Finite Group Databasebeta

← #10690421

Subgroups — #10690421

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#79004C211yesmaximal, characteristic
2#78468C211yesmaximal, characteristic
3#70429C211yesmaximal, characteristic, complemented
4#69493C211yesmaximal, characteristic, complemented
5#69469C211yesmaximal, characteristic
6C3×#1279C211yesmaximal, characteristic, complemented
7#67656C211yesmaximal, characteristic, complemented
8#9055—31nomaximal, Sylow
9C2×#901—21no—
10#5477C2211yescharacteristic
11#5470C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5114—22no—
16#5110—21no—
17#4925C2211yescharacteristic
18C4×#208C2211yescharacteristic
19#4838—21no—
20#4836—22no—
21#4834—21no—
22#4694C411yescharacteristic, complemented
23#4693C411yescharacteristic, complemented
24C4×#204—22no—
25#4681C411yescharacteristic, complemented
26#4680C411yescharacteristic, complemented
27#4523—21no—
28#4523—21no—
29#4484—21no—
30#4484—21no—
31#4451—22no—
32#3167—31no—
33#2935—31no—
34#1986—31no—
35#1502—31no—
36#1481—31no—
37#1279D311yescharacteristic, complemented
38#1278—31no—
39C2×#235—21no—
40C22×C3×C8—21no—
41C22×C3×C8—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C42—21no—
44C2×C3×C42C2311yescharacteristic
45#930—21no—
46#930D412yes—
47#926—21no—
48#924—21no—
49#924—21no—
50C2×#208D411yescharacteristic
51C2×#208D411yescharacteristic
52C2×#204—22no—
53#901—42no—
54#901—42no—
55C2×ΓL(1,𝔽25)—22no—
56C2×D3×C8—22no—
57#875—21no—
58#874C2×C411yescharacteristic
59C4×D12—21no—
60C4×D12D412yes—
61#870—21no—
62#869C2×C411yescharacteristic
63#869C2×C411yescharacteristic
64#869C2×C411yescharacteristic
65C4×Dic6—21no—
66C4×Dic6—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D411yescharacteristic
69C3×C4⋊C8D411yescharacteristic
70C3×C4×C8—21no—
71C3×C4×C8C2×C411yescharacteristic
72C3×C4×C8C2×C411yescharacteristic
73C3×#95—21no—
74#830—21no—
75#821—22no—
76(C2×D3)⋊C8—22no—
77(C2×D3)⋊C8—22no—
78#817—22no—
79#816—22no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82C4×C3⋊C8—21no—
83C4×C3⋊C8—21no—
84C2×#131—61no—
85#533—31no—
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#431—31no—
97C2×C4⋊C8C2×D311yescharacteristic
98#408—61no—
99C4×M16—31no—
100C2×C4×C8C2×D311yescharacteristic
101C4×C4⋊C4C2×D311yescharacteristic
102#339—61no—
103#339—61no—
104#327—62no—
105Q8⋊C8—61no—
106Q8⋊C8—61no—
107C22×C3×C4—41no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C2×#67—22no—
112#235—41no—
113#235—41no—
114#235—41no—
115#235—41no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12C22⋊C412yes—
119C2×C4×D3—22no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×Dic6—21no—
123C2×C3×C8—21no—
124C2×C3×C8—21no—
125C2×C3×C8—21no—
126C2×C3×C8—21no—
127C2×C3×C8—42no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—41no—
132C3×C42—41no—
133C3×C42—41no—
134C3×C42C22×C411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138#217—41no—
139(C2×D3)⋊C4—22no—
140(C2×D3)⋊C4—22no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3C22⋊C412yes—
147Dic3⋊C4—41no—
148Dic3⋊C4—41no—
149C4×Dic3—41no—
150#208—21no—
151#208—21no—
152#208—21no—
153#208SD1612yescomplemented
154C2×C3⋊C8—21no—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—41no—
157#204—44no—
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—21no—
169C22×C8—62no—
170C4⋊Q8—31no—
171C4⋊Q8—31no—
172C4⋊Q8—31no—
173#127—31no—
174#125—61no—
175#124—61no—
176C22⋊Q8—61no—
177C22⋊Q8—61no—
178#121—32no—
179#121—61no—
180C4×Q8—61no—
181C4×Q8—61no—
182C4×D4—32no—
183C4×D4—61no—
184#117—61no—
185C2×C4⋊C4—21no—
186C2×C42—21no—
187C2×C42C22×D311yescharacteristic
188#106—62no—
189C4⋊C8—21no—
190C4⋊C8—61no—
191C4⋊C8D1211yescharacteristic
192C4⋊C8D1211yescharacteristic
193#103—62no—
194#102—62no—
195C22⋊C8—62no—
196C22⋊C8—62no—
197C22⋊C8—61no—
198#97—61no—
