Finite Group Databasebeta

← #10778264

Subgroups — #10778264

Order 768.

714 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
1#83215C211yesmaximal, characteristic
2C2×#5092C211yesmaximal, characteristic, complemented
3#79268C211yesmaximal, characteristic
4#79041C211yesmaximal, characteristic
5C3×#1546C211yesmaximal, characteristic, complemented
6#70745C211yesmaximal, characteristic, complemented
7#70506C211yesmaximal, characteristic, complemented
8#68449C214yesmaximal, complemented
9#68441C214yesmaximal, complemented
10#19188—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C22×#216—21no—
13C22×#216—21no—
14C2×#907—21no—
15#5477C2211yescharacteristic
16C2×#869C2211yescharacteristic
17C3×#431C2211yescharacteristic
18C3×#423C2211yescharacteristic
19#5129—21no—
20#5092—21no—
21#5092—21no—
22#5092C2212yes—
23#5092C2212yes—
24#5088—21no—
25#5087C2214yes—
26#5078—21no—
27#4967—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×#332C2214yes—
39#4482C2214yes—
40#4480C2212yes—
41#4479C2212yes—
42#3226—31no—
43#3226—31no—
44C2×#494—31no—
45#2931—31no—
46#1689—34no—
47#1669—34no—
48#1564—31no—
49#1546—31no—
50#1546D311yescharacteristic, complemented
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
62C2×#216—22no—
63C2×#216—22no—
64C2×#216—22no—
65C2×#216—22no—
66C2×#216—44no—
67C2×#216—44no—
68#930—21no—
69#930D412yes—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C22×C3⋊C8D411yescharacteristic
73C22×C3⋊C8D411yescharacteristic
74#907—44no—
75C2×ΓL(1,𝔽25)—22no—
76C2×D3×C8—22no—
77#875—21no—
78#874D411yescharacteristic
79C4×D12—21no—
80C4×D12C2312yes—
81#869—21no—
82#869—21no—
83#869—21no—
84#869D411yescharacteristic
85#869C2312yes—
86C3×C4⋊C8—21no—
87C3×C4⋊C8C2312yes—
88C3×C4⋊C8C2312yes—
89C3×C22⋊C8—21no—
90C3×C4×C8—21no—
91#830—21no—
92#821—24no—
93(C2×D3)⋊C8—22no—
94(C2×D3)⋊C8—22no—
95#818—24no—
96#809—24no—
97#807—24no—
98C4⋊(C3⋊C8)C2312yes—
99C4×C3⋊C8—21no—
100C4×C3⋊C8D411yescharacteristic
101C4×C3⋊C8D411yescharacteristic
102C4×C3⋊C8C2311yescharacteristic
103C4×C3⋊C8C2311yescharacteristic
104C22×Q16—61no—
105C22×Q16—61no—
106C2×#131—61no—
107#532—31no—
108C2×C4⋊Q8—31no—
109C2×C4⋊Q8C2×D311yescharacteristic
110#500—61no—
111#500—61no—
112#494—32no—
113#494—32no—
114#494—61no—
115#494—61no—
116#486—61no—
117#486—61no—
118#480—62no—
119#480—62no—
120#474—34no—
121#474—34no—
122#436—32no—
123#436—61no—
124C8×D4—32no—
125C8×D4—61no—
126#431—31no—
127#431C2×D311yescharacteristic
128#423C2×D311yescharacteristic
129#408—61no—
130C2×C4×C8—31no—
131#335—32no—
132#334—32no—
133#332—34no—
134#332C2×D314yes—
135C2×C3×Q8—22no—
136C2×C3×Q8—41no—
137C2×C3×Q8—42no—
138C2×C3×Q8—42no—
139C22×C3×C4—41no—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C2×#67—22no—
144C22×Dic3—41no—
145#235—42no—
146#235—42no—
147C2×D12—21no—
148C2×D12—21no—
149C2×D12C2×D412yes—
150C2×C4×D3—22no—
151C2×Dic6—21no—
152C2×Dic6—22no—
153C2×Dic6—42no—
154C2×Dic6—42no—
155C2×Dic6—41no—
156C3×M16—42no—
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—41no—
162C3×C4⋊C4—41no—
163C3×C42C2×D411yescharacteristic
164C3×C42C2×D411yescharacteristic
165C3×C42#4111yescharacteristic
166C3×C42C2411yescharacteristic
167#216—44no—
168#216—44no—
169#216—44no—
170#216—44no—
171(C2×D3)⋊C4—22no—
172(C2×D3)⋊C4—22no—
173C4⋊Dic3—21no—
174C4⋊Dic3—22no—
175C4⋊Dic3—22no—
176C4⋊Dic3—41no—
177C4⋊Dic3—41no—
178C4⋊Dic3C2×D412yes—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8D814yes—
182C2×C3⋊C8C2×D411yescharacteristic
183C2×C3⋊C8C2×D411yescharacteristic
184C2×C3⋊C8C2×D411yescharacteristic
185C2×C3⋊C8C2×D411yescharacteristic
186ΓL(1,𝔽25)—44no—
187D3×C8—44no—
188C2×#41—61no—
189C22×Q8—21no—
190C22×Q8—61no—
191C2×Q16—62no—
192C2×Q16—62no—
193C2×Q16—62no—
194C2×Q16—62no—
195C2×Q16—124no—
196C2×Q16—124no—
197#131—124no—
198C2×M16—21no—
199C2×M16—62no—
200C22×C8—31no—
201C22×C8—31no—
202C22×C8—62no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—31no—
207C4⋊Q8—32no—
208C4⋊Q8—61no—
209C4⋊Q8—61no—
210C4⋊Q8—61no—
211C4⋊Q8C22×D312yes—
212#127—31no—
213#124—61no—
214#121—32no—
215#121—61no—
216C4×D4—32no—
217C4×D4—61no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—61no—
222C2×C42C22×D311yescharacteristic
