Finite Group Databasebeta

← #10777927

Subgroups — #10777927

Order 768.

642 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3169C211yesmaximal, characteristic, complemented
2#83299C212yesmaximal, complemented
3#78447C211yesmaximal, characteristic, complemented
4#71805C212yesmaximal, complemented
5#71380C211yesmaximal, characteristic, complemented
6#71374C211yesmaximal, characteristic, complemented
7#71287C211yesmaximal, characteristic, complemented
8#71283C211yesmaximal, characteristic, complemented
9#71210C212yesmaximal, complemented
10#71202C212yesmaximal, complemented
11#69729C211yesmaximal, characteristic, complemented
12#19160—31nomaximal, Sylow
13C3×#532C2212yes—
14C4×#235C2211yescharacteristic
15C3×#500C2211yescharacteristic
16C3×#493C2211yescharacteristic
17C3×#486C2212yes—
18C3×#466—21no—
19C3×#466—21no—
20C3×C4×D8—21no—
21C3×C4×D8—21no—
22C2×C3×C4×C8C2211yescharacteristic
23#5111—21no—
24#5111—21no—
25#5091C2212yes—
26#5084C2212yes—
27#5075C2212yes—
28#5072C2212yes—
29#5037—22no—
30#5035—22no—
31C4×#214—22no—
32C4×#213—22no—
33#4925C2212yes—
34#4925C2211yescharacteristic
35#4688C2211yescharacteristic
36#4688C2211yescharacteristic
37C4×ΓL(1,𝔽25)—21no—
38#4685—21no—
39#4679C2211yescharacteristic
40#4679C2211yescharacteristic
41#4546C2212yes—
42#4546C2212yes—
43#4544C2212yes—
44#4544C2212yes—
45#4458C2212yes—
46#4457C2211yescharacteristic
47#4457C2211yescharacteristic
48#3169D311yescharacteristic, complemented
49#3106—32no—
50#2965—31no—
51#1698—32no—
52#1697—32no—
53#1696—31no—
54#1695—31no—
55#1688—31no—
56#1687—31no—
57#1564—31no—
58#1543—32no—
59C2×C3×#41—22no—
60C2×#235D411yescharacteristic
61C2×C3×D8—21no—
62C2×C3×D8—21no—
63C22×C3×C8D411yescharacteristic
64C22×C3×C8D411yescharacteristic
65C3×C4⋊Q8C2312yes—
66C3×#127C2312yes—
67C3×#124—22no—
68C3×#121—22no—
69C3×#121—22no—
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C2×C3×C42C2311yescharacteristic
73C2×#214—22no—
74C2×#213—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×#102—22no—
89C3×#102—22no—
90C3×C4×C8D411yescharacteristic
91C3×C4×C8D411yescharacteristic
92C3×C4×C8C2311yescharacteristic
93C3×C4×C8C2311yescharacteristic
94#832—22no—
95#832—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#532C2×D312yes—
115C4×#41—31no—
116#500C2×D311yescharacteristic
117#493C2×D311yescharacteristic
118#486C2×D312yes—
119#482—62no—
120#477—32no—
121#476—32no—
122#466—21no—
123#466—21no—
124#460—32no—
125#459—32no—
126#449—62no—
127#446—31no—
128#446—31no—
129C4×SD16—62no—
130C4×D8—21no—
131C4×D8—21no—
132C4×D8—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#333—32no—
146#333—32no—
147#331—32no—
148#331—32no—
149C3×#41—84no—
150C2×C3×Q8—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—42no—
155C22×C3×C4—42no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4#4111yescharacteristic
159C2×#67—21no—
160#235—42no—
161C2×D12C2×D411yescharacteristic
162C2×C4×D3—21no—
163C2×C4×D3—21no—