199C4×C8—21no—
200C4×C8—61no—
201C4×C8—61no—
202C4×C8C4×D311yescharacteristic
203C4×C8C4×D311yescharacteristic
204#95—21no—
205C23×C3#12111yescharacteristic
206C22×D3—22no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—81no—
212C2×C3×C4—81no—
213C2×C3×C4C2×C22⋊C411yescharacteristic
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12311yescharacteristic
219C2×C3×C4C2×SD1611yescharacteristic
220C2×C3×C4#13511yescharacteristic
221C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
222C2×C3×C4#13711yescharacteristic
223#67—42no—
224#67—42no—
225C2×Dic3—22no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228D12—42no—
229D12—41no—
230D12—41no—
231D12—41no—
232D12—41no—
233C4×D3—42no—
234C4×D3—42no—
235Dic6—41no—
236Dic6—41no—
237Dic6—42no—
238Dic6—41no—
239Dic6—41no—
240C3×C8—22no—
241C3×C8—41no—
242C3×C8—41no—
243C3×C8—41no—
244C3⋊C8—22no—
245C3⋊C8—41no—
246C3⋊C8—81no—
247#41—121no—
248#41—121no—
249#41—121no—
250#41—121no—
251C2×Q8—61no—
252C2×Q8—61no—
253C2×Q8—61no—
254C2×D4—32no—
255C2×D4—62no—
256C2×D4—61no—
257C2×D4—61no—
258C22×C4—41no—
259C22×C4—62no—
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—21no—
272C2×C8—21no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—61no—
277C2×C8—61no—
278C2×C8—124no—
279C2×C8—121no—
280C4⋊C4—22no—
281C4⋊C4—32no—
282C4⋊C4—41no—
283C4⋊C4—61no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4—121no—
289C4⋊C4—121no—
290C22⋊C4—62no—
291C22⋊C4—62no—
292C22⋊C4—121no—
293C42—41no—
294C42—41no—
295C42—121no—
296C42C2×C4×D311yescharacteristic
297C42C2×D1211yescharacteristic
298C42D3×D411yescharacteristic
299C42#23911yescharacteristic
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—82no—
312C3×C4C4×SD1611yescharacteristic
313C3×C4#44011yescharacteristic
314C3×C4#44311yescharacteristic
315C3×C4#44411yescharacteristic
316C3×C4#45011yescharacteristic
317C3×C4#45211yescharacteristic
318Dic3—42no—
319Dic3—81no—
320Dic3—81no—
321C23—62no—
322C23#88411yescharacteristic
323Q8—122no—
324Q8—121no—
325Q8—121no—
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—42no—
339C2×C4—62no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—122no—
343C2×C4—122no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4C4×D1211yescharacteristic
347C2×C4D3×C22⋊C411yescharacteristic
348C2×C4#88211yescharacteristic
349C2×C4#89311yescharacteristic
350C2×C4#89411yescharacteristic
351C2×C4#89511yescharacteristic
352C2×C4#91111yescharacteristic
353C2×C4D3×SD1611yescharacteristic
354C2×C4#91611yescharacteristic
355C2×C4#91811yescharacteristic
356C8—22no—
357C8—41no—
358C8—41no—
359C8—41no—
360C8—62no—
361C8—121no—
362C8—241no—
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#466511yescharacteristic
375C22#478711yescharacteristic
376C22#481811yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—82no—
381C4—122no—
382C4—241no—
383C4—241no—
384C4#470011yescharacteristic
385C4#471711yescharacteristic
386C4#477611yescharacteristic
387C4#480811yescharacteristic
388C4#485211yescharacteristic
389C4#488911yescharacteristic
390C3#905511yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—242no—
394C2#6869511yescharacteristic
395C2#6877711yescharacteristic
396C2#6889611yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1279C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1279D3
hypocenterC3#9055
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
3. #70429C2semidirect
4. #69493C2semidirect
6. C3×#1279C2semidirect
7. #67656C2semidirect
22. #4694C4semidirect
23. #4693C4semidirect
25. #4681C4semidirect
26. #4680C4semidirect
37. #1279D3semidirect
153. #208SD16semidirect
390. C3#9055semidirect

Maximal subgroups

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

subgroupindex
#790042
#784682
#704292
#694932
#694692
C3×#12792
#676562
#90553