223#107—64no—
224#107—64no—
225C4⋊C8—21no—
226C4⋊C8—32no—
227C4⋊C8C22×D312yes—
228C4⋊C8C22×D312yes—
229#102—64no—
230#102—64no—
231C22⋊C8—21no—
232C22⋊C8—62no—
233C22⋊C8—62no—
234C22⋊C8—61no—
235C4×C8—21no—
236C4×C8—31no—
237C4×C8—31no—
238C4×C8—31no—
239C4×C8—31no—
240C4×C8—61no—
241C23×C3#12111yescharacteristic
242C22×D3—22no—
243C3×Q8—42no—
244C3×Q8—42no—
245C3×Q8—82no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—42no—
249C2×C3×C4—82no—
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12111yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4C2×D812yes—
254C2×C3×C4AGL(1,ℤ/8)12yes—
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C22×D411yescharacteristic
257C2×C3×C4C2×#4111yescharacteristic
258#67—44no—
259C2×Dic3—22no—
260C2×Dic3—42no—
261C2×Dic3—82no—
262D12—24no—
263D12—42no—
264D12—42no—
265C4×D3—42no—
266C4×D3—42no—
267Dic6—42no—
268Dic6—42no—
269Dic6—42no—
270Dic6—82no—
271C3×C8—42no—
272C3×C8—42no—
273C3⋊C8—21no—
274C3⋊C8—21no—
275C3⋊C8—42no—
276C3⋊C8C2×D812yes—
277C3⋊C8C2×D812yes—
278#41—122no—
279#41—122no—
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—61no—
285C2×Q8—62no—
286C2×Q8—121no—
287C2×Q8—122no—
288C2×Q8—122no—
289C2×D4—32no—
290C2×D4—62no—
291C2×D4—61no—
292C2×D4—61no—
293C22×C4—41no—
294C22×C4—62no—
295C22×C4—121no—
296C22×C4#23511yescharacteristic
297C22×C4D3×D411yescharacteristic
298C22×C4D3×D411yescharacteristic
299Q16—124no—
300Q16—124no—
301Q16—124no—
302Q16—124no—
303M16—42no—
304M16—124no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—34no—
308C2×C8—31no—
309C2×C8—31no—
310C2×C8—31no—
311C2×C8—31no—
312C2×C8—62no—
313C2×C8—62no—
314C2×C8—124no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—32no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—61no—
321C4⋊C4—62no—
322C4⋊C4—62no—
323C4⋊C4—121no—
324C4⋊C4—121no—
325C22⋊C4—62no—
326C22⋊C4—62no—
327C42#23511yescharacteristic
328C42D3×D411yescharacteristic
329C42D3×D411yescharacteristic
330C42C23×D311yescharacteristic
331C22×C3—21no—
332C22×C3—21no—
333C22×C3#45912yes—
334C22×C3C2×#12111yescharacteristic
335C2×D3—42no—
336C2×D3—42no—
337C2×D3—42no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—82no—
341C3×C4#45912yes—
342C3×C4#47312yes—
343C3×C4C22×D811yescharacteristic
344C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
345C3×C4#57812yes—
346Dic3—42no—
347Dic3—82no—
348C23—62no—
349C23#88411yescharacteristic
350Q8—42no—
351Q8—42no—
352Q8—82no—
353Q8—122no—
354Q8—122no—
355Q8—122no—
356Q8—242no—
357D4—64no—
358D4—124no—
359D4—122no—
360D4—122no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—42no—
364C2×C4—62no—
365C2×C4—82no—
366C2×C4—122no—
367C2×C4—122no—
368C2×C4—122no—
369C2×C4—242no—
370C2×C4#88411yescharacteristic
371C2×C4#88411yescharacteristic
372C2×C4#88411yescharacteristic
373C2×C4D3×D812yes—
374C2×C4#91412yes—
375C2×C4C2×#23511yescharacteristic
376C2×C4C2×D3×D411yescharacteristic
377C2×C4C2×D3×D411yescharacteristic
378C8—32no—
379C8—32no—
380C8—42no—
381C8—42no—
382C8—61no—
383C8—61no—
384C8—122no—
385C2×C3—22no—
386C2×C3C2×#45911yescharacteristic
387C2×C3#305911yescharacteristic
388C2×C3C2×#47311yescharacteristic
389D3—82no—
390C22—21no—
391C22—21no—
392C22—122no—
393C22—122no—
394C22—122no—
395C22#478612yes—
396C22C2×#88411yescharacteristic
397C4—21no—
398C4—21no—
399C4—82no—
400C4—122no—
401C4—242no—
402C4#475512yes—
403C4#478612yes—
404C4C2×D3×D811yescharacteristic
405C4C2×#91411yescharacteristic
406C4#577912yes—
407C3#1918811yescharacteristic, complemented, Sylow
408C2—22no—
409C2—242no—
410C2C2×#475511yescharacteristic
411C2C2×#478611yescharacteristic
412C2#7896411yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1546C2
socleC22×C3C2×#121
prime radical#5477C22
hypercenter#1546D3
hypocenterC3#19188
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
2. C2×#5092C2semidirect
5. C3×#1546C2semidirect
6. #70745C2semidirect
7. #70506C2semidirect
8. #68449C2semidirect
9. #68441C2semidirect
50. #1546D3semidirect
407. C3#19188semidirect

Maximal subgroups

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

subgroupindex
#832152
C2×#50922
#792682
#790412
C3×#15462
#707452
#705062
#684492
#684412
#191883