Finite Group Databasebeta

← #10692492

Subgroups — #10692492

Order 768.

462 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83273C211yesmaximal, characteristic
2#74261C211yesmaximal, characteristic, complemented
3C3×#1514C211yesmaximal, characteristic
4#71818C211yesmaximal, characteristic
5#70812C211yesmaximal, characteristic, complemented
6#67750C211yesmaximal, characteristic, complemented
7#67633C211yesmaximal, characteristic, complemented
8#9382—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#870C2211yescharacteristic
11C3×#432C2211yescharacteristic
12C3×#383C2211yescharacteristic
13C2×#837—21no—
14C2×#837—21no—
15#5081C412yescomplemented
16#5071C412yescomplemented
17#5034—21no—
18#5033—21no—
19#5009—22no—
20C4×C4⋊Dic3C2211yescharacteristic
21#4926C2211yescharacteristic
22C4×#208C2211yescharacteristic
23#4655—21no—
24#4654—21no—
25#4653—21no—
26#4652—21no—
27#4649—21no—
28#4647—21no—
29C3×#327—21no—
30C3×#327—21no—
31#4465—22no—
32#3234—31no—
33#2336—31no—
34#1980—31no—
35#1528—31no—
36#1514D311yescharacteristic
37#1288—31no—
38#1285—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×C42C2311yescharacteristic
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3D411yescharacteristic
50C2×C4⋊Dic3D411yescharacteristic
51C2×Dic3⋊C4—21no—
52C2×Dic3⋊C4—21no—
53C2×Dic3⋊C4—21no—
54C2×Dic3⋊C4—21no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3—21no—
57C2×C4×Dic3D411yescharacteristic
58C2×C4×Dic3D411yescharacteristic
59C2×#208—21no—
60#870—21no—
61#870—21no—
62#870C2×C412yes—
63#870D412yes—
64C3×C4⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×#97—21no—
67C3×#95—21no—
68C3×#95—21no—
69C3×#95D411yescharacteristic
70C3×#95D411yescharacteristic
71#837—42no—
72#837—42no—
73#832—22no—
74#832—22no—
75C3⋊#95—21no—
76C3⋊#95—21no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80#830—21no—
81#808—22no—
82#807—22no—
83C4⋊(C3⋊C8)—21no—
84#803—21no—
85#803C2×C412yes—
86C4×C3⋊C8—21no—
87#532C2×D311yescharacteristic
88C2×#125—31no—
89#501—32no—
90#490—32no—
91#481—61no—
92#480—61no—
93#478—62no—
94#432—31no—
95#432C2×D311yescharacteristic
96C2×#104—61no—
97C2×#104—61no—
98C4×M16—31no—
99#389—61no—
100#387—61no—
101#387—61no—
102#387—61no—
103#387—61no—
104#383C2×D311yescharacteristic
105C4×C4⋊C4—31no—
106C4×C4⋊C4—61no—
107#330—62no—
108#327—21no—
109#327—21no—
110C3×#41—41no—
111C3×#41—41no—
112C3×#41—41no—
113C3×#41—41no—
114C2×C3×Q8—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—41no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×Dic3—21no—
126C22×Dic3—21no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C2×C3×C8—41no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C42C22×C411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42#4111yescharacteristic
137C3×C42#4111yescharacteristic
138C4⋊Dic3—21no—
139C4⋊Dic3—21no—
140C4⋊Dic3—42no—
141C4⋊Dic3C22⋊C412yes—
142C4⋊Dic3C22⋊C412yes—
143Dic3⋊C4—22no—
144Dic3⋊C4—22no—
145Dic3⋊C4—22no—
146Dic3⋊C4—22no—
147Dic3⋊C4—42no—
148Dic3⋊C4—81no—
149C4×Dic3—42no—
150C4×Dic3—41no—
151C4×Dic3—41no—
152C4×Dic3—81no—
153#208—22no—
154#208—41no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—41no—
157C2×#41—21no—
158C2×M16—61no—
159C4⋊Q8C4×D311yescharacteristic
160#127C4×D311yescharacteristic
161#125—32no—
162#125—32no—
163#125—61no—
164#125—61no—
165#124—21no—
166#121—21no—
167#121—21no—
168C4×D4—21no—
169C4×D4—21no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—31no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C42—31no—
178C2×C42—31no—
179C2×C42—61no—
180C2×C42—61no—
181C2×C42C22×D311yescharacteristic
182#107—62no—
183#106—62no—
184C4⋊C8—21no—
185C4⋊C8—61no—
