Finite Group Databasebeta

← #10690412

Subgroups — #10690412

Order 768.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83290C211yesmaximal, characteristic
2#78508C211yesmaximal, characteristic, complemented
3C3×#1980C211yesmaximal, characteristic
4#71800C211yesmaximal, characteristic
5#69471C211yesmaximal, characteristic, complemented
6#67667C211yesmaximal, characteristic, complemented
7#67619C211yesmaximal, characteristic, complemented
8#9054—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C2×C3×#104—21no—
12C2×C3×#104—21no—
13C3×C4×M16C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5122—21no—
16#5119—21no—
17#5093C411yescharacteristic, complemented
18#5080C411yescharacteristic, complemented
19#5076C411yescharacteristic, complemented
20#5070C411yescharacteristic, complemented
21#5041—22no—
22C4×#214—22no—
23#4925C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26#4701—22no—
27#4679—21no—
28C8×Dic6—21no—
29D4⋊(C3⋊C8)—21no—
30D4⋊(C3⋊C8)—21no—
31#4452—22no—
32#3167—31no—
33#2962—31no—
34#1980D311yescharacteristic
35#1503—31no—
36#1480—31no—
37#1279—31no—
38#1278—31no—
39C2×C3×#41—21no—
40C2×C3×M16D411yescharacteristic
41C2×C3×M16D411yescharacteristic
42C3×C4⋊Q8C2×C411yescharacteristic
43C3×#127C2×C411yescharacteristic
44C3×#124—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#214—22no—
53#926—21no—
54#924—21no—
55#924D412yes—
56C22×C3⋊C8—21no—
57C22×C3⋊C8—21no—
58#870—21no—
59#869C2×C411yescharacteristic
60#869C2×C411yescharacteristic
61C4×Dic6—21no—
62C4×Dic6D412yes—
63C3×#104—42no—
64C3×#104—42no—
65C3×#97—21no—
66C3×C4×C8—21no—
67C3×#95—21no—
68#832—22no—
69#830—21no—
70#815—22no—
71Dic3⋊C8—22no—
72Dic3⋊C8—22no—
73C8×Dic3—22no—
74#810—22no—
75#808—22no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)D411yescharacteristic
79C4⋊(C3⋊C8)D411yescharacteristic
80C4×C3⋊C8—21no—
81C4×C3⋊C8—21no—
82C4×C3⋊C8C2×C411yescharacteristic
83C4×C3⋊C8C2×C411yescharacteristic
84#533—31no—
85#532C2×D311yescharacteristic
86#498—31no—
87#492—31no—
88#487—31no—
89#486—31no—
90#465—62no—
91#446—61no—
92#446—62no—
93C8×Q8—61no—
94C4×SD16—62no—
95#431—31no—
96#424—61no—
97C2×C4⋊C8—31no—
98C2×#104—21no—
99C2×#104—21no—
100#405—61no—
101C4×M16C2×D311yescharacteristic
102C2×C4×C8—31no—
103C4×C4⋊C4C2×D311yescharacteristic
104#328—62no—
105D4⋊C8—61no—
106D4⋊C8—61no—
107C3×#41—41no—
108C3×#41—41no—
109C3×#41—41no—
110C3×#41—41no—
111C2×C3×Q8—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C2×Dic6—21no—
122C2×Dic6C22⋊C412yes—
123C3×M16—21no—
124C3×M16—21no—
125C3×M16—21no—
126C3×M16SD1612yescomplemented
127C2×C3×C8—21no—
128C2×C3×C8—21no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C42—41no—
136C3×C42—41no—
137C3×C42C22×C411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141#217—22no—
142#214—44no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3C22⋊C412yes—
147Dic3⋊C4—22no—
148Dic3⋊C4—22no—
149C4×Dic3—22no—
150C2×C3⋊C8—21no—
151C2×C3⋊C8—21no—
152C2×C3⋊C8—21no—
153C2×C3⋊C8—21no—
154C2×C3⋊C8—42no—
155C2×C3⋊C8—41no—
156C2×C3⋊C8—41no—
157C2×C3⋊C8—41no—
158C2×#41—21no—
159C2×SD16—62no—
160C2×M16#6711yescharacteristic
161C2×M16#6711yescharacteristic
162C22×C8—61no—
163C22×C8—61no—
164C4⋊Q8—31no—
165C4⋊Q8—31no—
166C4⋊Q8C4×D311yescharacteristic
167#127C4×D311yescharacteristic
168#125—61no—
169#124—21no—
170C22⋊Q8—32no—
171C22⋊Q8—61no—
172#121—21no—
173#121—21no—
174C4×Q8—32no—
175C4×Q8—61no—
176C4×D4—21no—
177C4×D4—21no—
178#117—61no—
179C2×C4⋊C4—21no—
180C2×C42—21no—
181C2×C42C22×D311yescharacteristic
182#106—62no—
183C4⋊C8—31no—
184C4⋊C8—31no—
185C4⋊C8—62no—
186C4⋊C8—62no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189#104—42no—
190#104—42no—
191#103—62no—
192#102—62no—
193C22⋊C8—61no—
194#97—21no—
195#97—62no—
196C4×C8—21no—
197C4×C8—31no—
198C4×C8—31no—
199C4×C8—62no—
200C4×C8—61no—
201C4×C8—61no—
202#95—21no—
203C23×C3—41no—
204C23×C3#12111yescharacteristic
205C3×Q8—41no—
206C3×Q8—41no—
207C3×D4—42no—
208C3×D4—41no—
209C3×D4—41no—
210C3×D4—41no—
211C3×D4—41no—
