Finite Group Databasebeta

← C3×#29685 #10648292

Subgroups — C3×#29685 #10648292

Order 768.

1176 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1176 conjugacy classes of subgroups — too many to draw. Only its 242 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×#3459C211yesmaximal, characteristic
2C3×#2486C213yesmaximal, complemented
3C3×#2030C212yesmaximal, complemented
4C3×#2022C216yesmaximal, complemented
5C3×#1799C213yesmaximal, complemented
6#29685C311yesmaximal, characteristic, direct product factor, Sylow
7C3×#543C2211yescharacteristic
8C3×#539—23no—
9C3×#538—23no—
10C3×#534C2213yes—
11C3×#529—23no—
12C3×#525—23no—
13C2×C3×#123C2216yes—
14C2×C3×C4×D4C2213yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C2×C3×#99C2216yes—
17C3×#398—23no—
18C3×#397—23no—
19C3×#389—23no—
20C3×#381C2216yes—
21C3×#380—23no—
22C3×#380C2213yes—
23C3×#342C2216yes—
24#3459C2×C311yescharacteristic
25#2486C2×C313yescomplemented
26#2030C2×C312yescomplemented
27#2022C2×C316yescomplemented
28#1799C2×C313yescomplemented
29C2×C3×#41—21no—
30C22×C3×D4C2313yes—
31C23×C3×C4D413yes—
32C23×C3×C4C2312yes—
33C23×C3×C4C2311yescharacteristic
34C3×#126—26no—
35C3×#125—23no—
36C3×#124—23no—
37C3×#123—26no—
38C3×#123—26no—
39C3×#123—412no—
40C3×C22⋊Q8—23no—
41C3×C22⋊Q8—23no—
42C3×C22⋊Q8—23no—
43C3×#121—23no—
44C3×#121—23no—
45C3×#121—23no—
46C3×#120—26no—
47C3×C4×Q8—21no—
48C3×C4×D4—23no—
49C3×C4×D4—26no—
50C3×C4×D4—26no—
51C3×#117—23no—
52C2×C3×C4⋊C4—23no—
53C2×C3×C4⋊C4—23no—
54C2×C3×C4⋊C4—26no—
55C2×C3×C4⋊C4—23no—
56C2×C3×C4⋊C4D413yes—
57C2×C3×C22⋊C4—26no—
58C2×C3×C22⋊C4D413yes—
59C2×C3×C22⋊C4C2316yes—
60C2×C3×C22⋊C4C2313yes—
61C2×C3×C42D413yes—
62C3×#99—46no—
63C3×#95—26no—
64C3×#95—23no—
65C3×#95—26no—
66#543C22×C311yescharacteristic
67#539—23no—
68#538—23no—
69#534C22×C313yes—
70#529—23no—
71#525—23no—
72C2×#123C22×C316yes—
73C2×C4×D4C22×C313yes—
74C22×C4⋊C4C22×C311yescharacteristic
75C2×#99C22×C316yes—
76#398—23no—
77#397—23no—
78#389—23no—
79#381C22×C316yes—
80#380—23no—
81#380C22×C313yes—
82#342C22×C316yes—
83C24×C3C2×D413yes—
84C24×C3C2411yescharacteristic
85C3×#41—82no—
86C2×C3×Q8—21no—
87C2×C3×Q8—21no—
88C2×C3×D4—26no—
89C2×C3×D4—23no—
90C2×C3×D4—26no—
91C2×C3×D4—23no—
92C2×C3×D4—23no—
93C2×C3×D4—23no—
94C22×C3×C4—23no—
95C22×C3×C4—23no—
96C22×C3×C4—26no—
97C22×C3×C4—26no—
98C22×C3×C4—26no—
99C22×C3×C4—22no—
100C22×C3×C4—42no—
101C22×C3×C4—46no—
102C22×C3×C4C2×D413yes—
103C22×C3×C4C2×D413yes—
104C22×C3×C4C2×D413yes—
105C22×C3×C4C2×D413yes—
106C22×C3×C4C2×D413yes—
107C3×C4⋊C4—23no—
108C3×C4⋊C4—23no—
109C3×C4⋊C4—23no—
110C3×C4⋊C4—23no—
111C3×C4⋊C4—26no—
112C3×C4⋊C4—46no—
113C3×C4⋊C4—412no—
114C3×C4⋊C4—43no—
115C3×C4⋊C4—41no—
116C3×C22⋊C4—26no—
117C3×C22⋊C4—26no—
118C3×C22⋊C4—26no—
119C3×C22⋊C4—26no—
120C3×C22⋊C4—212no—
121C3×C22⋊C4—412no—
122C3×C42—23no—
123C3×C42—23no—
124C2×#41—21no—
125C22×D4C23×C313yes—
126C23×C4C3×D413yes—
127C23×C4C23×C312yes—
128C23×C4C23×C311yescharacteristic
129#126—26no—
130#125—23no—
131#124—23no—
132#123—26no—
133#123—26no—
134#123—412no—
135C22⋊Q8—23no—
136C22⋊Q8—23no—
137C22⋊Q8—23no—
138#121—23no—
139#121—23no—
140#121—23no—
141#120—26no—
142C4×Q8—21no—
143C4×D4—23no—
144C4×D4—26no—
145C4×D4—26no—
146#117—23no—
147C2×C4⋊C4—23no—
148C2×C4⋊C4—23no—
149C2×C4⋊C4—26no—
150C2×C4⋊C4—23no—
151C2×C4⋊C4C3×D413yes—
152C2×C22⋊C4—26no—
153C2×C22⋊C4C3×D413yes—
154C2×C22⋊C4C23×C316yes—
155C2×C22⋊C4C23×C313yes—
