Finite Group Databasebeta

← #10698799

Subgroups — #10698799

Order 768.

408 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 101 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×#462C211yesmaximal, characteristic
2#71773C211yesmaximal, characteristic
3#71773C211yesmaximal, characteristic
4#71637C211yesmaximal, characteristic
5#71621C211yesmaximal, characteristic
6#70421C211yesmaximal, characteristic
7#67707C211yesmaximal, characteristic
8#10447—31nomaximal, Sylow
9C22×C3×Q16—21no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12C3×#462—22no—
13C3×#462—22no—
14C2×C3×C4⋊C8C2211yescharacteristic
15C2×C3×#103—21no—
16C2×#836—21no—
17#4925C2211yescharacteristic
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20#4924C2211yescharacteristic
21#4547—22no—
22#4547—22no—
23#4535—22no—
24#4530—21no—
25#4480—22no—
26C2×#462D311yescharacteristic
27#1801—31no—
28#1570—31no—
29#1546—31no—
30#1546—31no—
31#1479—31no—
32#1301—31no—
33C22×C3×Q8—21no—
34C22×C3×Q8D411yescharacteristic
35C2×C3×Q16—24no—
36C2×C3×Q16—42no—
37C2×C3×Q16—42no—
38C22×C3×C8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8D412yes—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#208—21no—
54C2×#208—21no—
55C3×C4⋊C8—21no—
56C3×C4⋊C8D412yes—
57C3×#103—24no—
58#836—42no—
59#830—21no—
60#830—21no—
61C4⋊(C3⋊C8)—21no—
62C4⋊(C3⋊C8)—21no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)—21no—
65C4⋊(C3⋊C8)—21no—
66C4⋊(C3⋊C8)D412yes—
67C4×C3⋊C8—21no—
68C4×C3⋊C8—21no—
69C22×Q16—21no—
70C2×C4⋊Q8C2×D311yescharacteristic
71C2×C4×Q8C2×D311yescharacteristic
72#462—22no—
73#462—22no—
74#431—31no—
75#431—31no—
76#423—31no—
77#423—31no—
78C2×C4⋊C8C2×D311yescharacteristic
79C2×#103—21no—
80C2×#101—61no—
81#343—61no—
82#334—62no—
83#332—62no—
84#332—62no—
85Q8⋊C8—62no—
86C2×C3×Q8—21no—
87C2×C3×Q8—21no—
88C2×C3×Q8—22no—
89C2×C3×Q8—21no—
90C2×C3×Q8—21no—
91C2×C3×Q8—21no—
92C2×C3×Q8—41no—
93C2×C3×Q8—41no—
94C2×C3×Q8—41no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C2×C3×Q8SD1612yes—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—41no—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4111yescharacteristic
103C22×C3×C4#4111yescharacteristic
104C3×Q16—48no—
105C2×C3×C8—22no—
106C2×C3×C8—41no—
107C2×C3×C8—41no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—41no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—41no—
116C3×C42—41no—
117C3×C42—41no—
118C3×C42C2×D411yescharacteristic
119C3×C42C2×D411yescharacteristic
120C3×C42C2×D411yescharacteristic
121C3×C42#4111yescharacteristic
122#208—41no—
123#208—41no—
124#208—41no—
125#208—41no—
126C2×C3⋊C8—22no—
127C2×C3⋊C8—41no—
128C2×C3⋊C8—41no—
129C2×C3⋊C8—41no—
130C22×Q8—21no—
131C22×Q8#6711yescharacteristic
132C2×Q16—24no—
133C2×Q16—42no—
134C2×Q16—42no—
135C2×M16—61no—
136C2×M16—61no—
137C22×C8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8#6712yes—
142C4×Q8—21no—
143C4×Q8—21no—
144C4×Q8—21no—
145C4×Q8—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4#6711yescharacteristic
150C2×C42—21no—
151C2×C42C22×D311yescharacteristic
152C4⋊C8—21no—
153C4⋊C8—32no—
154C4⋊C8—61no—
155C4⋊C8—61no—
156C4⋊C8—61no—
157C4⋊C8—61no—
158C4⋊C8—61no—
159C4⋊C8#6712yes—
160#103—24no—
161#101—122no—
162C22⋊C8—61no—
163C22⋊C8—61no—
164C4×C8—61no—
165C4×C8—61no—
166C23×C3#12311yescharacteristic
167C3×Q8—22no—
168C3×Q8—22no—
