Finite Group Databasebeta

← #10697093

Subgroups — #10697093

Order 768.

556 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×C3×#486C211yesmaximal, characteristic
2#83198C211yesmaximal, characteristic, complemented
3#83169C211yesmaximal, characteristic
4#71773C211yesmaximal, characteristic, complemented
5#71766C211yesmaximal, characteristic
6#69738C211yesmaximal, characteristic, complemented
7#67656C211yesmaximal, characteristic, complemented
8#10169—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#127C2211yescharacteristic
11#5470C2211yescharacteristic
12C3×#486—22no—
13C3×#486—22no—
14C2×C3×#102—22no—
15C2×C3×C4×C8C2211yescharacteristic
16#5225—22no—
17#5218—22no—
18C2×#833—22no—
19#5082—22no—
20#5079—21no—
21#5079—21no—
22#5076—21no—
23#5075—22no—
24#5070—21no—
25#4925C2211yescharacteristic
26#4924C2211yescharacteristic
27C4×#208C2211yescharacteristic
28#4547—22no—
29#4456—22no—
30#3208—31no—
31#3139—31no—
32C2×#486D311yescharacteristic
33#1965—31no—
34#1546—31no—
35#1493—31no—
36#1278—31no—
37C22×C3×Q8—21no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4D412yes—
41C22×C3×C8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8D412yes—
46C3×#127—21no—
47C3×#127—21no—
48C3×#127—21no—
49C3×#127—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#214—24no—
54#926—21no—
55#924—21no—
56#924—21no—
57C2×#208—21no—
58C2×#208D412yes—
59#870—21no—
60#869D411yescharacteristic
61#869D411yescharacteristic
62C4×Dic6—21no—
63C4×Dic6—21no—
64C3×#102—28no—
65C3×C4×C8—21no—
66C3×C4×C8D412yes—
67#835—24no—
68#835—24no—
69#833—44no—
70#832—24no—
71#830—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)D412yes—
75#803—21no—
76C4×C3⋊C8—21no—
77C4×C3⋊C8—21no—
78#533—31no—
79C2×C4⋊Q8C2×D311yescharacteristic
80C2×#127C2×D311yescharacteristic
81#492—61no—
82#491—62no—
83#487—61no—
84#487—61no—
85#486—22no—
86#486—22no—
87#486—61no—
88#460—62no—
89#450—62no—
90#431—31no—
91#423—31no—
92#419—62no—
93C2×#102—22no—
94C2×#100—62no—
95C4×M16—31no—
96C2×C4×C8C2×D311yescharacteristic
97#332—62no—
98Q8⋊C8—62no—
99C24×C3—22no—
100C2×C3×Q8—22no—
101C2×C3×Q8—41no—
102C2×C3×Q8—42no—
103C2×C3×Q8—42no—
104C2×C3×D4—22no—
105C2×C3×D4—22no—
106C2×C3×D4—44no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C2×C3×D4—41no—
111C2×C3×D4—42no—
112C2×C3×D4—42no—
113C2×C3×D4—82no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C2×Dic6—21no—
119C2×Dic6—21no—
120C2×C3×C8—22no—
121C2×C3×C8—42no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C42C2×D411yescharacteristic
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D411yescharacteristic
129C3×C42C2×D411yescharacteristic
130#217—41no—
131#214—84no—
132C4⋊Dic3—21no—
133C4⋊Dic3—21no—
134C4⋊Dic3—21no—
135C4⋊Dic3—21no—
136Dic3⋊C4—41no—
137Dic3⋊C4—41no—
138C4×Dic3—41no—
139#208—22no—
140#208—22no—
141#208—42no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—41no—
145C22×Q8—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4#6712yes—
149C2×SD16—64no—
150C2×M16—32no—
151C2×M16—61no—
152C22×C8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—31no—
157C4⋊Q8—31no—
158C4⋊Q8#6712yes—
159#127—21no—
160#127—21no—
161#127—21no—
162#127—21no—
163#125—61no—
164C22⋊Q8—61no—
165C22⋊Q8—61no—
166C4×Q8—61no—
167C4×Q8—61no—
168#117—61no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C42C22×D311yescharacteristic
172C4⋊C8—32no—
173C4⋊C8—61no—
174C4⋊C8—61no—
175#103—64no—
176#103—64no—
177#102—28no—
178#102—64no—
179#100—124no—
