Finite Group Databasebeta

← #10690425

Subgroups — #10690425

Order 768.

500 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#83343C211yesmaximal, characteristic
2#79004C211yesmaximal, characteristic
3C3×#1481C211yesmaximal, characteristic, complemented
4#71841C211yesmaximal, characteristic, complemented
5#70789C211yesmaximal, characteristic
6#67739C211yesmaximal, characteristic, complemented
7#67704C211yesmaximal, characteristic, complemented
8#9055—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×#901—21no—
11#5477C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#5110—21no—
15#5088C411yescharacteristic, complemented
16#5085C411yescharacteristic, complemented
17#5080C411yescharacteristic, complemented
18#5079C411yescharacteristic, complemented
19#5048—22no—
20C4×#215—22no—
21C4×C4⋊Dic3C2211yescharacteristic
22C4×#208C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24#4838—21no—
25#4836—22no—
26#4834—21no—
27C3×Q8⋊C8—21no—
28C3×Q8⋊C8—21no—
29#4532—21no—
30#4532—21no—
31#4470—22no—
32#3167—31no—
33#2935—31no—
34#1986—31no—
35#1502—31no—
36#1481D311yescharacteristic, complemented
37#1279—31no—
38#1278—31no—
39C2×#235—21no—
40C22×C3×C8—21no—
41C3×C4⋊Q8C2×C411yescharacteristic
42C3×C4⋊Q8C2×C411yescharacteristic
43C3×#125—21no—
44C3×C22⋊Q8—21no—
45C3×C22⋊Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×#117—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#215—22no—
51#930—21no—
52#930D412yes—
53C2×C4⋊Dic3—21no—
54C2×C4×Dic3—21no—
55C2×#208D411yescharacteristic
56C2×#208D411yescharacteristic
57C22×C3⋊C8—21no—
58#901—42no—
59#901—42no—
60C2×ΓL(1,𝔽25)—22no—
61C2×D3×C8—22no—
62#875—21no—
63#874C2×C411yescharacteristic
64C4×D12—21no—
65C4×D12D412yes—
66#869C2×C411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8D411yescharacteristic
70C3×C4⋊C8D411yescharacteristic
71C3×C22⋊C8—21no—
72C3×C4×C8—21no—
73#835—22no—
74C3⋊#95—21no—
75(C2×D3)⋊C8—22no—
76(C2×D3)⋊C8—22no—
77#809—22no—
78#808—22no—
79#803—21no—
80C4×C3⋊C8—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8C2×C411yescharacteristic
83C4×C3⋊C8C2×C411yescharacteristic
84C2×#131—61no—
85#533C2×D311yescharacteristic
86#532—31no—
87#498—31no—
88#492—31no—
89#487—31no—
90#486—31no—
91#465—62no—
92C4×SD16—62no—
93#436—61no—
94#435—62no—
95C8×D4—61no—
96#431C2×D311yescharacteristic
97C2×C4⋊C8C2×D311yescharacteristic
98#408—61no—
99C4×M16—31no—
100C2×C4×C8—31no—
101C4×C4⋊C4—31no—
102#339—61no—
103#339—61no—
104#327—62no—
105Q8⋊C8—21no—
106Q8⋊C8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C2×#67—22no—
113C22×Dic3—41no—
114#235—41no—
115#235—41no—
116#235—41no—
117#235—41no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12C22⋊C412yes—
121C2×C4×D3—22no—
122C2×Dic6—21no—
123C2×C3×C8—21no—
124C2×C3×C8—21no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C2×C3×C8—41no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C4⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42—41no—
136C3×C42C22×C411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42#4111yescharacteristic
140#215—44no—
141(C2×D3)⋊C4—22no—
142(C2×D3)⋊C4—22no—
143C4⋊Dic3—21no—
144C4⋊Dic3—22no—
145C4⋊Dic3—41no—
146C4⋊Dic3C22⋊C412yes—
147C4×Dic3—41no—
148C4×Dic3—41no—
149#208—21no—
150#208—21no—
151#208—21no—
152#208SD1612yescomplemented
153C2×C3⋊C8—21no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—21no—
157C2×C3⋊C8—42no—
158ΓL(1,𝔽25)—44no—
159D3×C8—44no—
160C2×#41—61no—
161C2×SD16—62no—
162#131—122no—
163#131—122no—
164C2×M16—31no—
165C2×M16—31no—
166C2×M16—62no—
167C22×C8—21no—
168C22×C8—62no—
169C22×C8—61no—
170C4⋊Q8—31no—
171C4⋊Q8C4×D311yescharacteristic
172C4⋊Q8C4×D311yescharacteristic
173#127—31no—
174#125—21no—
175#124—61no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178#121—32no—
179#121—61no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×D4—32no—
183C4×D4—61no—
184#117—21no—
185C2×C4⋊C4—61no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188#106—62no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8#6711yescharacteristic
192C4⋊C8#6711yescharacteristic
193#103—62no—
194#102—62no—
195C22⋊C8—21no—
196C22⋊C8—62no—
197C22⋊C8—62no—
198#97—61no—
199C4×C8—21no—
200C4×C8—31no—
201C4×C8—31no—
202C4×C8—61no—
203C4×C8—61no—