212C3×D4—81no—
213C3×D4—81no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—41no—
219C2×C3×C4—81no—
220C2×C3×C4—81no—
221C2×C3×C4—81no—
222C2×C3×C4—81no—
223C2×C3×C4C2×C22⋊C411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4C4×D411yescharacteristic
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12311yescharacteristic
229C2×C3×C4C2×SD1611yescharacteristic
230C2×C3×C4#13511yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C3×C4#13711yescharacteristic
233C2×Dic3—22no—
234C2×Dic3—22no—
235Dic6—41no—
236Dic6—41no—
237Dic6—42no—
238C3×C8—22no—
239C3×C8—41no—
240C3⋊C8—22no—
241C3⋊C8—41no—
242C3⋊C8—41no—
243C3⋊C8—41no—
244C3⋊C8—81no—
245#41—41no—
246#41—41no—
247#41—41no—
248#41—41no—
249C2×Q8—21no—
250C2×Q8—32no—
251C2×Q8—61no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4D3×D411yescharacteristic
259C22×C4#23711yescharacteristic
260C22×C4C2×#6711yescharacteristic
261SD16—124no—
262M16—21no—
263M16—21no—
264M16—21no—
265M16#21412yescomplemented
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—61no—
269C2×C8—61no—
270C2×C8—61no—
271C2×C8—61no—
272C2×C8—122no—
273C2×C8—121no—
274C2×C8—121no—
275C2×C8—121no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—32no—
280C4⋊C4—41no—
281C4⋊C4—61no—
282C4⋊C4—62no—
283C4⋊C4—62no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—62no—
289C42—41no—
290C42—41no—
291C42—62no—
292C42C2×C4×D311yescharacteristic
293C42#23511yescharacteristic
294C42D3×D411yescharacteristic
295C42D3×D411yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—81no—
299C22×C3—81no—
300C22×C3—81no—
301C22×C3#38611yescharacteristic
302C22×C3#46011yescharacteristic
303C22×C3#46411yescharacteristic
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—82no—
308C3×C4—81no—
309C3×C4C4×SD1611yescharacteristic
310C3×C4#44011yescharacteristic
311C3×C4#44311yescharacteristic
312C3×C4#44411yescharacteristic
313C3×C4#44811yescharacteristic
314C3×C4#44911yescharacteristic
315Dic3—42no—
316Dic3—42no—
317C23—41no—
318C23#93811yescharacteristic
319Q8—41no—
320Q8—41no—
321Q8—122no—
322Q8—121no—
323Q8—121no—
324D4—42no—
325D4—41no—
326D4—41no—
327D4—41no—
328D4—41no—
329D4—81no—
330D4—81no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4D3×C22⋊C411yescharacteristic
343C2×C4#67⋊C411yescharacteristic
344C2×C4#88311yescharacteristic
345C2×C4#88411yescharacteristic
346C2×C4C4×#6711yescharacteristic
347C2×C4#93211yescharacteristic
348C2×C4C2×#21411yescharacteristic
349C2×C4#93711yescharacteristic
350C2×C4#95011yescharacteristic
351C2×C4#95111yescharacteristic
352C8—22no—
353C8—41no—
354C8—62no—
355C8—121no—
356C8—121no—
357C8—121no—
358C8—241no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—161no—
362C2×C3#187611yescharacteristic
363C2×C3#188011yescharacteristic
364C2×C3#188411yescharacteristic
365C22—21no—
366C22—21no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#495611yescharacteristic
371C22#505411yescharacteristic
372C22#507511yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—82no—
377C4—81no—
378C4—122no—
379C4—122no—
380C4#482311yescharacteristic
381C4#482411yescharacteristic
382C4C4×#21411yescharacteristic
383C4#501811yescharacteristic
384C4#503711yescharacteristic
385C4#504011yescharacteristic
386C3#905411yescharacteristic, complemented, Sylow
387C2—21no—
388C2—21no—
389C2—161no—
390C2#6966011yescharacteristic
391C2#7039011yescharacteristic
392C2#7041711yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1980C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1980D3
hypocenterC3#9054
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
2. #78508C2semidirect
5. #69471C2semidirect
6. #67667C2semidirect
7. #67619C2semidirect
17. #5093C4semidirect
18. #5080C4semidirect
19. #5076C4semidirect
20. #5070C4semidirect
126. C3×M16SD16semidirect
265. M16#214semidirect
386. C3#9054semidirect

Maximal subgroups

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

subgroupindex
#832902
#785082
C3×#19802
#718002
#694712
#676672
#676192
#90543