164C2×C4×D3—21no—
165C2×Dic6C2×D411yescharacteristic
166C3×D8—42no—
167C3×D8—42no—
168C2×C3×C8—22no—
169C2×C3×C8—41no—
170C2×C3×C8C2×D411yescharacteristic
171C2×C3×C8C2×D411yescharacteristic
172C2×C3×C8C2×D411yescharacteristic
173C2×C3×C8C2×D411yescharacteristic
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C22⋊C4—42no—
177C3×C22⋊C4—42no—
178C3×C42C2×D411yescharacteristic
179C3×C42C2×D411yescharacteristic
180C3×C42#4111yescharacteristic
181C3×C42C2411yescharacteristic
182#217—21no—
183#214—44no—
184#213—44no—
185(C2×D3)⋊C4—21no—
186(C2×D3)⋊C4—21no—
187C4⋊Dic3C2×D411yescharacteristic
188C4⋊Dic3C2×D411yescharacteristic
189Dic3⋊C4—21no—
190Dic3⋊C4—21no—
191C4×Dic3—21no—
192C4×Dic3—21no—
193C4×Dic3—21no—
194C2×C3⋊C8—22no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—21no—
198ΓL(1,𝔽25)—42no—
199ΓL(1,𝔽25)—42no—
200C2×#41—22no—
201C2×#41—31no—
202C2×SD16—62no—
203C2×D8—21no—
204C2×D8—21no—
205C2×D8—62no—
206C2×M16—61no—
207C2×M16—61no—
208C22×C8#6711yescharacteristic
209C22×C8#6711yescharacteristic
210C4⋊Q8C22×D312yes—
211#127C22×D312yes—
212#124—22no—
213#121—22no—
214#121—22no—
215C4×Q8—31no—
216C4×Q8—31no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4—31no—
220C4×D4—31no—
221C4×D4—61no—
222C4×D4—61no—
223#117—31no—
224#117—61no—
225C2×C42—61no—
226C2×C42C22×D311yescharacteristic
227#107—21no—
228#107—21no—
229#107—62no—
230#106—62no—
231C4⋊C8—31no—
232C4⋊C8—31no—
233C4⋊C8—32no—
234C4⋊C8—32no—
235C4⋊C8—61no—
236C4⋊C8—61no—
237#103—62no—
238#102—22no—
239#102—22no—
240#102—62no—
241#102—62no—
242#102—62no—
243C22⋊C8—62no—
244C22⋊C8—61no—
245C22⋊C8—61no—
246C22⋊C8—61no—
247#97—61no—
248#97—61no—
249C4×C8—62no—
250C4×C8—61no—
251C4×C8#6711yescharacteristic
252C4×C8#6711yescharacteristic
253C4×C8C22×D311yescharacteristic
254C4×C8C22×D311yescharacteristic
255C23×C3—42no—
256C23×C3#12111yescharacteristic
257C22×D3—21no—
258C3×Q8—82no—
259C3×D4—42no—
260C3×D4—42no—
261C3×D4—42no—
262C3×D4—42no—
263C3×D4—82no—
264C3×D4—84no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—42no—
270C2×C3×C4—84no—
271C2×C3×C4#12111yescharacteristic
272C2×C3×C4#12111yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4C22×D411yescharacteristic
275C2×C3×C4C22×D411yescharacteristic
276C2×C3×C4C2×#4111yescharacteristic
277#67—42no—
278C2×Dic3—21no—
279C2×Dic3—21no—
280C2×Dic3—21no—
281D12—22no—
282C4×D3—42no—
283C4×D3—41no—
284C4×D3—41no—
285C4×D3—42no—
286Dic6—22no—
287C3×C8—21no—
288C3×C8—21no—
289C3×C8—21no—
290C3×C8#13512yes—
291C3⋊C8—41no—
292C3⋊C8—41no—
293C3⋊C8—42no—
294C3⋊C8—42no—
295#41—84no—
296#41—122no—
297C2×Q8—22no—
298C2×Q8—31no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—31no—
303C2×D4—42no—
304C2×D4—61no—
305C22×C4—42no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4D3×D411yescharacteristic
310C22×C4#23711yescharacteristic
311C22×C4C2×#6711yescharacteristic
312SD16—124no—
313D8—42no—
314D8—42no—
315D8—124no—
316M16—122no—
