Finite Group Databasebeta

← #10697834

Subgroups — #10697834

Order 768.

626 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 109 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×#3227C211yesmaximal, characteristic
2#83317C211yesmaximal, characteristic
3#83253C211yesmaximal, characteristic, complemented
4#71815C211yesmaximal, characteristic
5#71775C211yesmaximal, characteristic, complemented
6#70779C211yesmaximal, characteristic, complemented
7#67736C211yesmaximal, characteristic, complemented
8#10283—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#944—22no—
12C2×#869C2211yescharacteristic
13C3×#498—21no—
14C3×#486—21no—
15C3×#481—22no—
16C3×#476—22no—
17C3×#431C2211yescharacteristic
18C2×#837—22no—
19#5093—21no—
20#5076—21no—
21#5040—22no—
22#5019—22no—
23#4975—22no—
24#4975—22no—
25C42×Dic3C2211yescharacteristic
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4547—22no—
29#4481—22no—
30#3227D311yescharacteristic
31#3206—31no—
32#2868—31no—
33#1962—31no—
34#1549—31no—
35#1548—31no—
36#1276—31no—
37C2×C3×#41—21no—
38C22×C3×Q8—21no—
39C22×Dic6D412yes—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8D411yescharacteristic
45C3×#127D411yescharacteristic
46C3×#124—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D412yes—
53C2×C3×C42C2311yescharacteristic
54#944—48no—
55C2×#214—24no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×Dic3⋊C4—28no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3D412yes—
62C2×#208—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67C3×#106—24no—
68C3×C4⋊C8D412yes—
69C3×#102—24no—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72#837—44no—
73#830—21no—
74#810—24no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D412yes—
78C4×C3⋊C8—21no—
79#532C2×D311yescharacteristic
80C2×C4⋊Q8—31no—
81C2×C4⋊Q8—62no—
82C2×C4⋊Q8C2×D311yescharacteristic
83#498—21no—
84#492—61no—
85#487—61no—
86#486—21no—
87#481—22no—
88#476—22no—
89#461—62no—
90#448—62no—
91#431—31no—
92#431C2×D311yescharacteristic
93#423—31no—
94C2×#104—62no—
95#384—62no—
96#384—62no—
97C43—31no—
98#333—62no—
99#332—62no—
100C3×#41—42no—
101C3×#41—42no—
102C2×C3×Q8—21no—
103C2×C3×Q8—44no—
104C2×C3×Q8—41no—
105C2×C3×Q8—41no—
106C2×C3×D4—21no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—41no—
110C22×C3×C4—22no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×Dic3—22no—
116C22×Dic3—22no—
117C2×Dic6—22no—
118C2×Dic6—22no—
119C2×Dic6—44no—
120C2×C3×C8—22no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C42C2×D411yescharacteristic
130C3×C42C2×D411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133#214—84no—
134C4⋊Dic3—41no—
135C4⋊Dic3—41no—
136C4⋊Dic3—41no—
137C4⋊Dic3—41no—
138Dic3⋊C4—416no—
139C4×Dic3—24no—
140C4×Dic3—24no—
141C4×Dic3—22no—
142C4×Dic3SD1614yescomplemented
143#208—42no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—41no—
146C2×#41—21no—
147C22×Q8—21no—
148C22×Q8—32no—
149C2×SD16—64no—
150C2×M16—61no—
151C4⋊Q8—21no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—61no—
156C4⋊Q8—61no—
157C4⋊Q8—61no—
158C4⋊Q8—61no—
159C4⋊Q8—128no—
160C4⋊Q8#6711yescharacteristic
161#127#6711yescharacteristic
162#124—21no—
163#121—21no—
164#121—21no—
165C4×D4—21no—
166C4×D4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—68no—
169C2×C4⋊C4—61no—
170C2×C4⋊C4—61no—
171C2×C4⋊C4#6712yes—
172C2×C42—32no—
173C2×C42—61no—
174C2×C42—61no—
175C2×C42C22×D311yescharacteristic
176#106—24no—
177C4⋊C8—32no—
178C4⋊C8—61no—
179C4⋊C8—61no—
180C4⋊C8#6712yes—
181#104—124no—
182#103—64no—
183#102—24no—
