Finite Group Databasebeta

← #10778357

Subgroups — #10778357

Order 768.

650 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#83216C211yesmaximal, characteristic
2C2×#5093C211yesmaximal, characteristic, complemented
3#79268C211yesmaximal, characteristic
4#79041C211yesmaximal, characteristic
5C3×#1548C211yesmaximal, characteristic, complemented
6#70745C211yesmaximal, characteristic, complemented
7#70506C211yesmaximal, characteristic, complemented
8#68503C214yesmaximal, complemented
9#68499C214yesmaximal, complemented
10#19199—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×#907—21no—
13#5477C2211yescharacteristic
14C2×#869C2211yescharacteristic
15C3×#431C2211yescharacteristic
16C3×#423C2211yescharacteristic
17#5129—21no—
18#5093—21no—
19#5093—21no—
20#5093C2212yes—
21#5093C2212yes—
22#5088—21no—
23#5084C2214yes—
24#5080—21no—
25C2×#810—21no—
26C2×#810—21no—
27#4968—22no—
28#4925C2211yescharacteristic
29C2×C4×C3⋊C8C2211yescharacteristic
30#4846C2214yes—
31#4838—21no—
32#4838C2212yes—
33#4834—21no—
34#4834C2212yes—
35#4736—22no—
36#4694—21no—
37#4682—21no—
38C3×#333C2214yes—
39#4482C2214yes—
40#4480C2212yes—
41#4479C2212yes—
42#3227—31no—
43#3226—31no—
44C2×#487—31no—
45#2931—31no—
46#1703—34no—
47#1683—34no—
48#1564—31no—
49#1548D311yescharacteristic, complemented
50#1546—31no—
51C22×C3×Q8—21no—
52C2×#235—21no—
53C22×Dic6—21no—
54C2×C3×M16—21no—
55C3×C4⋊Q8—21no—
56C3×C4⋊Q8—21no—
57C3×C4⋊Q8—21no—
58C3×C4⋊Q8C2312yes—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62#930—21no—
63#930D412yes—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C22×C3⋊C8D411yescharacteristic
67C22×C3⋊C8D411yescharacteristic
68#907—44no—
69C2×ΓL(1,𝔽25)—22no—
70C2×D3×C8—22no—
71#875—21no—
72#874D411yescharacteristic
73C4×D12—21no—
74C4×D12C2312yes—
75#869—21no—
76#869—21no—
77#869—21no—
78#869D411yescharacteristic
79#869C2312yes—
80C3×C4⋊C8—21no—
81C3×C4⋊C8C2312yes—
82C3×C4⋊C8C2312yes—
83C3×C22⋊C8—21no—
84C3×C4×C8—21no—
85#830—21no—
86#821—24no—
87(C2×D3)⋊C8—22no—
88(C2×D3)⋊C8—22no—
89#818—24no—
90#810—22no—
91#810—22no—
92#810—22no—
93#810—22no—
94#809—24no—
95#808—24no—
96C4⋊(C3⋊C8)C2312yes—
97C4×C3⋊C8—21no—
98C4×C3⋊C8D411yescharacteristic
99C4×C3⋊C8D411yescharacteristic
100C4×C3⋊C8C2311yescharacteristic
101C4×C3⋊C8C2311yescharacteristic
102C2×#131—61no—
103#532—31no—
104C2×C4⋊Q8—31no—
105C2×C4⋊Q8C2×D311yescharacteristic
106#500—61no—
107#498—61no—
108#487—32no—
109#487—32no—
110#487—61no—
111#487—61no—
112#486—61no—
113#486—61no—
114#481—62no—
115#480—62no—
116#476—34no—
117#474—34no—
118#436—32no—
119#436—61no—
120C8×D4—32no—
121C8×D4—61no—
122#431—31no—
123#431C2×D311yescharacteristic
124#423C2×D311yescharacteristic
125C2×#103—61no—
126C2×#103—61no—
127#408—61no—
128C2×C4×C8—31no—
129#335—32no—
130#334—32no—
131#333C2×D314yes—
132#332—34no—
133C2×C3×Q8—22no—
134C2×C3×Q8—41no—
135C2×C3×Q8—82no—
136C22×C3×C4—41no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C2×#67—22no—
141C22×Dic3—41no—
142#235—42no—
143#235—42no—
144C2×D12—21no—
145C2×D12—21no—
146C2×D12C2×D412yes—
147C2×C4×D3—22no—
148C2×Dic6—21no—
149C2×Dic6—22no—
150C2×Dic6—42no—
151C2×Dic6—42no—
152C2×Dic6—41no—
153C3×M16—42no—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—41no—
159C3×C4⋊C4—41no—
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42#4111yescharacteristic
163C3×C42C2411yescharacteristic
164(C2×D3)⋊C4—22no—
165(C2×D3)⋊C4—22no—
166C4⋊Dic3—21no—
167C4⋊Dic3—22no—
168C4⋊Dic3—22no—
169C4⋊Dic3—41no—
170C4⋊Dic3—41no—
171C4⋊Dic3C2×D412yes—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8C2×D411yescharacteristic
176C2×C3⋊C8C2×D411yescharacteristic
177C2×C3⋊C8C2×D411yescharacteristic
178C2×C3⋊C8C2×D411yescharacteristic
179ΓL(1,𝔽25)—44no—
180D3×C8—44no—
181C2×#41—61no—
182C22×Q8—21no—
183C22×Q8—61no—
184#131—124no—
185C2×M16—21no—
186C2×M16—62no—
187C22×C8—31no—
188C22×C8—31no—
189C22×C8—62no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—31no—
194C4⋊Q8—32no—
195C4⋊Q8—61no—
196C4⋊Q8—61no—
197C4⋊Q8—61no—
198C4⋊Q8C22×D312yes—
199#127—31no—
200#124—61no—
201#121—32no—
202#121—61no—
203C4×D4—32no—
204C4×D4—61no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—61no—
