Finite Group Databasebeta

← #10767980

Subgroups — #10767980

Order 768.

970 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 970 conjugacy classes of subgroups — too many to draw. Only its 183 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
1C3×#3501C211yesmaximal, characteristic
2#83369C212yesmaximal, complemented
3#82348C211yesmaximal, characteristic
4#71893C214yesmaximal, complemented
5#71890C214yesmaximal, complemented
6#71836C212yesmaximal, complemented
7#71603C211yesmaximal, characteristic
8#18181—31nomaximal, Sylow
9C3×#554—22no—
10C3×#554—24no—
11C3×D4×Q8—22no—
12C3×#544—21no—
13C3×#543C2214yes—
14C3×#533C2212yes—
15C2×C3×C22⋊Q8—22no—
16C3×#518—21no—
17C2×C3×C4×D4C2211yescharacteristic
18C2×C4×Dic6C2211yescharacteristic
19#5222—21no—
20C4×#217—22no—
21#5094—22no—
22#5090—22no—
23#5050C2214yes—
24#5043C2214yes—
25#5030—24no—
26#4985—21no—
27#4953C2212yes—
28#4947C2212yes—
29#4932C2211yescharacteristic
30#4926C2212yes—
31#4538C2218yes—
32#4525C2214yes—
33#3501D311yescharacteristic
34#3113—32no—
35#2398—31no—
36#1608—34no—
37#1602—34no—
38#1506—32no—
39#1466—31no—
40C22×C3×Q8—22no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4D412yes—
43C22×Dic6C2311yescharacteristic
44C3×C4⋊Q8—24no—
45C3×C4⋊Q8D412yes—
46C3×C4⋊Q8D412yes—
47C3×#125—24no—
48C3×#125—22no—
49C3×C22⋊Q8—24no—
50C3×C22⋊Q8—24no—
51C3×C22⋊Q8—28no—
52C3×C22⋊Q8—22no—
53C3×C22⋊Q8—24no—
54C3×C22⋊Q8—28no—
55C3×C22⋊Q8C2314yes—
56C3×C4×Q8—22no—
57C3×C4×Q8D414yes—
58C3×C4×D4—24no—
59C3×C4×D4—24no—
60C3×C4×D4—22no—
61C3×C4×D4—22no—
62C3×#117—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4C2312yes—
68C2×C3×C42C2311yescharacteristic
69C2×#217—22no—
70C2×#217—22no—
71C2×#216—24no—
72C2×C4⋊Dic3C2311yescharacteristic
73C2×Dic3⋊C4—21no—
74C2×C4×Dic3—21no—
75C2×C4×Dic3—21no—
76C4×Dic6—22no—
77C4×Dic6—22no—
78#835—24no—
79C3⋊#95—21no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95C2312yes—
83#830C2314yes—
84#810—24no—
85#808—24no—
86C4⋊(C3⋊C8)—22no—
87#803—22no—
88#554—22no—
89#554—24no—
90D4×Q8—22no—
91#544—21no—
92#543C2×D314yes—
93#533C2×D312yes—
94C2×C22⋊Q8—22no—
95#518—21no—
96C2×C4×Q8—31no—
97C2×C4×D4C2×D311yescharacteristic
98#483—34no—
99#475—62no—
100#462—62no—
101#451—34no—
102#441—64no—
103#432—32no—
104#400—61no—
105#397—32no—
106#394—61no—
107#393—32no—
108C4×C22⋊C4—62no—
109#376—31no—
110#328—38no—
111#323—34no—
112C24×C3C2×D412yes—
113C2×C3×Q8—24no—
114C2×C3×Q8—22no—
115C2×C3×Q8—84no—
116C2×C3×Q8C2×D414yes—
117C2×C3×D4—24no—
118C2×C3×D4—22no—
119C22×C3×C4—24no—
120C22×C3×C4—24no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—44no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D412yes—
127C22×C3×C4C2411yescharacteristic
128C22×Dic3—21no—
129C22×Dic3—22no—
130C22×Dic3—21no—
131C2×Dic6—22no—
132C2×Dic6—42no—
133C3×C4⋊C4—24no—
134C3×C4⋊C4—28no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—24no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—24no—
141C3×C4⋊C4C2×D414yes—
142C3×C22⋊C4—28no—
143C3×C22⋊C4—24no—
144C3×C22⋊C4—24no—
145C3×C42—24no—
146C3×C42C2×D412yes—
147C3×C42C2×D412yes—
148#217—44no—
149#217—44no—
150#216—48no—
151C4⋊Dic3—22no—
152Dic3⋊C4—42no—
153C4×Dic3—42no—
154C4×Dic3—41no—
155C4×Dic3—41no—
156C2×C3⋊C8—24no—
157C22×Q8—22no—
158C22×Q8—31no—
159C22×D4#6711yescharacteristic
160C23×C4#6712yes—
161C2×Q16—64no—
162C4⋊Q8—24no—
163C4⋊Q8#6712yes—
164C4⋊Q8#6712yes—
165#125—24no—
166#125—22no—
167C22⋊Q8—24no—
168C22⋊Q8—24no—
169C22⋊Q8—28no—
170C22⋊Q8—22no—
171C22⋊Q8—24no—
172C22⋊Q8—28no—
173C22⋊Q8C22×D314yes—
174C4×Q8—22no—
175C4×Q8—62no—
176C4×Q8—62no—
177C4×Q8#6714yes—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—22no—
181C4×D4—22no—