204#95—61no—
205C23×C3#12111yescharacteristic
206C22×D3—22no—
207C3×Q8—42no—
208C3×Q8—41no—
209C3×Q8—41no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4C2×C22⋊C411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4C4×D411yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12311yescharacteristic
221C2×C3×C4C2×SD1611yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C4#13711yescharacteristic
225#67—42no—
226#67—42no—
227C2×Dic3—22no—
228C2×Dic3—42no—
229C2×Dic3—81no—
230C2×Dic3—81no—
231D12—42no—
232D12—41no—
233D12—41no—
234D12—41no—
235D12—41no—
236C4×D3—42no—
237C4×D3—42no—
238Dic6—41no—
239Dic6—41no—
240C3×C8—41no—
241C3×C8—41no—
242C3×C8—81no—
243C3⋊C8—22no—
244C3⋊C8—22no—
245C3⋊C8—41no—
246C3⋊C8—41no—
247#41—121no—
248#41—121no—
249#41—121no—
250#41—121no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—61no—
254C2×D4—32no—
255C2×D4—62no—
256C2×D4—61no—
257C2×D4—61no—
258C22×C4—62no—
259C22×C4—121no—
260C22×C4#23511yescharacteristic
261C22×C4D3×D411yescharacteristic
262C22×C4D3×D411yescharacteristic
263SD16—124no—
264M16—32no—
265M16—61no—
266M16—61no—
267M16—61no—
268M16—124no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—61no—
275C2×C8—61no—
276C2×C8—61no—
277C2×C8—61no—
278C2×C8—124no—
279C2×C8—122no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—32no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—62no—
288C4⋊C4—61no—
289C4⋊C4—121no—
290C22⋊C4—41no—
291C22⋊C4—62no—
292C22⋊C4—62no—
293C42—41no—
294C42—121no—
295C42—121no—
296C42C2×C4×D311yescharacteristic
297C42D3×D411yescharacteristic
298C42#23711yescharacteristic
299C42C2×#6711yescharacteristic
300C22×C3—21no—
301C22×C3—21no—
302C22×C3#38611yescharacteristic
303C22×C3#46111yescharacteristic
304C22×C3#46311yescharacteristic
305C2×D3—42no—
306C2×D3—42no—
307C2×D3—42no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—81no—
312C3×C4—81no—
313C3×C4C4×SD1611yescharacteristic
314C3×C4#44011yescharacteristic
315C3×C4#44311yescharacteristic
316C3×C4#44411yescharacteristic
317C3×C4#45011yescharacteristic
318C3×C4#45211yescharacteristic
319Dic3—42no—
320Dic3—82no—
321C23—62no—
322C23#88411yescharacteristic
323Q8—42no—
324Q8—41no—
325Q8—41no—
326Q8—121no—
327Q8—121no—
328D4—122no—
329D4—122no—
330D4—122no—
331D4—121no—
332D4—121no—
333D4—121no—
334D4—121no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—62no—
341C2×C4—122no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4—241no—
345C2×C4—241no—
346C2×C4D3×C22⋊C411yescharacteristic
347C2×C4#67⋊C411yescharacteristic
348C2×C4#88311yescharacteristic
349C2×C4D3×SD1611yescharacteristic
350C2×C4#91411yescharacteristic
351C2×C4#91811yescharacteristic
352C2×C4#91911yescharacteristic
353C2×C4C4×#6711yescharacteristic
354C2×C4#93711yescharacteristic
355C2×C4#93811yescharacteristic
356C8—41no—
357C8—41no—
358C8—62no—
359C8—62no—
360C8—81no—
361C8—121no—
362C8—121no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3#187611yescharacteristic
366C2×C3#187711yescharacteristic
367C2×C3#188411yescharacteristic
368D3—82no—
369C22—21no—
370C22—21no—
371C22—122no—
372C22—122no—
373C22—122no—
374C22#481811yescharacteristic
375C22#481911yescharacteristic
376C22#495611yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—81no—
381C4—81no—
382C4—122no—
383C4—242no—
384C4#478811yescharacteristic
385C4#481711yescharacteristic
386C4#514411yescharacteristic
387C4#514511yescharacteristic
388C4#516911yescharacteristic
389C4#518711yescharacteristic
390C3#905511yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—242no—
394C2#7001111yescharacteristic
395C2#7027611yescharacteristic
396C2#7041711yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1481C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1481D3
hypocenterC3#9055
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
3. C3×#1481C2semidirect
4. #71841C2semidirect
6. #67739C2semidirect
7. #67704C2semidirect
15. #5088C4semidirect
16. #5085C4semidirect
17. #5080C4semidirect
18. #5079C4semidirect
36. #1481D3semidirect
152. #208SD16semidirect
390. C3#9055semidirect

Maximal subgroups

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

subgroupindex
#833432
#790042
C3×#14812
#718412
#707892
#677392
#677042
#90553