208C2×C4⋊C4—61no—
209C2×C42C22×D311yescharacteristic
210#107—64no—
211#106—64no—
212C4⋊C8—21no—
213C4⋊C8—32no—
214C4⋊C8C22×D312yes—
215C4⋊C8C22×D312yes—
216#103—62no—
217#103—62no—
218#103—62no—
219#103—62no—
220#102—64no—
221#102—64no—
222C22⋊C8—21no—
223C22⋊C8—62no—
224C22⋊C8—62no—
225C22⋊C8—61no—
226C4×C8—21no—
227C4×C8—31no—
228C4×C8—31no—
229C4×C8—31no—
230C4×C8—31no—
231C4×C8—61no—
232C23×C3#12111yescharacteristic
233C22×D3—22no—
234C3×Q8—82no—
235C3×Q8—82no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—42no—
239C2×C3×C4—82no—
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#13512yes—
244C2×C3×C4AGL(1,ℤ/8)12yes—
245C2×C3×C4C22×D411yescharacteristic
246C2×C3×C4C22×D411yescharacteristic
247C2×C3×C4C2×#4111yescharacteristic
248#67—44no—
249C2×Dic3—22no—
250C2×Dic3—42no—
251C2×Dic3—82no—
252D12—24no—
253D12—42no—
254D12—42no—
255C4×D3—42no—
256C4×D3—42no—
257Dic6—42no—
258Dic6—42no—
259Dic6—42no—
260Dic6—82no—
261C3×C8—42no—
262C3×C8—42no—
263C3⋊C8—21no—
264C3⋊C8—21no—
265C3⋊C8—21no—
266C3⋊C8—21no—
267C3⋊C8—42no—
268#41—122no—
269#41—122no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—61no—
273C2×Q8—62no—
274C2×Q8—82no—
275C2×Q8—121no—
276C2×Q8—122no—
277C2×Q8—122no—
278C2×D4—32no—
279C2×D4—62no—
280C2×D4—61no—
281C2×D4—61no—
282C22×C4—41no—
283C22×C4—62no—
284C22×C4—121no—
285C22×C4#23511yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4D3×D411yescharacteristic
288M16—42no—
289M16—124no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—31no—
293C2×C8—31no—
294C2×C8—31no—
295C2×C8—31no—
296C2×C8—62no—
297C2×C8—62no—
298C2×C8—62no—
299C2×C8—124no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—32no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—61no—
306C4⋊C4—62no—
307C4⋊C4—62no—
308C4⋊C4—121no—
309C4⋊C4—121no—
310C22⋊C4—62no—
311C22⋊C4—62no—
312C42#23511yescharacteristic
313C42D3×D411yescharacteristic
314C42D3×D411yescharacteristic
315C42C23×D311yescharacteristic
316C22×C3—21no—
317C22×C3—21no—
318C22×C3#46312yes—
319C22×C3C2×#12111yescharacteristic
320C2×D3—42no—
321C2×D3—42no—
322C2×D3—42no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—82no—
326C3×C4#46312yes—
327C3×C4#47112yes—
328C3×C4C2×#13511yescharacteristic
329C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
330C3×C4#57812yes—
331Dic3—42no—
332Dic3—82no—
333C23—62no—
334C23#88411yescharacteristic
335Q8—82no—
336Q8—82no—
337Q8—122no—
338Q8—122no—
339Q8—122no—
340Q8—242no—
341D4—64no—
342D4—124no—
343D4—122no—
344D4—122no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—42no—
348C2×C4—62no—
349C2×C4—82no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4—242no—
354C2×C4#88411yescharacteristic
355C2×C4#88411yescharacteristic
356C2×C4#88411yescharacteristic
357C2×C4#91412yes—
358C2×C4#91912yes—
359C2×C4C2×#23511yescharacteristic
360C2×C4C2×D3×D411yescharacteristic
361C2×C4C2×D3×D411yescharacteristic
362C8—42no—
363C8—42no—
364C8—61no—
365C8—61no—
366C8—61no—
367C8—61no—
368C8—122no—
369C2×C3—22no—
370C2×C3C2×#46311yescharacteristic
371C2×C3#305911yescharacteristic
372C2×C3C2×#47111yescharacteristic
373D3—82no—
374C22—21no—
375C22—21no—
376C22—122no—
377C22—122no—
378C22—122no—
379C22#481912yes—
380C22C2×#88411yescharacteristic
381C4—21no—
382C4—21no—
383C4—82no—
384C4—122no—
385C4—242no—
386C4#474812yes—
387C4#481912yes—
388C4C2×#91411yescharacteristic
389C4C2×#91911yescharacteristic
390C4#577912yes—
391C3#1919911yescharacteristic, complemented, Sylow
392C2—22no—
393C2—242no—
394C2C2×#474811yescharacteristic
395C2C2×#481911yescharacteristic
396C2#7896411yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1548C2
socleC22×C3C2×#121
prime radical#5477C22
hypercenter#1548D3
hypocenterC3#19199
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
2. C2×#5093C2semidirect
5. C3×#1548C2semidirect
6. #70745C2semidirect
7. #70506C2semidirect
8. #68503C2semidirect
9. #68499C2semidirect
49. #1548D3semidirect
391. C3#19199semidirect

Maximal subgroups

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

subgroupindex
#832162
C2×#50932
#792682
#790412
C3×#15482
#707452
#705062
#685032
#684992
#191993