182#117—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—31no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4#6711yescharacteristic
188C2×C22⋊C4—24no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—62no—
191C2×C22⋊C4C22×D312yes—
192C2×C42—61no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195#106—64no—
196C4⋊C8—62no—
197#103—64no—
198#103—64no—
199C22⋊C8—34no—
200#97—62no—
201#95—32no—
202#95—61no—
203#95—62no—
204#95—62no—
205C23×C3—22no—
206C23×C3—22no—
207C23×C3—42no—
208C23×C3—42no—
209C23×C3#13714yes—
210C23×C3C22×D411yescharacteristic
211C3×Q8—44no—
212C3×Q8—44no—
213C3×Q8—84no—
214C3×D4—28no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—24no—
218C2×C3×C4—24no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—44no—
222C2×C3×C4—42no—
223C2×C3×C4—82no—
224C2×C3×C4—84no—
225C2×C3×C4—84no—
226C2×C3×C4#12011yescharacteristic
227C2×C3×C4#12012yes—
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4C22×D412yes—
230C2×Dic3—22no—
231C2×Dic3—41no—
232C2×Dic3—41no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—41no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238Dic6—24no—
239Dic6—42no—
240C3⋊C8—44no—
241C24C2×#6712yes—
242C2×Q8—24no—
243C2×Q8—22no—
244C2×Q8—62no—
245C2×Q8—84no—
246C2×Q8—122no—
247C2×Q8C2×#6714yes—
248C2×D4—24no—
249C2×D4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—44no—
255C22×C4—61no—
256C22×C4—62no—
257C22×C4—61no—
258C22×C4C2×#6711yescharacteristic
259C22×C4C2×#6711yescharacteristic
260C22×C4C2×#6712yes—
261C22×C4C23×D311yescharacteristic
262Q16—128no—
263C2×C8—64no—
264C4⋊C4—24no—
265C4⋊C4—28no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—62no—
273C4⋊C4—122no—
274C4⋊C4C2×#6714yes—
275C22⋊C4—28no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—124no—
279C22⋊C4—124no—
280C42—24no—
281C42—122no—
282C42—121no—
283C42—121no—
284C42C2×#6712yes—
285C42C2×#6712yes—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—24no—
289C22×C3—44no—
290C22×C3—42no—
291C22×C3—42no—
292C22×C3—82no—
293C22×C3C2×#12011yescharacteristic
294C22×C3C2×#13712yes—
295C3×C4—22no—
296C3×C4—21no—
297C3×C4—22no—
298C3×C4—42no—
299C3×C4—44no—
300C3×C4—44no—
301C3×C4#45512yes—
302Dic3—42no—
303Dic3—82no—
304Dic3—81no—
305C23—22no—
306C23—22no—
307C23—42no—
308C23—42no—
309C23#94214yes—
310C23C22×#6711yescharacteristic
311Q8—44no—
312Q8—44no—
313Q8—64no—
314Q8—84no—
315Q8—122no—
316D4—28no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—62no—
326C2×C4—82no—
327C2×C4—84no—
328C2×C4—84no—
329C2×C4—121no—
330C2×C4—121no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—121no—
336C2×C4#95312yes—
337C2×C4#95311yescharacteristic
338C2×C4#95311yescharacteristic
339C2×C4C22×#6712yes—
340C8—124no—
341C2×C3—22no—
342C2×C3—44no—
343C2×C3#300411yescharacteristic
344C2×C3C2×#45511yescharacteristic
345C2×C3#302511yescharacteristic
346C22—21no—
347C22—21no—
348C22—24no—
349C22—44no—
350C22—42no—
351C22—42no—
352C22—82no—
353C22C2×#94212yes—
354C22C2×#95311yescharacteristic
355C4—22no—
356C4—21no—
357C4—22no—
358C4—42no—
359C4—44no—
360C4—44no—
361C4—122no—
362C4—242no—
363C4—241no—
364C4#524212yes—
365C3#1818111yescharacteristic, complemented, Sylow
366C2—22no—
367C2—44no—
368C2#8405911yescharacteristic
369C2C2×#524211yescharacteristic
370C2#8412511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3501C2
socleC22×C3C2×#120
prime radicalC22×C3×D4D4
hypercenter#3501D3
hypocenterC3#18181
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
2. #83369C2semidirect
4. #71893C2semidirect
5. #71890C2semidirect
6. #71836C2semidirect
365. C3#18181semidirect

Maximal subgroups

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

subgroupindex
C3×#35012
#833692
#823482
#718932
#718902
#718362
#716032
#181813