Finite Group Databasebeta

← #10694569

Subgroups — #10694569

Order 768.

400 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83281C211yesmaximal, characteristic
2C3×#1517C211yesmaximal, characteristic
3#71814C211yesmaximal, characteristic
4#70793C211yesmaximal, characteristic, complemented
5#70740C211yesmaximal, characteristic, complemented
6#69547C211yesmaximal, characteristic, complemented
7#69546C211yesmaximal, characteristic, complemented
8#9757—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10#5470C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#382C2211yescharacteristic
13#5233—21no—
14#5233—21no—
15#5079C411yescharacteristic, complemented
16#5079C411yescharacteristic, complemented
17#5070C411yescharacteristic, complemented
18#5070C411yescharacteristic, complemented
19#5056—21no—
20#5056—21no—
21#4936C2211yescharacteristic
22#4925C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24C3×#327—22no—
25#4537—22no—
26#4529—21no—
27#4529—21no—
28#4467—22no—
29#4454—21no—
30C8×C3⋊C8—21no—
31#4452—22no—
32#3131—31no—
33#1835—31no—
34#1518—31no—
35#1518—31no—
36#1517—31no—
37#1517D311yescharacteristic
38#1450—31no—
39C2×C3×#41—21no—
40C3×C4⋊Q8C2×C411yescharacteristic
41C3×#127C2×C411yescharacteristic
42C3×#124—21no—
43C3×#121—21no—
44C3×#121—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49#926—21no—
50C2×C4⋊Dic3—21no—
51#924—21no—
52#924—21no—
53C22×C3⋊C8D411yescharacteristic
54C22×C3⋊C8D411yescharacteristic
55#870—21no—
56#869C2×C411yescharacteristic
57#869C2×C411yescharacteristic
58C4×Dic6—21no—
59C4×Dic6—21no—
60C3×C4⋊C8—21no—
61C3×C22⋊C8D412yes—
62C3×C4×C8—21no—
63C3×#95—21no—
64C3×#95—21no—
65#835—21no—
66#835—21no—
67#835—21no—
68#835—21no—
69#832—21no—
70#832—21no—
71#832—21no—
72#832—21no—
73C3⋊#95—21no—
74C3⋊#95—21no—
75#830D412yes—
76C4⋊(C3⋊C8)—21no—
77C4×C3⋊C8—21no—
78C4×C3⋊C8C2×C411yescharacteristic
79C4×C3⋊C8C2×C411yescharacteristic
80C4×C3⋊C8D411yescharacteristic
81C4×C3⋊C8D411yescharacteristic
82#533—31no—
83#532C2×D311yescharacteristic
84#488—61no—
85#488—61no—
86#487—31no—
87#487—31no—
88#486—31no—
89#486—31no—
90#431—31no—
91#431C2×D311yescharacteristic
92#416—61no—
93#416—61no—
94C2×C4×C8—31no—
95#382—31no—
96#382C2×D311yescharacteristic
97#340—61no—
98#340—61no—
99#328—62no—
100#328—62no—
101#327—22no—
102#327—62no—
103#322—61no—
104C82—61no—
105C3×#41—41no—
106C3×#41—41no—
107C3×#41—41no—
108C3×#41—41no—
109C2×C3×Q8—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—21no—
113C2×C3×D4—41no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×Dic3—41no—
120C2×Dic6—21no—
121C2×Dic6—21no—
122C2×C3×C8—22no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C42C22×C411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42#4111yescharacteristic
132C3×C42#4111yescharacteristic
133#217—41no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—41no—
139C4⋊Dic3—41no—
140Dic3⋊C4—41no—
141Dic3⋊C4—41no—
142C4×Dic3—41no—
143C2×C3⋊C8—21no—
144C2×C3⋊C8—21no—
145C2×C3⋊C8—21no—
146C2×C3⋊C8—21no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8C22⋊C411yescharacteristic
149C2×C3⋊C8C22⋊C411yescharacteristic
150C2×C3⋊C8C22⋊C411yescharacteristic
151C2×C3⋊C8C22⋊C411yescharacteristic
152C2×#41—21no—
153C22×C8—31no—
154C22×C8—31no—
155C4⋊Q8—31no—
156C4⋊Q8—31no—
157C4⋊Q8C4×D311yescharacteristic
158#127C4×D311yescharacteristic
159#125—61no—
160#124—21no—
161C22⋊Q8—61no—
162C22⋊Q8—61no—
163#121—21no—
164#121—21no—
165C4×Q8—61no—
166C4×Q8—61no—
167C4×D4—21no—
168C4×D4—21no—
169#117—61no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—61no—
172C2×C42C22×D311yescharacteristic
173C4⋊C8—21no—
174C4⋊C8—61no—
175#103—61no—
176#103—61no—
177#103—61no—
178#103—61no—
179#102—61no—
180#102—61no—
181#102—61no—
182#102—61no—
183C22⋊C8—32no—
184C22⋊C8#6712yes—
185C4×C8—21no—
186C4×C8—31no—
187C4×C8—31no—
188C4×C8—31no—
189C4×C8—31no—
190C4×C8—61no—
191#95—21no—
192#95—21no—
193#95—61no—
