Finite Group Databasebeta

← C3×#69155 #11798718

Subgroups — C3×#69155 #11798718

Order 1152.

487 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 487 conjugacy classes of subgroups — too many to draw. Only its 82 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×#5672C211yesmaximal, characteristic, complemented
2C3×#5061C211yesmaximal, characteristic
3C3×#4742C212yesmaximal
4C32×#354C211yesmaximal, characteristic, complemented
5C3×#4472C212yesmaximal, complemented
6C3×#2201—31nomaximal
7#69155C311yesmaximal, characteristic, direct product factor
8C3×#1004C2211yescharacteristic
9C32×#124C2211yescharacteristic
10C3×#944—21no—
11C3×#942—21no—
12C3×#935C2212yes—
13C3×#886—22no—
14C3×#879—22no—
15C3×#878—22no—
16C3×#870—21no—
17C32×#99C2212yes—
18C3×#833C2211yescharacteristic
19C3×#806—22no—
20C3×#805—21no—
21C3×#563—31no—
22#5672C2×C311yescharacteristic, complemented
23C3×#458—32no—
24C3×#455—31no—
25#5061C2×C311yescharacteristic
26#4742C2×C312yes—
27C3×#354—21no—
28C3×#354D311yescharacteristic, complemented
29C3×#354C2×C311yescharacteristic, complemented
30C3×#352—32no—
31#4472C2×C312yescomplemented
32C2×C32×Q8D411yescharacteristic
33C2×C32×D4C2311yescharacteristic
34C2×C3×#67—22no—
35C22×C3×Dic3D412yes—
36C3×#237—41no—
37C3×D3×D4—41no—
38C3×#235—21no—
39C2×C3×Dic6—21no—
40C32×C22⋊C4—22no—
41C32×C22⋊C4—22no—
42C32×C42D411yescharacteristic
43C3×#217—22no—
44C3×#217D412yes—
45C3×#216—41no—
46C3×#215—41no—
47C3×(C2×D3)⋊C4—42no—
48C3×C4⋊Dic3—21no—
49C3×C4⋊Dic3—21no—
50C3×C4⋊Dic3—42no—
51C3×Dic3⋊C4—22no—
52C3×Dic3⋊C4—22no—
53C3×Dic3⋊C4—22no—
54C3×C4×Dic3—21no—
55C3×#208—21no—
56#2201—31noSylow
57C3×H(5,𝔽2)—31no—
58#1004C22×C311yescharacteristic
59C3×#137—61no—
60C3×C4⋊Q8—61no—
61C3×#126—62no—
62C3×#125—61no—
63C3×#124—21no—
64C3×#124C2×D311yescharacteristic
65C3×#124C22×C311yescharacteristic
66C3×#123—32no—
67C3×#123—62no—
68C3×C22⋊Q8—62no—
69#944—21no—
70#942—21no—
71#935C22×C312yes—
72#886—22no—
73#879—22no—
74#878—22no—
75#870—21no—
76C3×#104—61no—
77C3×#100—31no—
78C3×#99—22no—
79C3×#99—62no—
80C3×#99C2×D312yes—
81C3×#99C22×C312yes—
82#833C22×C311yescharacteristic
83#806—22no—
84#805—21no—
85C23×C32C2×D412yes—
86C22×C3×D3—22no—
87C32×Q8—41no—
88C32×D4—41no—
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4—42no—
92C2×C32×C4C2×D411yescharacteristic
93C3×#67—42no—
94C3×#67—41no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—22no—
98C2×C3×Dic3—42no—
99C3×D12—21no—
100C3×C4×D3—41no—
101C3×Dic6—21no—
102C3×Dic6—81no—
103C3×C3⋊C8—41no—
104#563—31no—
105#458—32no—
106#455—31no—
107#354C3×D311yescharacteristic, complemented
108#352—32no—
109C3×#41—61no—
110C3×#41—121no—
111C2×C3×Q8—21no—
112C2×C3×Q8—61no—
113C2×C3×Q8D1211yescharacteristic
114C2×C3×Q8C3×D411yescharacteristic
115C2×C3×D4—21no—
116C2×C3×D4—62no—
117C2×C3×D4—121no—
118C2×C3×D4C22×D311yescharacteristic
119C2×C3×D4C23×C311yescharacteristic
120C22×C3×C4—32no—
121C2×#67—22no—
122C22×Dic3C3×D412yes—