180C22⋊C8—61no—
181#97—61no—
182C4×C8—21no—
183C4×C8—61no—
184C4×C8—61no—
185C4×C8#6712yes—
186C23×C3—42no—
187C23×C3—44no—
188C23×C3—82no—
189C23×C3—82no—
190C23×C3#12111yescharacteristic
191C3×Q8—44no—
192C3×Q8—81no—
193C3×Q8—81no—
194C3×D4—44no—
195C3×D4—44no—
196C3×D4—82no—
197C3×D4—82no—
198C3×D4—82no—
199C3×D4—81no—
200C3×D4—81no—
201C2×C3×C4—22no—
202C2×C3×C4—22no—
203C2×C3×C4—42no—
204C2×C3×C4—82no—
205C2×C3×C4#12011yescharacteristic
206C2×C3×C4#12011yescharacteristic
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12111yescharacteristic
210C2×C3×C4#12711yescharacteristic
211C2×C3×C4#13512yes—
212C2×C3×C4AGL(1,ℤ/8)12yes—
213C2×Dic3—41no—
214C2×Dic3—41no—
215Dic6—42no—
216Dic6—42no—
217C3×C8—42no—
218C3⋊C8—42no—
219C3⋊C8—42no—
220C3⋊C8—81no—
221C24—22no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—61no—
227C2×Q8—61no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—44no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—82no—
238C22×C4—41no—
239C22×C4D3×D411yescharacteristic
240C22×C4#23711yescharacteristic
241C22×C4C2×#6711yescharacteristic
242SD16—244no—
243M16—62no—
244M16—62no—
245M16—122no—
246C2×C8—22no—
247C2×C8—42no—
248C2×C8—62no—
249C2×C8—62no—
250C2×C8—121no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C22⋊C4—121no—
262C42—121no—
263C42D3×D411yescharacteristic
264C42D3×D411yescharacteristic
265C42C2×#6711yescharacteristic
266C42C2×#6711yescharacteristic
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—82no—
270C22×C3—82no—
271C22×C3—82no—
272C22×C3—164no—
273C22×C3#39211yescharacteristic
274C22×C3#46312yes—
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—82no—
278C3×C4#44912yes—
279C3×C4#45512yes—
280C3×C4#47112yes—
281C3×C4#49511yescharacteristic
282C3×C4#49611yescharacteristic
283Dic3—81no—
284Dic3—81no—
285C23—42no—
286C23—44no—
287C23—82no—
288C23—82no—
289C23#93911yescharacteristic
290Q8—44no—
291Q8—81no—
292Q8—81no—
293Q8—122no—
294Q8—122no—
295D4—44no—
296D4—44no—
297D4—82no—
298D4—81no—
299D4—81no—
300D4—82no—
301D4—82no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—42no—
305C2×C4—82no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4#91112yes—
309C2×C4#91212yes—
310C2×C4#93711yescharacteristic
311C2×C4#93711yescharacteristic
312C2×C4#93811yescharacteristic
313C2×C4#93911yescharacteristic
314C2×C4#94011yescharacteristic
315C2×C4#95311yescharacteristic
316C8—42no—
317C8—122no—
318C8—122no—
319C8—241no—
320C2×C3—22no—
321C2×C3—162no—
322C2×C3#200411yescharacteristic
323C2×C3#201211yescharacteristic
324C2×C3#201911yescharacteristic
325C22—21no—
326C22—21no—
327C22—82no—
328C22—82no—
329C22—82no—
330C22—164no—
331C22#515312yes—
332C22#522311yescharacteristic
333C4—21no—
334C4—21no—
335C4—82no—
336C4—241no—
337C4—241no—
338C4#506112yes—
339C4#515411yescharacteristic
340C4#515511yescharacteristic
341C4#515812yes—
342C4#517712yes—
343C3#1016911yescharacteristic, complemented, Sylow
344C2—22no—
345C2—162no—
346C2#7173111yescharacteristic
347C2#7204511yescharacteristic
348C2#7205911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#486C2
socleC22×C3#392
prime radicalC2×C3×#127C22
hypercenterC2×#486D3
hypocenterC3#10169
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
2. #83198C2semidirect
4. #71773C2semidirect
6. #69738C2semidirect
7. #67656C2semidirect
343. C3#10169semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4862
#831982
#831692
#717732
#717662
#697382
#676562
#101693