Finite Group Databasebeta

← #10692080

Subgroups — #10692080

Order 768.

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

normal subgroups only

This group has 612 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#82317C211yesmaximal, characteristic
2#82315C211yesmaximal, characteristic
3C3×#1490C211yesmaximal, characteristic
4#70836C211yesmaximal, characteristic, complemented
5#69868C211yesmaximal, characteristic, complemented
6#67749C211yesmaximal, characteristic, complemented
7#67665C211yesmaximal, characteristic, complemented
8#9323—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×D4×Dic3—21no—
11#5470C2211yescharacteristic
12C3×#432C2211yescharacteristic
13C3×C4×M16C2211yescharacteristic
14#5218—21no—
15C4×#217—21no—
16C2×C3⋊#95—21no—
17#5019C412yescomplemented
18#5018—22no—
19#5007C412yescomplemented
20Dic3×C4⋊C4—21no—
21#4970—21no—
22Dic3×C22⋊C4—22no—
23C4×C4⋊Dic3C2211yescharacteristic
24#4932C2211yescharacteristic
25#4926C2211yescharacteristic
26C3×D4⋊C8—22no—
27#4555—22no—
28#4467—21no—
29#4467—21no—
30#3110—31no—
31#2275—31no—
32#1986—31no—
33#1506—31no—
34#1490D311yescharacteristic
35#1300—31no—
36#1285—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C23×Dic3—22no—
40C2×C3×M16—21no—
41C3×C4×D4—21no—
42C3×C4×D4—21no—
43C3×C4×D4C2×C412yes—
44C3×C4×D4D412yes—
45C2×C3×C4⋊C4D411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#217—22no—
49D4×Dic3—28no—
50C2×#214—22no—
51#926—21no—
52C2×C4⋊Dic3—21no—
53#924—21no—
54#924—21no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3—22no—
57C2×C4×Dic3D412yes—
58#870—21no—
59#869C2×C411yescharacteristic, complemented
60#869C2×C411yescharacteristic, complemented
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×C4⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×#97—21no—
66C3×#97—21no—
67C3×C4×C8—21no—
68#832—22no—
69C3⋊#95—22no—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95D411yescharacteristic
73C3⋊#95D411yescharacteristic
74#830—21no—
75#810—22no—
76#807—22no—
77C4⋊(C3⋊C8)C2×C412yescomplemented
78#803—21no—
79#533—31no—
80C2×C4×D4—61no—
81C2×C4×D4C2×D311yescharacteristic
82#484—61no—
83#474—32no—
84#461—32no—
85#450—61no—
86#440—62no—
87#432—31no—
88#432C2×D311yescharacteristic
89C4×M16C2×D311yescharacteristic
90C4×C4⋊C4—31no—
91C4×C4⋊C4—61no—
92C4×C22⋊C4—61no—
93C4×C22⋊C4—62no—
94#376—31no—
95C2×#95—61no—
96#339—62no—
97#328—61no—
98#328—61no—
99D4⋊C8—22no—
100C24×C3—21no—
101C2×C3×D4—24no—
102C2×C3×D4—21no—
103C2×C3×D4C22⋊C412yes—
104C22×C3×C4—21no—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—82no—
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×Dic3—21no—
112C22×Dic3—22no—
113C22×Dic3—21no—
114C22×Dic3—28no—
115C22×Dic3—82no—
116C22×Dic3—82no—
117C2×Dic6—21no—
118C2×Dic6—21no—
119C3×M16—22no—
120C3×M16—41no—
121C2×C3×C8—41no—
122C2×C3×C8—41no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4C22⋊C412yes—
125C3×C22⋊C4—22no—
126C3×C22⋊C4—22no—
127C3×C42C22×C411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42#4111yescharacteristic
130C3×C42#4111yescharacteristic
131#217—28no—
132#217—41no—
133#214—44no—
134C4⋊Dic3—24no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139Dic3⋊C4—41no—
140Dic3⋊C4—41no—
141C4×Dic3—24no—
142C4×Dic3—41no—
143C4×Dic3—42no—
144C4×Dic3—82no—
145C2×C3⋊C8—22no—
146C22×D4D1211yescharacteristic
147C23×C4—21no—
148C23×C4—62no—
149C2×SD16—62no—
150C2×M16—21no—
151C4⋊Q8—31no—
152C4⋊Q8—31no—
153#125—61no—
154C22⋊Q8—61no—
155C22⋊Q8—61no—
156C4×Q8—61no—
157C4×Q8—61no—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—68no—
161C4×D4C4×D312yes—
162C4×D4#6712yes—
163#117—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4D1211yescharacteristic
166C2×C22⋊C4—21no—
167C2×C22⋊C4—62no—
168C2×C42—32no—
169C2×C42—61no—
170C2×C42—62no—
171C2×C42C22×D311yescharacteristic
172#107—62no—
173C4⋊C8—21no—
174C4⋊C8—32no—
175#103—62no—
176#102—62no—
177C22⋊C8—21no—
178C22⋊C8—61no—
179#97—21no—
180#97—21no—
181#97—61no—
