Finite Group Databasebeta

← #10692082

Subgroups — #10692082

Order 768.

612 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2275C211yesmaximal, characteristic
2#82315C211yesmaximal, characteristic
3#71599C211yesmaximal, characteristic
4#69493C211yesmaximal, characteristic, complemented
5#69433C211yesmaximal, characteristic, complemented
6#67621C211yesmaximal, characteristic, complemented
7#67615C211yesmaximal, characteristic, complemented
8#9323—31nomaximal, Sylow
9C2×C3×C4×D4—21no—
10C2×C3×C4×D4C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×C4×C4⋊C4—21no—
13C3×C4×C4⋊C4C2211yescharacteristic
14C3×C4×C22⋊C4—22no—
15C3×C4×C22⋊C4—21no—
16C3×#376C2211yescharacteristic
17C2×C3×#95—21no—
18#5218—21no—
19#5019C412yescomplemented
20#5018—22no—
21#5007C412yescomplemented
22#4970—21no—
23#4926C2211yescharacteristic
24#4926C2211yescharacteristic
25C4×#208C2211yescharacteristic
26D4⋊(C3⋊C8)—22no—
27#4523—22no—
28#4452—21no—
29#4452—21no—
30#3110—31no—
31#2275D311yescharacteristic
32#1986—31no—
33#1506—31no—
34#1490—31no—
35#1300—31no—
36#1285—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—22no—
39C23×C3×C4—21no—
40C3×C4×D4—28no—
41C3×C4×D4—21no—
42C3×C4×D4—21no—
43C3×C4×D4C2×C412yes—
44C3×C4×D4D412yes—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C42—21no—
50C2×C3×C42—22no—
51C2×C3×C42D412yes—
52C2×C3×C42C2311yescharacteristic
53C2×#214—22no—
54#926—21no—
55#924—21no—
56#924—21no—
57C2×#208—21no—
58#870—21no—
59#869C2×C411yescharacteristic, complemented
60#869C2×C411yescharacteristic, complemented
61C4×Dic6—21no—
62C4×Dic6—21no—
63C3×#95—22no—
64C3×#95—21no—
65C3×#95—21no—
66C3×#95D411yescharacteristic
67C3×#95D411yescharacteristic
68#832—22no—
69#830—21no—
70#830—21no—
71#810—22no—
72#807—22no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)C2×C412yescomplemented
75#803—21no—
76#803—21no—
77#803—21no—
78C4×C3⋊C8—21no—
79#533—31no—
80C2×C4×D4—21no—
81C2×C4×D4C2×D311yescharacteristic
82#484—61no—
83#474—32no—
84#461—32no—
85#450—61no—
86#440—62no—
87#432—31no—
88#432—31no—
89C4×M16—31no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4C2×D311yescharacteristic
92C4×C22⋊C4—22no—
93C4×C22⋊C4—21no—
94#376C2×D311yescharacteristic
95C2×#95—21no—
96#339—62no—
97#328—61no—
98#328—61no—
99D4⋊C8—62no—
100C24×C3—21no—
101C2×C3×D4—24no—
102C2×C3×D4—21no—
103C2×C3×D4C22⋊C412yes—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—22no—
107C22×C3×C4—21no—
108C22×C3×C4—28no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—82no—
112C22×C3×C4—82no—
113C22×C3×C4—82no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C2×Dic6—21no—
118C2×Dic6—21no—
119C3×C4⋊C4—24no—
120C3×C4⋊C4—21no—
121C3×C4⋊C4C22⋊C412yes—
122C3×C22⋊C4—28no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—22no—
125C3×C42—24no—
126C3×C42—42no—
127C3×C42—82no—
128C3×C42C22×C411yescharacteristic
129C3×C42C2×D411yescharacteristic
130C3×C42#4111yescharacteristic
131C3×C42#4111yescharacteristic
132#217—41no—
133#214—44no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138Dic3⋊C4—41no—
139Dic3⋊C4—41no—
140C4×Dic3—41no—
141#208—22no—
142#208—41no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—41no—
145C2×C3⋊C8—41no—
146C22×D4#6711yescharacteristic
147C23×C4—22no—
148C23×C4—21no—
149C2×SD16—62no—
150C2×M16—61no—
151C4⋊Q8—31no—
152C4⋊Q8—31no—
153#125—61no—
154C22⋊Q8—61no—
155C22⋊Q8—61no—
156C4×Q8—61no—
157C4×Q8—61no—
158C4×D4—28no—
159C4×D4—21no—
