Finite Group Databasebeta

← C3×#29710 #10648317

Subgroups — C3×#29710 #10648317

Order 768.

1116 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1116 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×#2487C213yesmaximal, complemented
3C3×#2062C212yesmaximal, complemented
4C3×#2044C216yesmaximal, complemented
5C3×#1799C213yesmaximal, complemented
6#29710C311yesmaximal, characteristic, direct product factor, Sylow
7C3×#543C2211yescharacteristic
8C3×#539—23no—
9C3×#538—23no—
10C3×#534C2213yes—
11C3×#529—23no—
12C3×#525—23no—
13C2×C3×C4×D4C2213yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×C3×#99C2216yes—
16C3×#400—23no—
17C3×#399—23no—
18C3×#389—23no—
19C3×#386C2216yes—
20C3×#385C2216yes—
21C3×#380—23no—
22C3×#380C2213yes—
23C3×#342C2216yes—
24#3459C2×C311yescharacteristic
25#2487C2×C313yescomplemented
26#2062C2×C312yescomplemented
27#2044C2×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×C22⋊Q8—23no—
40C3×C22⋊Q8—23no—
41C3×C22⋊Q8—23no—
42C3×#121—23no—
43C3×#121—23no—
44C3×#121—23no—
45C3×#120—26no—
46C3×C4×Q8—21no—
47C3×C4×D4—23no—
48C3×C4×D4—26no—
49C3×C4×D4—26no—
50C3×#117—23no—
51C2×C3×C4⋊C4—23no—
52C2×C3×C4⋊C4—23no—
53C2×C3×C4⋊C4—23no—
54C2×C3×C4⋊C4—26no—
55C2×C3×C4⋊C4D413yes—
56C2×C3×C22⋊C4—26no—
57C2×C3×C22⋊C4D413yes—
58C2×C3×C22⋊C4C2316yes—
59C2×C3×C22⋊C4C2313yes—
60C2×C3×C42D413yes—
61C3×#99—46no—
62C3×#95—26no—
63C3×#95—23no—
64C3×#95—26no—
65#543C22×C311yescharacteristic
66#539—23no—
67#538—23no—
68#534C22×C313yes—
69#529—23no—
70#525—23no—
71C2×C4×D4C22×C313yes—
72C22×C4⋊C4C22×C311yescharacteristic
73C2×#99C22×C316yes—
74#400—23no—
75#399—23no—
76#389—23no—
77#386C22×C316yes—
78#385C22×C316yes—
79#380—23no—
80#380C22×C313yes—
81#342C22×C316yes—
82C24×C3C2×D413yes—
83C24×C3C2411yescharacteristic
84C3×#41—82no—
85C2×C3×Q8—21no—
86C2×C3×Q8—21no—
87C2×C3×D4—26no—
88C2×C3×D4—23no—
89C2×C3×D4—26no—
90C2×C3×D4—23no—
91C2×C3×D4—23no—
92C2×C3×D4—23no—
93C22×C3×C4—23no—
94C22×C3×C4—23no—
95C22×C3×C4—26no—
96C22×C3×C4—26no—
97C22×C3×C4—42no—
98C22×C3×C4—46no—
99C22×C3×C4—82no—
100C22×C3×C4C2×D413yes—
101C22×C3×C4C2×D413yes—
102C22×C3×C4C2×D413yes—
103C22×C3×C4C2×D413yes—
104C22×C3×C4C2×D413yes—
105C3×C4⋊C4—23no—
106C3×C4⋊C4—23no—
107C3×C4⋊C4—23no—
108C3×C4⋊C4—23no—
109C3×C4⋊C4—26no—
110C3×C4⋊C4—46no—
111C3×C4⋊C4—43no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—86no—
114C3×C22⋊C4—26no—
115C3×C22⋊C4—26no—
116C3×C22⋊C4—26no—
117C3×C22⋊C4—26no—
118C3×C22⋊C4—26no—
119C3×C22⋊C4—26no—
120C3×C22⋊C4—86no—
121C3×C42—23no—
122C3×C42—23no—
123C2×#41—21no—
124C22×D4C23×C313yes—
125C23×C4C3×D413yes—
126C23×C4C23×C312yes—
127C23×C4C23×C311yescharacteristic
128#126—26no—
129#125—23no—
130#124—23no—
131#123—26no—
132#123—26no—
133C22⋊Q8—23no—
134C22⋊Q8—23no—
135C22⋊Q8—23no—
136#121—23no—
137#121—23no—
138#121—23no—
139#120—26no—
140C4×Q8—21no—
141C4×D4—23no—
142C4×D4—26no—
143C4×D4—26no—
144#117—23no—
145C2×C4⋊C4—23no—
146C2×C4⋊C4—23no—
147C2×C4⋊C4—23no—
148C2×C4⋊C4—26no—
149C2×C4⋊C4C3×D413yes—
150C2×C22⋊C4—26no—
151C2×C22⋊C4C3×D413yes—
152C2×C22⋊C4C23×C316yes—
153C2×C22⋊C4C23×C313yes—
154C2×C42C3×D413yes—
155#99—46no—
156#95—26no—
157#95—23no—
158#95—26no—
159C23×C3—23no—
160C23×C3—23no—
161C23×C3—23no—
162C23×C3—41no—
