Finite Group Databasebeta

← #10692000

Subgroups — #10692000

Order 768.

514 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3115C211yesmaximal, characteristic
2#82511C211yesmaximal, characteristic, complemented
3#71801C211yesmaximal, characteristic, complemented
4#71798C211yesmaximal, characteristic, complemented
5#71640C211yesmaximal, characteristic, complemented
6#67753C211yesmaximal, characteristic, complemented
7#67751C211yesmaximal, characteristic, complemented
8#9311—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C3×C4×#41C2211yescharacteristic
11C2×#948—21no—
12C3×#482—21no—
13C3×#475C411yescharacteristic
14C3×#474C411yescharacteristic
15C3×#460C411yescharacteristic
16C3×#459C411yescharacteristic
17C3×#449—21no—
18C3×#442—21no—
19C3×#440—21no—
20C3×#432C2211yescharacteristic
21#4997—21no—
22#4997—22no—
23#4926C2211yescharacteristic
24#4925C2211yescharacteristic
25C4×#208—21no—
26C4×#208C2211yescharacteristic
27C2×#803—21no—
28C2×#803C2211yescharacteristic
29#4540—21no—
30#4540—21no—
31#4535—21no—
32D4⋊(C3⋊C8)—21no—
33D4⋊(C3⋊C8)—22no—
34#3115D311yescharacteristic
35#2875—31no—
36#1530—31no—
37#1488—31no—
38#1484—31no—
39#1302—31no—
40#1291—31no—
41C2×C3×#41—21no—
42C2×C3×#41D411yescharacteristic
43C2×C3×SD16—21no—
44C2×C3×D8—21no—
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×#127C2×C411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×Q8C2×C411yescharacteristic
51C3×C4×Q8D411yescharacteristic
52C3×C4×D4—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4C2×C411yescharacteristic
57C3×C4×D4D411yescharacteristic
58C3×#117—21no—
59C3×#117D411yescharacteristic
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62#948—24no—
63#948—24no—
64C2×#208—22no—
65C2×#208—21no—
66C22×C3⋊C8—22no—
67C22×C3⋊C8—21no—
68C3×#107—21no—
69C3×#106—21no—
70C3×C4⋊C8C2×C411yescharacteristic
71C3×C4⋊C8C2×C411yescharacteristic
72C3×#103—21no—
73C3×#102—21no—
74C3×#102—21no—
75C3×#102—21no—
76C3×C22⋊C8—21no—
77C3×#97—21no—
78#830—21no—
79#830—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82#803—22no—
83#803—21no—
84#803—21no—
85#803—21no—
86#803D411yescharacteristic
87#803D411yescharacteristic
88C4×C3⋊C8—22no—
89C4×C3⋊C8—21no—
90C4×C3⋊C8D412yes—
91C2×#131—61no—
92#532C2×D311yescharacteristic
93C4×#41C2×D311yescharacteristic
94#482—21no—
95#475Dic311yescharacteristic
96#474Dic311yescharacteristic
97#460Dic311yescharacteristic
98#459Dic311yescharacteristic
99#449—21no—
100#442—21no—
101#440—21no—
102#432—31no—
103#432C2×D311yescharacteristic
104#431—31no—
105#405—62no—
106#405—61no—
107C4×M16—31no—
108C4×M16—61no—
109C2×#97—31no—
110C2×#97—61no—
111#329—61no—
112#329—61no—
113Q8⋊C8—61no—
114D4⋊C8—62no—
115D4⋊C8—61no—
116C3×#41—22no—
117C3×#41—22no—
118C3×#41—82no—
119C2×C3×Q8—21no—
120C2×C3×Q8C22⋊C411yescharacteristic
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—41no—
126C2×C3×D4C22⋊C411yescharacteristic
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—41no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C3×SD16—42no—
135C3×D8—42no—
136C2×C3×C8—21no—
137C2×C3×C8—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4C22⋊C411yescharacteristic
143C3×C4⋊C4C22⋊C411yescharacteristic
144C3×C22⋊C4—21no—
145C3×C22⋊C4—21no—
146C3×C22⋊C4—21no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C42—21no—
150C3×C42—21no—
151C3×C42—21no—
152C3×C42C22×C411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42C2×D411yescharacteristic
155C3×C42C2×D411yescharacteristic
156#208—28no—
157#208—22no—
158#208—22no—
159C2×C3⋊C8—28no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—22no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—21no—
165C2×C3⋊C8—41no—
166C2×C3⋊C8—41no—
167C2×#41—21no—
168C2×#41#6711yescharacteristic
169C2×SD16—21no—
170C2×D8—21no—
171#131—64no—
172#131—64no—
173C2×M16—62no—
174C2×M16—61no—
175C22×C8—62no—
176C22×C8—61no—
177C4⋊Q8C2×Dic311yescharacteristic
178#127C2×Dic311yescharacteristic
179#124—21no—
180#121—21no—
181#121—21no—
182C4×Q8C2×Dic311yescharacteristic
183C4×Q8#6711yescharacteristic