186#104—122no—
187#104—122no—
188#102—62no—
189#102—62no—
190C22⋊C8—21no—
191C22⋊C8—61no—
192#97—21no—
193#97—32no—
194#97—61no—
195C4×C8—61no—
196#95—21no—
197#95—21no—
198#95—61no—
199#95—61no—
200#95—61no—
201#95—61no—
202#95—61no—
203#95#6711yescharacteristic
204#95#6711yescharacteristic
205C23×C3—41no—
206C23×C3#12011yescharacteristic
207C3×Q8—41no—
208C3×Q8—41no—
209C3×D4—42no—
210C3×D4—41no—
211C3×D4—41no—
212C3×D4—81no—
213C3×D4—81no—
214C3×D4—81no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×C3×C4C2×C22⋊C411yescharacteristic
229C2×C3×C4C4×D411yescharacteristic
230C2×C3×C4C4×D411yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12311yescharacteristic
234C2×Dic3—22no—
235C2×Dic3—22no—
236C2×Dic3—22no—
237C2×Dic3—41no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C3×C8—81no—
247C3⋊C8—42no—
248C3⋊C8—42no—
249#41—41no—
250#41—41no—
251#41—41no—
252#41—41no—
253C2×Q8—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4D3×D411yescharacteristic
266C22×C4D3×D411yescharacteristic
267C22×C4C2×#6711yescharacteristic
268M16—62no—
269M16—121no—
270C2×C8—41no—
271C2×C8—62no—
272C2×C8—121no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—32no—
276C4⋊C4—32no—
277C4⋊C4—62no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—62no—
281C4⋊C4—62no—
282C4⋊C4—62no—
283C4⋊C4—122no—
284C4⋊C4—122no—
285C4⋊C4—241no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C42—121no—
289C42—121no—
290C42—122no—
291C42—241no—
292C42C2×C4×D311yescharacteristic
293C42#23511yescharacteristic
294C42D3×D411yescharacteristic
295C42#23711yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—81no—
299C22×C3—81no—
300C22×C3—81no—
301C22×C3#38611yescharacteristic
302C22×C3#45311yescharacteristic
303C22×C3#45411yescharacteristic
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—81no—
311C3×C4—81no—
312C3×C4—81no—
313Dic3—42no—
314Dic3—42no—
315Dic3—42no—
316Dic3—81no—
317C23—41no—
318C23#93711yescharacteristic
319Q8—41no—
320Q8—41no—
321D4—41no—
322D4—41no—
323D4—42no—
324D4—81no—
325D4—81no—
326D4—81no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—62no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—121no—
347C2×C4—121no—
348C2×C4—121no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4D3×C22⋊C411yescharacteristic
353C2×C4#67⋊C411yescharacteristic
354C2×C4#88311yescharacteristic
355C2×C4#88411yescharacteristic
356C2×C4C4×#6711yescharacteristic
357C2×C4#93811yescharacteristic
358C8—81no—
359C8—122no—
360C8—122no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—161no—
364C2×C3#188311yescharacteristic
365C2×C3#190611yescharacteristic
366C2×C3#191211yescharacteristic
367C22—21no—
368C22—21no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#495611yescharacteristic
373C22#519811yescharacteristic
374C22#520111yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—122no—
385C4—122no—
386C4—122no—
387C4—241no—
388C3#938211yescharacteristic, complemented, Sylow
389C2—21no—
390C2—21no—
391C2—161no—
392C2#7005511yescharacteristic
393C2#7032811yescharacteristic
394C2#7041111yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1514C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1514D3
hypocenterC3#9382
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1514

Complements

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

NH ≅ G/Ntype
2. #74261C2semidirect
5. #70812C2semidirect
6. #67750C2semidirect
7. #67633C2semidirect
15. #5081C4semidirect
16. #5071C4semidirect
388. C3#9382semidirect

Maximal subgroups

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

subgroupindex
#832732
#742612
C3×#15142
#718182
#708122
#677502
#676332
#93823