182C4×C8—21no—
183#95—31no—
184#95—31no—
185#95—62no—
186#95—61no—
187#95—61no—
188C23×C3—24no—
189C23×C3—22no—
190C23×C3—42no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3#12011yescharacteristic
194C3×D4—24no—
195C3×D4—22no—
196C3×D4#10414yes—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—41no—
202C2×C3×C4—42no—
203C2×C3×C4—82no—
204C2×C3×C4—82no—
205C2×C3×C4C2×C22⋊C411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4C4×D411yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12311yescharacteristic
211C2×Dic3—22no—
212C2×Dic3—22no—
213C2×Dic3—24no—
214C2×Dic3—41no—
215C2×Dic3—42no—
216C2×Dic3—41no—
217C2×Dic3—42no—
218C2×Dic3—41no—
219C2×Dic3—41no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—88no—
223Dic6—42no—
224Dic6—81no—
225C3×C8—42no—
226C3×C8—81no—
227C3⋊C8—42no—
228C24—21no—
229C2×Q8—61no—
230C2×Q8—61no—
231C2×D4—24no—
232C2×D4—21no—
233C2×D4(C2×D3)⋊C412yes—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—62no—
240C22×C4—68no—
241C22×C4—82no—
242C22×C4—242no—
243C22×C4—242no—
244C22×C4C2×D1211yescharacteristic
245C22×C4D3×D411yescharacteristic
246C22×C4D3×D411yescharacteristic
247SD16—124no—
248M16—22no—
249M16—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—62no—
253C4⋊C4—21no—
254C4⋊C4—64no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C4⋊C4(C2×D3)⋊C412yes—
262C22⋊C4—22no—
263C22⋊C4—22no—
264C22⋊C4—68no—
265C22⋊C4—121no—
266C42—64no—
267C42—122no—
268C42—121no—
269C42—242no—
270C42C2×C4×D311yescharacteristic
271C42#23711yescharacteristic
272C42#23711yescharacteristic
273C42C2×#6711yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—24no—
277C22×C3—44no—
278C22×C3—44no—
279C22×C3—42no—
280C22×C3—42no—
281C22×C3#38611yescharacteristic
282C22×C3#45511yescharacteristic
283C22×C3#45611yescharacteristic
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—42no—
290C3×C4C2×#10412yes—
291Dic3—42no—
292Dic3—44no—
293Dic3—42no—
294Dic3—81no—
295Dic3—81no—
296C23—24no—
297C23—22no—
298C23—42no—
299C23—41no—
300C23—41no—
301C23#88211yescharacteristic
302Q8—122no—
303Q8—241no—
304D4—24no—
305D4—22no—
306D4#82514yes—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—62no—
314C2×C4—62no—
315C2×C4—64no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—121no—
319C2×C4—122no—
320C2×C4—121no—
321C2×C4—122no—
322C2×C4—121no—
323C2×C4—121no—
324C2×C4—121no—
325C2×C4—121no—
326C2×C4—248no—
327C2×C4#67⋊C411yescharacteristic
328C2×C4#67⋊C411yescharacteristic
329C2×C4#88611yescharacteristic
330C2×C4C2×(C2×D3)⋊C411yescharacteristic
331C2×C4#93911yescharacteristic
332C2×C4#93911yescharacteristic
333C8—42no—
334C8—81no—
335C8—122no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—44no—
339C2×C3#188211yescharacteristic
340C2×C3#189611yescharacteristic
341C2×C3#190411yescharacteristic
342C22—21no—
343C22—21no—
344C22—24no—
345C22—44no—
346C22—44no—
347C22—42no—
348C22—42no—
349C22#482511yescharacteristic
350C22#482611yescharacteristic
351C22#495811yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—42no—
358C4—122no—
359C4—124no—
360C4—122no—
361C4—241no—
362C4—241no—
363C4C2×#82512yes—
364C3#932311yescharacteristic, complemented, Sylow
365C2—21no—
366C2—21no—
367C2—44no—
368C2#6996511yescharacteristic
369C2#7031411yescharacteristic
370C2#7036511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1490C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#1490D3
hypocenterC3#9323
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1490

Complements

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

NH ≅ G/Ntype
4. #70836C2semidirect
5. #69868C2semidirect
6. #67749C2semidirect
7. #67665C2semidirect
17. #5019C4semidirect
19. #5007C4semidirect
59. #869C2×C4semidirect
60. #869C2×C4semidirect
77. C4⋊(C3⋊C8)C2×C4semidirect
364. C3#9323semidirect

Maximal subgroups

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

subgroupindex
#823172
#823152
C3×#14902
#708362
#698682
#677492
#676652
#93233