no—
219C4×D4—61no—
220C4×D4—61no—
221#117—22no—
222#117—31no—
223#117—61no—
224C2×C42—61no—
225C2×C42C22×D311yescharacteristic
226#107—21no—
227#107—21no—
228#107—62no—
229#106—62no—
230C4⋊C8—31no—
231C4⋊C8—31no—
232C4⋊C8—32no—
233C4⋊C8—32no—
234C4⋊C8—61no—
235C4⋊C8—61no—
236#103—22no—
237#103—22no—
238#103—62no—
239#103—62no—
240#103—62no—
241#102—62no—
242C22⋊C8—62no—
243C22⋊C8—61no—
244C22⋊C8—61no—
245C22⋊C8—61no—
246#97—61no—
247#97—61no—
248C4×C8—62no—
249C4×C8—61no—
250C4×C8#6711yescharacteristic
251C4×C8#6711yescharacteristic
252C4×C8C22×D311yescharacteristic
253C4×C8C22×D311yescharacteristic
254C23×C3#12111yescharacteristic
255C22×D3—21no—
256C3×Q8—42no—
257C3×Q8—42no—
258C3×Q8—42no—
259C3×Q8—42no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4#12111yescharacteristic
267C2×C3×C4#12111yescharacteristic
268C2×C3×C4#12111yescharacteristic
269C2×C3×C4C22×D411yescharacteristic
270C2×C3×C4C22×D411yescharacteristic
271C2×C3×C4C2×#4111yescharacteristic
272#67—42no—
273C2×Dic3—21no—
274C2×Dic3—21no—
275C2×Dic3—21no—
276D12—22no—
277C4×D3—42no—
278C4×D3—41no—
279C4×D3—41no—
280C4×D3—42no—
281Dic6—22no—
282C3×C8—21no—
283C3×C8—21no—
284C3×C8—21no—
285C3×C8#13512yes—
286C3⋊C8—41no—
287C3⋊C8—41no—
288C3⋊C8—42no—
289C3⋊C8—42no—
290#41—122no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—31no—
294C2×D4—31no—
295C2×D4—61no—
296C22×C4—61no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4D3×D411yescharacteristic
300C22×C4#23711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302Q16—42no—
303Q16—42no—
304Q16—124no—
305SD16—124no—
306M16—122no—
307M16—122no—
308C2×C8—22no—
309C2×C8—41no—
310C2×C8—62no—
311C2×C8—62no—
312C2×C8—61no—
313C2×C8—61no—
314C2×C8C2×#6711yescharacteristic
315C2×C8C2×#6711yescharacteristic
316C2×C8C2×#6711yescharacteristic
317C2×C8C2×#6711yescharacteristic
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—31no—
323C4⋊C4—31no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C22⋊C4—42no—
329C22⋊C4—61no—
330C22⋊C4—61no—
331C22⋊C4—61no—
332C42—42no—
333C42—61no—
334C42—61no—
335C42—61no—
336C42D3×D411yescharacteristic
337C42#23711yescharacteristic
338C42C2×#6711yescharacteristic
339C42C23×D311yescharacteristic
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3C2×#12111yescharacteristic
344C2×D3—41no—
345C2×D3—41no—
346C2×D3—41no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—82no—
351C3×C4—82no—
352C3×C4#47312yes—
353C3×C4C2×#13511yescharacteristic
354C3×C4#57512yes—
355C3×C4#57511yescharacteristic
356Dic3—41no—
357Dic3—41no—
358Dic3—41no—
359C23—61no—
360C23#93811yescharacteristic
361Q8—42no—
362Q8—42no—
363Q8—42no—
364Q8—42no—
365Q8—62no—
366D4—62no—
367D4—122no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4#93811yescharacteristic
382C2×C4#93811yescharacteristic
383C2×C4#93811yescharacteristic
384C2×C4C2×D3×D411yescharacteristic
385C2×C4C2×#23711yescharacteristic
386C2×C4C22×#6711yescharacteristic
387C8—21no—
388C8—21no—
389C8—21no—
390C8—122no—
391C8—122no—
392C8—121no—
393C8—121no—
394C8#95012yes—
395C2×C3—41no—
396C2×C3#303511yescharacteristic
397C2×C3#305411yescharacteristic
398C2×C3C2×#47311yescharacteristic
399D3—81no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22C2×#93811yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—82no—
411C4—82no—
412C4—121no—
413C4—121no—
414C4—121no—
415C4#519212yes—
416C4#576812yes—
417C4C2×#95011yescharacteristic
418C4#582211yescharacteristic
419C3#1916111yescharacteristic, complemented, Sylow
420C2—41no—
421C2—241no—
422C2#8329611yescharacteristic
423C2C2×#519211yescharacteristic
424C2#8391511yescharacteristic
subgroupG/H
centerC2×C4#938
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3170C2
socleC22×C3C2×#121
prime radicalC2×#235D4
hypercenter#3170D3
hypocenterC3#19161
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
1. C3×#3170C2semidirect
4. #71829C2semidirect
5. #71379C2semidirect
6. #71373C2semidirect
7. #71309C2semidirect
8. #71306C2semidirect
9. #71242C2semidirect
10. #71234C2semidirect
11. #69729C2semidirect
48. #3170D3semidirect
419. C3#19161semidirect

Maximal subgroups

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

subgroupindex
C3×#31702
#833522
#784472
#718292
#713792
#713732
#713092
#713062
#712422
#712342
#697292
#191613