317M16—122no—
318C2×C8—22no—
319C2×C8—41no—
320C2×C8—62no—
321C2×C8—62no—
322C2×C8—61no—
323C2×C8—61no—
324C2×C8C2×#6711yescharacteristic
325C2×C8C2×#6711yescharacteristic
326C2×C8C2×#6711yescharacteristic
327C2×C8C2×#6711yescharacteristic
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—31no—
331C4⋊C4—31no—
332C4⋊C4—61no—
333C4⋊C4—61no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—61no—
337C22⋊C4—61no—
338C22⋊C4—61no—
339C42—61no—
340C42—61no—
341C42—61no—
342C42D3×D411yescharacteristic
343C42#23711yescharacteristic
344C42C2×#6711yescharacteristic
345C42C23×D311yescharacteristic
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—82no—
350C22×C3—82no—
351C22×C3—82no—
352C22×C3C2×#12111yescharacteristic
353C2×D3—41no—
354C2×D3—41no—
355C2×D3—41no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—82no—
360C3×C4#47212yes—
361C3×C4C2×#13511yescharacteristic
362C3×C4#57512yes—
363C3×C4#57511yescharacteristic
364Dic3—41no—
365Dic3—41no—
366Dic3—41no—
367C23—42no—
368C23—61no—
369C23#93811yescharacteristic
370Q8—62no—
371Q8—82no—
372D4—42no—
373D4—42no—
374D4—42no—
375D4—42no—
376D4—62no—
377D4—82no—
378D4—84no—
379D4—122no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—61no—
386C2×C4—61no—
387C2×C4—61no—
388C2×C4—84no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4#93811yescharacteristic
394C2×C4#93811yescharacteristic
395C2×C4#93811yescharacteristic
396C2×C4C2×D3×D411yescharacteristic
397C2×C4C2×#23711yescharacteristic
398C2×C4C22×#6711yescharacteristic
399C8—21no—
400C8—21no—
401C8—21no—
402C8—122no—
403C8—122no—
404C8—121no—
405C8—121no—
406C8#95012yes—
407C2×C3—41no—
408C2×C3—162no—
409C2×C3#303511yescharacteristic
410C2×C3#305411yescharacteristic
411C2×C3C2×#47211yescharacteristic
412D3—81no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—121no—
420C22—121no—
421C22—121no—
422C22C2×#93811yescharacteristic
423C4—21no—
424C4—21no—
425C4—21no—
426C4—82no—
427C4—121no—
428C4—121no—
429C4—121no—
430C4#516012yes—
431C4#576812yes—
432C4C2×#95011yescharacteristic
433C4#582211yescharacteristic
434C3#1916011yescharacteristic, complemented, Sylow
435C2—41no—
436C2—162no—
437C2—241no—
438C2#8329611yescharacteristic
439C2C2×#516011yescharacteristic
440C2#8391511yescharacteristic
subgroupG/H
centerC2×C4#938
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#3169C2
socleC22×C3C2×#121
hypercenter#3169D3
hypocenterC3#19160
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. C3×#3169C2semidirect
2. #83299C2semidirect
3. #78447C2semidirect
4. #71805C2semidirect
5. #71380C2semidirect
6. #71374C2semidirect
7. #71287C2semidirect
8. #71283C2semidirect
9. #71210C2semidirect
10. #71202C2semidirect
11. #69729C2semidirect
48. #3169D3semidirect
434. C3#19160semidirect

Maximal subgroups

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

subgroupindex
C3×#31692
#832992
#784472
#718052
#713802
#713742
#712872
#712832
#712102
#712022
#697292
#191603