160C4×D4—21no—
161C4×D4C4×D312yes—
162C4×D4D1212yes—
163#117—61no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4#6711yescharacteristic
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42#6712yes—
171C2×C42C22×D311yescharacteristic
172#107—62no—
173C4⋊C8—32no—
174C4⋊C8—61no—
175#103—62no—
176#102—62no—
177C22⋊C8—61no—
178C22⋊C8—61no—
179#97—61no—
180#97—61no—
181#97—61no—
182C4×C8—61no—
183#95—22no—
184#95—21no—
185#95—21no—
186#95#6711yescharacteristic
187#95#6711yescharacteristic
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—22no—
202C2×C3×C4—24no—
203C2×C3×C4—22no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—82no—
213C2×C3×C4—82no—
214C2×C3×C4—88no—
215C2×C3×C4C2×C22⋊C411yescharacteristic
216C2×C3×C4C4×D411yescharacteristic
217C2×C3×C4C4×D411yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12311yescharacteristic
221C2×Dic3—41no—
222C2×Dic3—41no—
223Dic6—42no—
224Dic6—81no—
225C3⋊C8—42no—
226C3⋊C8—42no—
227C3⋊C8—81no—
228C24—21no—
229C2×Q8—61no—
230C2×Q8—61no—
231C2×D4—24no—
232C2×D4—21no—
233C2×D4(C2×D3)⋊C412yes—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—28no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—82no—
242C22×C4—82no—
243C22×C4—82no—
244C22×C4C2×#6711yescharacteristic
245C22×C4C2×#6711yescharacteristic
246C22×C4C2×#6711yescharacteristic
247SD16—124no—
248M16—62no—
249M16—121no—
250C2×C8—62no—
251C2×C8—121no—
252C2×C8—121no—
253C4⋊C4—24no—
254C4⋊C4—21no—
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—28no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—121no—
266C42—24no—
267C42—42no—
268C42—82no—
269C42—121no—
270C42C2×C4×D311yescharacteristic
271C42C2×D1211yescharacteristic
272C42#23511yescharacteristic
273C42#23511yescharacteristic
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×C4—44no—
291C3×C4—42no—
292C3×C4—42no—
293C3×C4C2×#10412yes—
294Dic3—81no—
295Dic3—81no—
296C23—24no—
297C23—22no—
298C23—42no—
299C23—41no—
300C23—41no—
301C23#95311yescharacteristic
302Q8—122no—
303Q8—241no—
304D4—24no—
305D4—22no—
306D4#80514yes—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—88no—
325C2×C4—121no—
326C2×C4—121no—
327C2×C4C2×(C2×D3)⋊C411yescharacteristic
328C2×C4C4×#6711yescharacteristic
329C2×C4C4×#6711yescharacteristic
330C2×C4#92911yescharacteristic
331C2×C4#93011yescharacteristic
332C2×C4#93011yescharacteristic
333C8—122no—
334C8—122no—
335C8—241no—
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#521311yescharacteristic
350C22#524211yescharacteristic
351C22#524611yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—42no—
358C4—44no—
359C4—42no—
360C4—42no—
361C4—241no—
362C4—241no—
363C4C2×#80512yes—
364C3#932311yescharacteristic, complemented, Sylow
365C2—21no—
366C2—21no—
367C2—44no—
368C2#7153211yescharacteristic
369C2#7165111yescharacteristic
370C2#7166611yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2275C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#2275D3
hypocenterC3#9323
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2275

Complements

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

NH ≅ G/Ntype
4. #69493C2semidirect
5. #69433C2semidirect
6. #67621C2semidirect
7. #67615C2semidirect
19. #5019C4semidirect
21. #5007C4semidirect
59. #869C2×C4semidirect
60. #869C2×C4semidirect
74. 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
C3×#22752
#823152
#715992
#694932
#694332
#676212
#676152
#93233