156C2×C42C3×D413yes—
157#99—46no—
158#95—26no—
159#95—23no—
160#95—26no—
161C23×C3—23no—
162C23×C3—23no—
163C23×C3—23no—
164C23×C3—41no—
165C23×C3—41no—
166C23×C3—43no—
167C23×C3—43no—
168C23×C3C22×D413yes—
169C23×C3H(5,𝔽2)13yes—
170C23×C32−1+411yescharacteristic
171C3×Q8—81no—
172C3×Q8—81no—
173C3×D4—46no—
174C3×D4—46no—
175C3×D4—83no—
176C3×D4—83no—
177C2×C3×C4—26no—
178C2×C3×C4—23no—
179C2×C3×C4—23no—
180C2×C3×C4—23no—
181C2×C3×C4—23no—
182C2×C3×C4—23no—
183C2×C3×C4—43no—
184C2×C3×C4—46no—
185C2×C3×C4—43no—
186C2×C3×C4—43no—
187C2×C3×C4—46no—
188C2×C3×C4—46no—
189C2×C3×C4—42no—
190C2×C3×C4—46no—
191C2×C3×C4—46no—
192C2×C3×C4—46no—
193C2×C3×C4—83no—
194C2×C3×C4#12013yes—
195C2×C3×C4#12011yescharacteristic
196C24C2×C3×D413yes—
197C24C24×C311yescharacteristic
198#41—82no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×D4—26no—
202C2×D4—23no—
203C2×D4—26no—
204C2×D4—23no—
205C2×D4—23no—
206C2×D4—23no—
207C22×C4—23no—
208C22×C4—23no—
209C22×C4—26no—
210C22×C4—26no—
211C22×C4—26no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—46no—
215C22×C4C2×C3×D413yes—
216C22×C4C2×C3×D413yes—
217C22×C4C2×C3×D413yes—
218C22×C4C2×C3×D413yes—
219C22×C4C2×C3×D413yes—
220C4⋊C4—23no—
221C4⋊C4—23no—
222C4⋊C4—23no—
223C4⋊C4—23no—
224C4⋊C4—26no—
225C4⋊C4—412no—
226C4⋊C4—46no—
227C4⋊C4—43no—
228C4⋊C4—41no—
229C22⋊C4—26no—
230C22⋊C4—26no—
231C22⋊C4—26no—
232C22⋊C4—26no—
233C22⋊C4—212no—
234C22⋊C4—412no—
235C42—23no—
236C42—23no—
237C22×C3—23no—
238C22×C3—23no—
239C22×C3—43no—
240C22×C3—41no—
241C22×C3—46no—
242C22×C3—43no—
243C22×C3—43no—
244C22×C3—43no—
245C22×C3—83no—
246C22×C3C2×#12011yescharacteristic
247C22×C3#53513yes—
248C22×C3#53713yes—
249C3×C4—23no—
250C3×C4—43no—
251C3×C4—43no—
252C3×C4—43no—
253C3×C4—86no—
254C3×C4—82no—
255C3×C4#45812yes—
256C23—23no—
257C23—23no—
258C23—23no—
259C23—41no—
260C23—41no—
261C23—43no—
262C23—43no—
263C23C22×C3×D413yes—
264C23C3×H(5,𝔽2)13yes—
265C23C3×2−1+411yescharacteristic
266Q8—81no—
267Q8—81no—
268D4—46no—
269D4—46no—
270D4—83no—
271D4—83no—
272C2×C4—26no—
273C2×C4—23no—
274C2×C4—23no—
275C2×C4—23no—
276C2×C4—23no—
277C2×C4—23no—
278C2×C4—43no—
279C2×C4—46no—
280C2×C4—43no—
281C2×C4—43no—
282C2×C4—46no—
283C2×C4—46no—
284C2×C4—42no—
285C2×C4—46no—
286C2×C4—46no—
287C2×C4—46no—
288C2×C4—83no—
289C2×C4C3×#12013yes—
290C2×C4C3×#12011yescharacteristic
291C2×C3—23no—
292C2×C3—23no—
293C2×C3—83no—
294C2×C3#261511yescharacteristic
295C2×C3C2×#45811yescharacteristic
296C2×C3#302811yescharacteristic
297C22—23no—
298C22—23no—
299C22—43no—
300C22—41no—
301C22—46no—
302C22—43no—
303C22—43no—
304C22—43no—
305C22—83no—
306C22C2×C3×#12011yescharacteristic
307C22C3×#53513yes—
308C22C3×#53713yes—
309C4—23no—
310C4—43no—
311C4—43no—
312C4—43no—
313C4—86no—
314C4—82no—
315C4C3×#45812yes—
316C3#2968511yescharacteristic, direct product factor, Sylow
317C2—23no—
318C2—23no—
319C2—83no—
320C2C3×#261511yescharacteristic
321C2C2×C3×#45811yescharacteristic
322C2C3×#302811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC3×#3459C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2486C2semidirect
3. C3×#2030C2semidirect
4. C3×#2022C2semidirect
5. C3×#1799C2semidirect
6. #29685C3direct (N × H)
25. #2486C2×C3semidirect
26. #2030C2×C3semidirect
27. #2022C2×C3semidirect
28. #1799C2×C3semidirect
316. C3#29685direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34592
C3×#24862
C3×#20302
C3×#20222
C3×#17992
#296853