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4C2×Dic311yescharacteristic
189C4×D4#6711yescharacteristic
190#117—21no—
191#117#6711yescharacteristic
192C2×C42—21no—
193C2×C42C22×D311yescharacteristic
194#107—21no—
195#106—21no—
196C4⋊C8—61no—
197C4⋊C8—61no—
198C4⋊C8C2×Dic311yescharacteristic
199C4⋊C8C2×Dic311yescharacteristic
200#103—21no—
201#102—21no—
202#102—21no—
203#102—21no—
204C22⋊C8—21no—
205C22⋊C8—61no—
206C22⋊C8—61no—
207#97—21no—
208#97—31no—
209#97—31no—
210#97—62no—
211#97—61no—
212#97—61no—
213#97—61no—
214C4×C8—32no—
215C4×C8—62no—
216C4×C8—61no—
217C23×C3—21no—
218C23×C3—41no—
219C23×C3#12311yescharacteristic
220C3×Q8—21no—
221C3×Q8—81no—
222C3×Q8#10412yes—
223C3×D4—24no—
224C3×D4—21no—
225C3×D4—42no—
226C3×D4—41no—
227C3×D4—41no—
228C3×D4—81no—
229C3×D4—82no—
230C3×D4#10412yes—
231C2×C3×C4—24no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—81no—
241C2×C3×C4—81no—
242C2×C3×C4—82no—
243C2×C3×C4C2×C22⋊C411yescharacteristic
244C2×C3×C4C4×D411yescharacteristic
245C2×C3×C4C4×D411yescharacteristic
246C2×C3×C4#12011yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12111yescharacteristic
249C3×C8—41no—
250C3×C8—41no—
251C3⋊C8—22no—
252C3⋊C8—22no—
253C3⋊C8—24no—
254C3⋊C8—81no—
255C3⋊C8—81no—
256#41—22no—
257#41—22no—
258#41—82no—
259C2×Q8—21no—
260C2×Q8#21711yescharacteristic
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4#21711yescharacteristic
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4#23711yescharacteristic
272C22×C4#23711yescharacteristic
273C22×C4C2×#6711yescharacteristic
274SD16—42no—
275D8—42no—
276M16—68no—
277M16—62no—
278M16—62no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—68no—
282C2×C8—62no—
283C2×C8—62no—
284C2×C8—62no—
285C2×C8—61no—
286C2×C8—61no—
287C2×C8—121no—
288C2×C8—121no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4#21711yescharacteristic
294C4⋊C4#21711yescharacteristic
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42D3×D411yescharacteristic
304C42D3×D411yescharacteristic
305C42C22×Dic311yescharacteristic
306C42C2×#6711yescharacteristic
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—22no—
310C22×C3—21no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3#38611yescharacteristic
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—22no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4—81no—
326C3×C4C2×#10411yescharacteristic
327C3×C4C2×#10411yescharacteristic
328C23—21no—
329C23—41no—
330C23#93511yescharacteristic
331Q8—21no—
332Q8—81no—
333Q8#83712yes—
334D4—24no—
335D4—21no—
336D4—42no—
337D4—41no—
338D4—41no—
339D4—81no—
340D4—82no—
341D4#83712yes—
342C2×C4—24no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4D4×Dic311yescharacteristic
355C2×C4D4×Dic311yescharacteristic
356C2×C4#93711yescharacteristic
357C2×C4#93911yescharacteristic
358C2×C4#93911yescharacteristic
359C2×C4C2×#21711yescharacteristic
360C8—41no—
361C8—41no—
362C8—62no—
363C8—64no—
364C8—62no—
365C8—241no—
366C8—241no—
367C2×C3—41no—
368C2×C3—42no—
369C2×C3—161no—
370C2×C3#188111yescharacteristic
371C2×C3#189611yescharacteristic
372C2×C3#190311yescharacteristic
373C22—21no—
374C22—22no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22#522011yescharacteristic
383C4—21no—
384C4—22no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—81no—
392C4C2×#83711yescharacteristic
393C4C2×#83711yescharacteristic
394C3#931111yescharacteristic, complemented, Sylow
395C2—41no—
396C2—42no—
397C2—161no—
398C2#7171111yescharacteristic
399C2#7209011yescharacteristic
400C2#7211611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3115C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3115D3
hypocenterC3#9311
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3115

Complements

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

NH ≅ G/Ntype
2. #82511C2semidirect
3. #71801C2semidirect
4. #71798C2semidirect
5. #71640C2semidirect
6. #67753C2semidirect
7. #67751C2semidirect
394. C3#9311semidirect

Maximal subgroups

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

subgroupindex
C3×#31152
#825112
#718012
#717982
#716402
#677532
#677512
#93113