Finite Group Databasebeta

← #10696844

Subgroups — #10696844

Order 768.

536 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3215C211yesmaximal, characteristic
2#83324C211yesmaximal, characteristic, complemented
3C2×#5060C211yesmaximal, characteristic
4#71822C211yesmaximal, characteristic, complemented
5#71720C211yesmaximal, characteristic
6#70823C211yesmaximal, characteristic, complemented
7#67751C211yesmaximal, characteristic, complemented
8#10133—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C22×#216—21no—
12C22×#214—21no—
13C2×#869C2211yescharacteristic
14C3×#497—21no—
15C3×#489—21no—
16C3×#464—22no—
17C3×#451—21no—
18C3×#450—21no—
19C3×#432C2211yescharacteristic
20#5079—21no—
21#5078—21no—
22#5060—22no—
23#5060—22no—
24#5044—22no—
25#5021—22no—
26#4926C2211yescharacteristic
27#4925C2211yescharacteristic
28C2×#803C2211yescharacteristic
29#4543—22no—
30#4525—22no—
31#4482—22no—
32#3230—31no—
33#3215D311yescharacteristic
34C2×#497—31no—
35#1556—31no—
36#1531—31no—
37#1469—31no—
38#1302—31no—
39C22×C3×Q8D411yescharacteristic
40C22×C3×D4D411yescharacteristic
41C22×Dic6—21no—
42C2×C3×Q16—22no—
43C2×C3×SD16—22no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×#124—21no—
48C3×#124—21no—
49C3×#124—21no—
50C3×#124—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×#117—21no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#216—24no—
60C2×#216—44no—
61C2×#214—24no—
62C2×#214—44no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3—21no—
65C22×C3⋊C8—21no—
66#869—21no—
67#869—21no—
68#869—21no—
69#869D412yes—
70C3×C4⋊C8D412yes—
71C3×#103—22no—
72C3×#102—22no—
73C3×C22⋊C8—21no—
74C3×#97—21no—
75#835—24no—
76#830—21no—
77#830D412yes—
78#807—24no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81#803—21no—
82#803—21no—
83#803D412yes—
84C4×C3⋊C8—21no—
85C22×Q16—61no—
86C22×SD16—61no—
87#533C2×D311yescharacteristic
88C2×C4⋊Q8—31no—
89C2×#124C2×D311yescharacteristic
90#500—61no—
91#497—21no—
92#497—62no—
93#497—62no—
94#489—21no—
95#487—61no—
96#484—62no—
97#477—62no—
98#464—22no—
99#451—21no—
100#450—21no—
101#432—31no—
102#432C2×D311yescharacteristic
103#431—31no—
104C2×#97—31no—
105#332—62no—
106#330—62no—
107#323—62no—
108C24×C3—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—42no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—42no—
117C22×C3×C4—21no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×Dic3—41no—
122C2×Dic6—22no—
123C2×Dic6—42no—
124C2×Dic6—42no—
125C2×Dic6—41no—
126C3×Q16—82no—
127C3×SD16—82no—
128C2×C3×C8—22no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C42—41no—
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145#216—48no—
146#214—48no—
147C4⋊Dic3—22no—
148C4⋊Dic3—22no—
149C4⋊Dic3—41no—
150C4⋊Dic3—41no—
151C2×C3⋊C8—24no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—41no—
155C22×Q8—61no—
156C22×Q8#6711yescharacteristic
157C22×D4#6711yescharacteristic
158C2×Q16—22no—
159C2×Q16—64no—
160C2×Q16—124no—
161C2×SD16—22no—
162C2×SD16—64no—
163C2×SD16—124no—
164C22×C8—61no—
165C4⋊Q8—32no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8—61no—
169C4⋊Q8#6711yescharacteristic
170C4⋊Q8#6711yescharacteristic
171#125—21no—
172#124—21no—
173#124—21no—
174#124—21no—
175#124—21no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180#117—21no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C42C22×D311yescharacteristic
186#107—64no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8#6712yes—
190#103—22no—
191#103—64no—
192#102—22no—
193C22⋊C8—21no—
194C22⋊C8—32no—
195C22⋊C8—61no—
196#97—21no—
197#97—32no—
198#97—61no—
199#97—61no—
200C4×C8—61no—
201C23×C3—41no—
202C23×C3—42no—
203C23×C3—81no—
204C23×C3—81no—
205C23×C3#12111yescharacteristic
206C3×Q8—42no—
207C3×Q8—42no—
208C3×Q8—42no—
209C3×Q8—42no—
210C3×D4—42no—
211C3×D4—42no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—81no—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12711yescharacteristic
225C2×Dic3—42no—
226C2×Dic3—82no—
227Dic6—42no—
228Dic6—42no—
229Dic6—82no—
230C3×C8—42no—
231C3⋊C8—24no—
232C3⋊C8—42no—
233C3⋊C8—81no—
234C24—21no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—42no—
240C2×Q8—62no—
241C2×Q8—122no—
242C2×Q8—122no—
243C2×Q8—121no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—42no—
247C22×C4—21no—
248C22×C4—121no—
249C22×C4D3×D411yescharacteristic
250C22×C4#23711yescharacteristic
251C22×C4C2×#6711yescharacteristic
252Q16—82no—
253Q16—128no—
254SD16—82no—
255SD16—128no—
256C2×C8—22no—
257C2×C8—64no—
258C2×C8—62no—
259C2×C8—62no—
260C2×C8—121no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—62no—
268C4⋊C4—62no—
269C4⋊C4—121no—
270C4⋊C4—121no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C42—41no—
277C42D3×D411yescharacteristic
278C42D3×D411yescharacteristic
279C42C2×#6711yescharacteristic
280C42C2×#6711yescharacteristic
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3—162no—
287C22×C3#39211yescharacteristic
288C22×C3#47312yes—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—81no—
295C3×C4—81no—
296C3×C4—81no—
297C3×C4#45312yes—
298Dic3—82no—
299C23—41no—
300C23—42no—
301C23—81no—
302C23—81no—
303C23#93911yescharacteristic
304Q8—42no—
305Q8—42no—
306Q8—42no—
307Q8—42no—
308Q8—122no—
309Q8—122no—
310Q8—242no—
311D4—42no—
312D4—42no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—122no—
321C2×C4—242no—
322C2×C4#93711yescharacteristic
323C2×C4#93711yescharacteristic
324C2×C4#93811yescharacteristic
325C2×C4#93911yescharacteristic
326C2×C4#94011yescharacteristic
327C2×C4#95311yescharacteristic
328C8—42no—
329C8—64no—
330C8—122no—
331C8—241no—
332C2×C3—22no—
333C2×C3—161no—
334C2×C3#200211yescharacteristic
335C2×C3#200911yescharacteristic
336C2×C3#202011yescharacteristic
337C22—21no—
338C22—21no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—162no—
343C22#520412yes—
344C22#522311yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—242no—
354C4#524112yes—
355C3#1013311yescharacteristic, complemented, Sylow
356C2—22no—
357C2—161no—
358C2#7185611yescharacteristic
359C2#7210911yescharacteristic
360C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3215C2
socleC22×C3#392
prime radicalC22×C3×D4D4
hypercenter#3215D3
hypocenterC3#10133
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3215

Complements

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

NH ≅ G/Ntype
2. #83324C2semidirect
4. #71822C2semidirect
6. #70823C2semidirect
7. #67751C2semidirect
355. C3#10133semidirect

Maximal subgroups

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

subgroupindex
C3×#32152
#833242
C2×#50602
#718222
#717202
#708232
#677512
#101333