184C22⋊C8—21no—
185C22⋊C8—61no—
186C4×C8—21no—
187C4×C8—61no—
188C23×C3—41no—
189C23×C3#12011yescharacteristic
190C3×Q8—42no—
191C3×Q8—82no—
192C3×Q8—82no—
193C3×D4—42no—
194C3×D4—42no—
195C3×D4—42no—
196C3×D4—82no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—44no—
200C2×C3×C4—41no—
201C2×C3×C4—42no—
202C2×C3×C4—81no—
203C2×C3×C4—81no—
204C2×C3×C4#12011yescharacteristic
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12111yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12711yescharacteristic
210C2×C3×C4C2×SD1612yes—
211C2×C3×C4#13712yes—
212C2×Dic3—24no—
213C2×Dic3—24no—
214C2×Dic3—24no—
215C2×Dic3—44no—
216C2×Dic3—42no—
217Dic6—44no—
218Dic6—82no—
219C3×C8—42no—
220C3⋊C8—42no—
221C3⋊C8—81no—
222#41—42no—
223#41—42no—
224C2×Q8—21no—
225C2×Q8—44no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—62no—
229C2×Q8—62no—
230C2×Q8—124no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—41no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—62no—
238C22×C4—62no—
239C22×C4D3×D411yescharacteristic
240C22×C4D3×D411yescharacteristic
241C22×C4C2×#6711yescharacteristic
242SD16—244no—
243M16—122no—
244C2×C8—22no—
245C2×C8—62no—
246C2×C8—121no—
247C4⋊C4—21no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—1216no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C42—34no—
261C42—64no—
262C42—64no—
263C42—62no—
264C42D3×D411yescharacteristic
265C42#23711yescharacteristic
266C42C2×#6711yescharacteristic
267C42C2×#6711yescharacteristic
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—81no—
271C22×C3—81no—
272C22×C3—81no—
273C22×C3#39211yescharacteristic
274C22×C3#44812yes—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—82no—
278C3×C4—81no—
279C3×C4#45312yes—
280C3×C4#45612yes—
281C3×C4#46112yes—
282C3×C4#49211yescharacteristic
283C3×C4#49711yescharacteristic
284Dic3—28no—
285Dic3—82no—
286C23—41no—
287C23#93711yescharacteristic
288Q8—42no—
289Q8—82no—
290Q8—82no—
291Q8—124no—
292Q8—242no—
293D4—42no—
294D4—42no—
295D4—42no—
296D4—82no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—64no—
303C2×C4—64no—
304C2×C4—64no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—124no—
308C2×C4—122no—
309C2×C4D3×SD1612yes—
310C2×C4#91512yes—
311C2×C4#93711yescharacteristic
312C2×C4#93811yescharacteristic
313C2×C4#93911yescharacteristic
314C2×C4#93911yescharacteristic
315C2×C4#94011yescharacteristic
316C2×C4#95311yescharacteristic
317C8—42no—
318C8—122no—
319C8—241no—
320C2×C3—22no—
321C2×C3—161no—
322C2×C3#200211yescharacteristic
323C2×C3#200511yescharacteristic
324C2×C3#201311yescharacteristic
325C22—21no—
326C22—21no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22#517012yes—
331C22#522311yescharacteristic
332C4—21no—
333C4—21no—
334C4—68no—
335C4—81no—
336C4—82no—
337C4—242no—
338C4#516212yes—
339C4#516711yescharacteristic
340C4#517511yescharacteristic
341C4#519812yes—
342C4#524612yes—
343C3#1028311yescharacteristic, complemented, Sylow
344C2—22no—
345C2—161no—
346C2#7186811yescharacteristic
347C2#7205711yescharacteristic
348C2#7209411yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3227C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3227D3
hypocenterC3#10283
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
3. #83253C2semidirect
5. #71775C2semidirect
6. #70779C2semidirect
7. #67736C2semidirect
142. C4×Dic3SD16semidirect
343. C3#10283semidirect

Maximal subgroups

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

subgroupindex
C3×#32272
#833172
#832532
#718152
#717752
#707792
#677362
#102833