194#95—61no—
195C23×C3—41no—
196C23×C3#12711yescharacteristic
197C3×Q8—41no—
198C3×Q8—41no—
199C3×D4—41no—
200C3×D4—41no—
201C3×D4—81no—
202C3×D4—81no—
203C3×D4—81no—
204C3×D4—81no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—81no—
209C2×C3×C4—81no—
210C2×C3×C4—81no—
211C2×C3×C4—81no—
212C2×C3×C4—81no—
213C2×C3×C4C2×C22⋊C411yescharacteristic
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12411yescharacteristic
219C2×C3×C4#13511yescharacteristic
220C2×C3×C4#13511yescharacteristic
221C2×C3×C4#13511yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×Dic3—41no—
224C2×Dic3—41no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227C2×Dic3—81no—
228Dic6—81no—
229Dic6—81no—
230C3×C8—42no—
231C3⋊C8—21no—
232C3⋊C8—21no—
233C3⋊C8—21no—
234C3⋊C8—21no—
235C3⋊C8—42no—
236#41—41no—
237#41—41no—
238#41—41no—
239#41—41no—
240C2×Q8—21no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—121no—
250C22×C4D3×D411yescharacteristic
251C22×C4D3×D411yescharacteristic
252C22×C4C2×#6711yescharacteristic
253C2×C8—22no—
254C2×C8—31no—
255C2×C8—31no—
256C2×C8—31no—
257C2×C8—31no—
258C2×C8—61no—
259C2×C8—61no—
260C2×C8—61no—
261C2×C8—61no—
262C2×C8—62no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—61no—
268C4⋊C4—61no—
269C4⋊C4—61no—
270C4⋊C4—61no—
271C4⋊C4—121no—
272C4⋊C4—121no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—121no—
278C42—121no—
279C42C2×C4×D311yescharacteristic
280C42#23511yescharacteristic
281C42D3×D411yescharacteristic
282C42#23711yescharacteristic
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3—81no—
288C22×C3#39011yescharacteristic
289C22×C3#49511yescharacteristic
290C22×C3#49511yescharacteristic
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—81no—
296C3×C4—161no—
297C3×C4#41611yescharacteristic
298C3×C4#41611yescharacteristic
299C3×C4#48811yescharacteristic
300C3×C4#48811yescharacteristic
301Dic3—81no—
302Dic3—81no—
303Dic3—161no—
304C23—41no—
305C23#94011yescharacteristic
306Q8—41no—
307Q8—41no—
308Q8—241no—
309Q8—241no—
310D4—41no—
311D4—41no—
312D4—81no—
313D4—81no—
314D4—81no—
315D4—81no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—241no—
327C2×C4—241no—
328C2×C4—241no—
329C2×C4D3×C22⋊C411yescharacteristic
330C2×C4#67⋊C411yescharacteristic
331C2×C4#88411yescharacteristic
332C2×C4#88511yescharacteristic
333C2×C4#91211yescharacteristic
334C2×C4#91211yescharacteristic
335C2×C4#91611yescharacteristic
336C2×C4#91611yescharacteristic
337C2×C4C4×#6711yescharacteristic
338C2×C4#93911yescharacteristic
339C8—42no—
340C8—61no—
341C8—61no—
342C8—61no—
343C8—61no—
344C8—122no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—161no—
348C2×C3#195811yescharacteristic
349C2×C3#197611yescharacteristic
350C2×C3#197611yescharacteristic
351C22—21no—
352C22—21no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#495711yescharacteristic
357C22#515511yescharacteristic
358C22#516811yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—81no—
364C4—161no—
365C4—241no—
366C4—241no—
367C4—481no—
368C4#476811yescharacteristic
369C4#476811yescharacteristic
370C4#477211yescharacteristic
371C4#477211yescharacteristic
372C3#975711yescharacteristic, complemented, Sylow
373C2—21no—
374C2—21no—
375C2—161no—
376C2#7004211yescharacteristic
377C2#7015811yescharacteristic
378C2#7039711yescharacteristic
subgroupG/H
centerC22#4957
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1517C2
socleC22×C3#390
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1517D3
hypocenterC3#9757
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
4. #70793C2semidirect
5. #70740C2semidirect
6. #69547C2semidirect
7. #69546C2semidirect
15. #5079C4semidirect
16. #5079C4semidirect
17. #5070C4semidirect
18. #5070C4semidirect
372. C3#9757semidirect

Maximal subgroups

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

subgroupindex
#832812
C3×#15172
#718142
#707932
#707402
#695472
#695462
#97573