Finite Group Databasebeta

← #10692101

Subgroups — #10692101

Order 768.

574 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 107 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#82450C211yesmaximal, characteristic, complemented
2#82444C211yesmaximal, characteristic
3C3×#1491C211yesmaximal, characteristic, complemented
4#70832C211yesmaximal, characteristic
5#69867C211yesmaximal, characteristic
6#67749C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#9326—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×Q8×Dic3—21no—
11#5477C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5227—21no—
15C2×#825—22no—
16#5027C412yescomplemented
17#5026—22no—
18#5021C412yescomplemented
19Dic3×C4⋊C4—21no—
20Dic3×C4⋊C4—22no—
21#4967—21no—
22C4×C4⋊Dic3—21no—
23C4×C4⋊Dic3C2211yescharacteristic
24#4932—21no—
25#4932C2211yescharacteristic
26#4926C2211yescharacteristic
27C3×Q8⋊C8—22no—
28#4470—21no—
29#4470—21no—
30#3112—31no—
31#2276—31no—
32#1980—31no—
33#1505—31no—
34#1491D311yescharacteristic, complemented
35#1300—31no—
36#1285—31no—
37C22×C3×Q8D411yescharacteristic
38C2×#235—21no—
39C2×C3×M16—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8C2×C412yes—
43C3×C4×Q8D412yes—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4D411yescharacteristic
46C2×C3×C42—21no—
47C2×C3×C42C2311yescharacteristic
48Q8×Dic3—28no—
49C2×#215—22no—
50#930—21no—
51#930—21no—
52C2×C4⋊Dic3—22no—
53C2×C4⋊Dic3—21no—
54C2×C4×Dic3—22no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3—22no—
57C2×C4×Dic3D412yes—
58#875—21no—
59#874C2×C411yescharacteristic
60C4×D12—21no—
61C4×D12—21no—
62#869C2×C411yescharacteristic
63C3×C4⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#97—21no—
66C3×#97—21no—
67C3×C4×C8—21no—
68#835—22no—
69C3⋊#95—22no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95D411yescharacteristic
73C3⋊#95D411yescharacteristic
74#830—21no—
75#825—44no—
76#809—22no—
77#807—22no—
78C4⋊(C3⋊C8)C2×C412yes—
79#803—21no—
80#532—31no—
81C2×C4×Q8—61no—
82C2×C4×Q8C2×D311yescharacteristic
83#480—61no—
84#477—32no—
85#460—32no—
86#448—61no—
87#440—62no—
88#432—31no—
89#432C2×D311yescharacteristic
90C2×#104—62no—
91C4×M16C2×D311yescharacteristic
92C4×C4⋊C4—31no—
93C4×C4⋊C4—61no—
94C4×C4⋊C4—62no—
95C4×C4⋊C4—61no—
96#376—31no—
97#376—61no—
98#327—61no—
99#327—61no—
100Q8⋊C8—22no—
101C2×C3×Q8—24no—
102C2×C3×Q8—21no—
103C2×C3×Q8C22⋊C412yes—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C2×#67—41no—
110C22×Dic3—21no—
111C22×Dic3—22no—
112C22×Dic3—21no—
113#235—42no—
114#235—42no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×C4×D3—41no—
119C2×Dic6—21no—
120C3×M16—22no—
121C3×M16—41no—
122C2×C3×C8—41no—
123C2×C3×C8—41no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4C22⋊C412yes—
128C3×C42—22no—
129C3×C42—22no—
130C3×C42C22×C411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133C3×C42#4111yescharacteristic
134#215—44no—
135(C2×D3)⋊C4—41no—
136(C2×D3)⋊C4—41no—
137C4⋊Dic3—28no—
138C4⋊Dic3—24no—
139C4⋊Dic3—21no—
140C4⋊Dic3—21no—
141C4×Dic3—28no—
142C4×Dic3—24no—
143C4×Dic3—42no—
144C4×Dic3—82no—
145C2×C3⋊C8—22no—
146C2×#41—61no—
147C22×Q8D1211yescharacteristic
148C2×SD16—62no—
149C2×M16—21no—
150C4⋊Q8—31no—
151#127—31no—
152#124—61no—
153#121—61no—
154#121—61no—
155C4×Q8—21no—
156C4×Q8—21no—
157C4×Q8—68no—
158C4×Q8C4×D312yes—
159C4×Q8#6712yes—
160C4×D4—61no—
161C4×D4—61no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—62no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4D1211yescharacteristic
166C2×C42—21no—
167C2×C42—32no—
168C2×C42—62no—
169C2×C42—61no—
170C2×C42—62no—
171C2×C42C22×D311yescharacteristic
172#107—62no—
173C4⋊C8—21no—
174C4⋊C8—32no—
175#104—124no—
176#103—62no—
177#102—62no—
178C22⋊C8—21no—
179C22⋊C8—61no—
180#97—21no—
181#97—21no—
182#97—61no—
183C4×C8—21no—
184#95—31no—
185#95—31no—