169C3×Q8—41no—
170C3×Q8—41no—
171C3×Q8—42no—
172C3×Q8—42no—
173C3×Q8—81no—
174C3×Q8—81no—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—42no—
182C2×C3×C4—81no—
183C2×C3×C4—81no—
184C2×C3×C4—81no—
185C2×C3×C4#12011yescharacteristic
186C2×C3×C4#12111yescharacteristic
187C2×C3×C4#12111yescharacteristic
188C2×C3×C4#12111yescharacteristic
189C2×C3×C4#12311yescharacteristic
190C2×C3×C4#12411yescharacteristic
191C2×C3×C4C2×SD1611yescharacteristic
192C2×C3×C4#13511yescharacteristic
193C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
194C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
195C3×C8—42no—
196C3⋊C8—42no—
197C3⋊C8—81no—
198C3⋊C8—81no—
199C3⋊C8—81no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—21no—
203C2×Q8—21no—
204C2×Q8—21no—
205C2×Q8—22no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8#21412yes—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—41no—
215C22×C4#23711yescharacteristic
216C22×C4#23711yescharacteristic
217C22×C4C2×#6711yescharacteristic
218Q16—48no—
219M16—121no—
220M16—121no—
221M16—121no—
222M16—121no—
223C2×C8—22no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—62no—
227C2×C8—121no—
228C2×C8—121no—
229C2×C8—121no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C42—41no—
239C42—41no—
240C42D3×D411yescharacteristic
241C42#23711yescharacteristic
242C42C2×#6711yescharacteristic
243C42C2×#6711yescharacteristic
244C22×C3—21no—
245C22×C3—21no—
246C22×C3#39411yescharacteristic
247C22×C3#48111yescharacteristic
248C22×C3#48211yescharacteristic
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—42no—
253C3×C4—81no—
254C3×C4—82no—
255C3×C4#44911yescharacteristic
256C3×C4#45011yescharacteristic
257C3×C4#46511yescharacteristic
258C3×C4#46911yescharacteristic
259C3×C4#47312yes—
260C23#93511yescharacteristic
261Q8—22no—
262Q8—22no—
263Q8—41no—
264Q8—41no—
265Q8—42no—
266Q8—42no—
267Q8—81no—
268Q8—81no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4#93211yescharacteristic
280C2×C4C2×#21411yescharacteristic
281C2×C4#93511yescharacteristic
282C2×C4#93611yescharacteristic
283C2×C4#93811yescharacteristic
284C2×C4#93911yescharacteristic
285C2×C4#93911yescharacteristic
286C2×C4#94911yescharacteristic
287C2×C4#95011yescharacteristic
288C2×C4#95311yescharacteristic
289C8—42no—
290C8—122no—
291C8—241no—
292C8—241no—
293C8—241no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3#203411yescharacteristic
297C2×C3#204711yescharacteristic
298C2×C3#204911yescharacteristic
299C22—21no—
300C22—21no—
301C22#503411yescharacteristic
302C22#503511yescharacteristic
303C22#522211yescharacteristic
304C4—21no—
305C4—21no—
306C4—21no—
307C4—42no—
308C4—82no—
309C4—81no—
310C4#503911yescharacteristic
311C4#504111yescharacteristic
312C4#519212yes—
313C4#521811yescharacteristic
314C4#523811yescharacteristic
315C3#1044711yescharacteristic, complemented, Sylow
316C2—21no—
317C2—21no—
318C2#7168011yescharacteristic
319C2#7207011yescharacteristic
320C2#7210411yescharacteristic
subgroupG/H
centerC22#5222
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#462C2
socleC22×C3#394
prime radicalC23×C3#123
hypercenterC2×#462D3
hypocenterC3#10447
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
315. C3#10447semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4622
#717732
#716372
#716212
#704212
#677072
#104473