163C23×C3—41no—
164C23×C3—43no—
165C23×C3—43no—
166C23×C3C22×D413yes—
167C23×C3H(5,𝔽2)11yescharacteristic
168C23×C32−1+413yes—
169C3×Q8—81no—
170C3×Q8—81no—
171C3×D4—46no—
172C3×D4—46no—
173C3×D4—83no—
174C3×D4—83no—
175C2×C3×C4—26no—
176C2×C3×C4—23no—
177C2×C3×C4—23no—
178C2×C3×C4—23no—
179C2×C3×C4—23no—
180C2×C3×C4—23no—
181C2×C3×C4—43no—
182C2×C3×C4—46no—
183C2×C3×C4—43no—
184C2×C3×C4—43no—
185C2×C3×C4—46no—
186C2×C3×C4—46no—
187C2×C3×C4—46no—
188C2×C3×C4—46no—
189C2×C3×C4—42no—
190C2×C3×C4—46no—
191C2×C3×C4—83no—
192C2×C3×C4#12013yes—
193C2×C3×C4#12011yescharacteristic
194C24C2×C3×D413yes—
195C24C24×C311yescharacteristic
196#41—82no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×D4—26no—
200C2×D4—23no—
201C2×D4—26no—
202C2×D4—23no—
203C2×D4—23no—
204C2×D4—23no—
205C22×C4—23no—
206C22×C4—23no—
207C22×C4—26no—
208C22×C4—26no—
209C22×C4—42no—
210C22×C4—46no—
211C22×C4—82no—
212C22×C4C2×C3×D413yes—
213C22×C4C2×C3×D413yes—
214C22×C4C2×C3×D413yes—
215C22×C4C2×C3×D413yes—
216C22×C4C2×C3×D413yes—
217C4⋊C4—23no—
218C4⋊C4—23no—
219C4⋊C4—23no—
220C4⋊C4—23no—
221C4⋊C4—26no—
222C4⋊C4—46no—
223C4⋊C4—43no—
224C4⋊C4—41no—
225C4⋊C4—86no—
226C22⋊C4—26no—
227C22⋊C4—26no—
228C22⋊C4—26no—
229C22⋊C4—26no—
230C22⋊C4—26no—
231C22⋊C4—26no—
232C22⋊C4—86no—
233C42—23no—
234C42—23no—
235C22×C3—23no—
236C22×C3—23no—
237C22×C3—41no—
238C22×C3—43no—
239C22×C3—46no—
240C22×C3—43no—
241C22×C3—43no—
242C22×C3—43no—
243C22×C3—83no—
244C22×C3C2×#12011yescharacteristic
245C22×C3#53613yes—
246C22×C3#53713yes—
247C3×C4—23no—
248C3×C4—43no—
249C3×C4—43no—
250C3×C4—43no—
251C3×C4—86no—
252C3×C4—82no—
253C3×C4#45812yes—
254C23—23no—
255C23—23no—
256C23—23no—
257C23—41no—
258C23—41no—
259C23—43no—
260C23—43no—
261C23C22×C3×D413yes—
262C23C3×H(5,𝔽2)11yescharacteristic
263C23C3×2−1+413yes—
264Q8—81no—
265Q8—81no—
266D4—46no—
267D4—46no—
268D4—83no—
269D4—83no—
270C2×C4—26no—
271C2×C4—23no—
272C2×C4—23no—
273C2×C4—23no—
274C2×C4—23no—
275C2×C4—23no—
276C2×C4—43no—
277C2×C4—46no—
278C2×C4—43no—
279C2×C4—43no—
280C2×C4—46no—
281C2×C4—46no—
282C2×C4—46no—
283C2×C4—46no—
284C2×C4—42no—
285C2×C4—46no—
286C2×C4—83no—
287C2×C4C3×#12013yes—
288C2×C4C3×#12011yescharacteristic
289C2×C3—23no—
290C2×C3—23no—
291C2×C3—83no—
292C2×C3#261611yescharacteristic
293C2×C3C2×#45811yescharacteristic
294C2×C3#302711yescharacteristic
295C22—23no—
296C22—23no—
297C22—41no—
298C22—43no—
299C22—46no—
300C22—43no—
301C22—43no—
302C22—43no—
303C22—83no—
304C22C2×C3×#12011yescharacteristic
305C22C3×#53613yes—
306C22C3×#53713yes—
307C4—23no—
308C4—43no—
309C4—43no—
310C4—43no—
311C4—86no—
312C4—82no—
313C4C3×#45812yes—
314C3#2971011yescharacteristic, direct product factor, Sylow
315C2—23no—
316C2—23no—
317C2—83no—
318C2C3×#261611yescharacteristic
319C2C2×C3×#45811yescharacteristic
320C2C3×#302711yescharacteristic
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×#2487C2semidirect
3. C3×#2062C2semidirect
4. C3×#2044C2semidirect
5. C3×#1799C2semidirect
6. #29710C3direct (N × H)
25. #2487C2×C3semidirect
26. #2062C2×C3semidirect
27. #2044C2×C3semidirect
28. #1799C2×C3semidirect
314. C3#29710direct (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×#24872
C3×#20622
C3×#20442
C3×#17992
#297103