186#95—62no—
187#95—61no—
188#95—61no—
189C23×C3#12011yescharacteristic
190C22×D3—41no—
191C3×Q8—24no—
192C3×Q8—22no—
193C3×Q8#10414yescomplemented
194C2×C3×C4—24no—
195C2×C3×C4—22no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4C2×C22⋊C411yescharacteristic
204C2×C3×C4C4×D411yescharacteristic
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12311yescharacteristic
209#67—81no—
210#67—81no—
211C2×Dic3—22no—
212C2×Dic3—24no—
213C2×Dic3—22no—
214C2×Dic3—41no—
215C2×Dic3—42no—
216C2×Dic3—41no—
217C2×Dic3—42no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221D12—42no—
222D12—42no—
223D12—81no—
224C4×D3—82no—
225Dic6—42no—
226C3×C8—42no—
227C3×C8—81no—
228C3⋊C8—42no—
229#41—122no—
230#41—122no—
231C2×Q8—24no—
232C2×Q8—21no—
233C2×Q8—61no—
234C2×Q8(C2×D3)⋊C412yes—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—121no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4—62no—
244C22×C4—121no—
245C22×C4C2×D1211yescharacteristic
246C22×C4D3×D411yescharacteristic
247C22×C4D3×D411yescharacteristic
248SD16—124no—
249M16—22no—
250M16—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—62no—
254C4⋊C4—22no—
255C4⋊C4—21no—
256C4⋊C4—22no—
257C4⋊C4—68no—
258C4⋊C4—64no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4(C2×D3)⋊C412yes—
262C22⋊C4—121no—
263C22⋊C4—121no—
264C42—22no—
265C42—22no—
266C42—68no—
267C42—64no—
268C42—122no—
269C42—242no—
270C42C2×C4×D311yescharacteristic
271C42#23711yescharacteristic
272C42#23711yescharacteristic
273C42C2×#6711yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3#38611yescharacteristic
277C22×C3#45311yescharacteristic
278C22×C3#45411yescharacteristic
279C2×D3—81no—
280C2×D3—81no—
281C2×D3—81no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—24no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—42no—
289C3×C4C2×#10412yes—
290Dic3—44no—
291Dic3—42no—
292Dic3—42no—
293Dic3—81no—
294C23—121no—
295C23#88211yescharacteristic
296Q8—24no—
297Q8—22no—
298Q8—122no—
299Q8#82514yescomplemented
300D4—122no—
301D4—122no—
302D4—241no—
303D4—241no—
304D4—241no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—62no—
315C2×C4—62no—
316C2×C4—64no—
317C2×C4—121no—
318C2×C4—121no—
319C2×C4—122no—
320C2×C4—122no—
321C2×C4—121no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—242no—
325C2×C4#67⋊C411yescharacteristic
326C2×C4#67⋊C411yescharacteristic
327C2×C4#88611yescharacteristic
328C2×C4C2×(C2×D3)⋊C411yescharacteristic
329C2×C4#93911yescharacteristic
330C2×C4#93911yescharacteristic
331C8—42no—
332C8—81no—
333C8—122no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3#188311yescharacteristic
337C2×C3#189611yescharacteristic
338C2×C3#190511yescharacteristic
339D3—161no—
340C22—21no—
341C22—21no—
342C22—241no—
343C22—241no—
344C22—241no—
345C22#482211yescharacteristic
346C22#482711yescharacteristic
347C22#495811yescharacteristic
348C4—24no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—42no—
355C4—124no—
356C4—122no—
357C4—122no—
358C4—241no—
359C4C2×#82512yes—
360C3#932611yescharacteristic, complemented, Sylow
361C2—21no—
362C2—21no—
363C2—481no—
364C2#6996511yescharacteristic
365C2#7033411yescharacteristic
366C2#7036611yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1491C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#1491D3
hypocenterC3#9326
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1491

Complements

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

NH ≅ G/Ntype
1. #82450C2semidirect
3. C3×#1491C2semidirect
6. #67749C2semidirect
7. #67665C2semidirect
16. #5027C4semidirect
18. #5021C4semidirect
34. #1491D3semidirect
193. C3×Q8#104semidirect
299. Q8#825semidirect
360. C3#9326semidirect

Maximal subgroups

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

subgroupindex
#824502
#824442
C3×#14912
#708322
#698672
#677492
#676652
#93263