123#237—41no—
124D3×D4—41no—
125#235—21no—
126C2×Dic6—21no—
127C3×Q16—121no—
128C3×SD16—121no—
129C3×M16—61no—
130C3×C4⋊C4—62no—
131C3×C4⋊C4—62no—
132C3×C4⋊C4—61no—
133C3×C4⋊C4—61no—
134C3×C4⋊C4—62no—
135C3×C4⋊C4—122no—
136C3×C22⋊C4—22no—
137C3×C22⋊C4—22no—
138C3×C22⋊C4—22no—
139C3×C22⋊C4—22no—
140C3×C22⋊C4—32no—
141C3×C22⋊C4—42no—
142C3×C22⋊C4—42no—
143C3×C22⋊C4—62no—
144C3×C22⋊C4—122no—
145C3×C42—21no—
146C3×C42—61no—
147C3×C42D1211yescharacteristic
148C3×C42C3×D411yescharacteristic
149#217—22no—
150#217C3×D412yes—
151#216—41no—
152#215—41no—
153(C2×D3)⋊C4—42no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—42no—
157Dic3⋊C4—22no—
158Dic3⋊C4—22no—
159Dic3⋊C4—22no—
160C4×Dic3—21no—
161#208—21no—
162C22×C32—22no—
163C22×C32—42no—
164C22×C32#12011yescharacteristic
165C2×C3×D3—41no—
166C2×C3×D3—82no—
167C32×C4—21no—
168C32×C4—41no—
169C32×C4—41no—
170C32×C4—82no—
171C3×Dic3—42no—
172C3×Dic3—41no—
173C3×Dic3—42no—
174H(5,𝔽2)—31no—
175#137—61no—
176C4⋊Q8—61no—
177#126—62no—
178#125—61no—
179#124C2×C3×D311yescharacteristic
180#123—32no—
181#123—62no—
182C22⋊Q8—62no—
183#104—61no—
184#100—31no—
185#99—62no—
186#99C2×C3×D312yes—
187C23×C3—22no—
188C23×C3—62no—
189C23×C3D3×D412yes—
190C23×C3C2×C3×D412yes—
191C22×D3—22no—
192C3×Q8—41no—
193C3×Q8—41no—
194C3×Q8—61no—
195C3×Q8—81no—
196C3×Q8—241no—
197C3×D4—41no—
198C3×D4—41no—
199C3×D4—61no—
200C3×D4—81no—
201C3×D4—122no—
202C3×D4—121no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—62no—
213C2×C3×C4—62no—
214C2×C3×C4—62no—
215C2×C3×C4—82no—
216C2×C3×C4—121no—
217C2×C3×C4—122no—
218C2×C3×C4C2×D1211yescharacteristic
219C2×C3×C4C2×C3×D411yescharacteristic
220#67—42no—
221#67—41no—
222C2×Dic3—22no—
223C2×Dic3—22no—
224C2×Dic3—22no—
225C2×Dic3—42no—
226D12—21no—
227C4×D3—41no—
228Dic6—21no—
229Dic6—81no—
230C3×C8—121no—
231C3⋊C8—41no—
232C2×C32—21no—
233C2×C32—42no—
234C2×C32B(4,𝔽2)11yescharacteristic
235C3×D3—81no—
236#41—61no—
237#41—121no—
238C2×Q8—61no—
239C2×Q8C3×D1211yescharacteristic
240C2×D4—62no—
241C2×D4—121no—
242C2×D4C22×C3×D311yescharacteristic
243C22×C4—32no—
244Q16—121no—
245SD16—121no—
246M16—61no—
247C4⋊C4—61no—
248C4⋊C4—62no—
249C4⋊C4—62no—
250C4⋊C4—61no—
251C4⋊C4—62no—
252C4⋊C4—122no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—32no—
256C22⋊C4—62no—
257C22⋊C4—122no—
258C42—61no—
259C42C3×D1211yescharacteristic
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—21no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3—82no—
267C22×C3—121no—
268C22×C3—242no—
269C22×C3#88211yescharacteristic
270C22×C3C3×#12011yescharacteristic
271C2×D3—41no—
272C2×D3—82no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—81no—
281C3×C4—81no—
282C3×C4—82no—
283C3×C4—82no—
284C3×C4—122no—
285C3×C4—121no—
286C3×C4—122no—
287C3×C4—162no—
288Dic3—42no—
289Dic3—41no—
290Dic3—42no—
291C32#220111yescharacteristic, complemented, Sylow
292C23—62no—
293C23C3×D3×D412yes—
294Q8—41no—
295Q8—61no—
296Q8—241no—
297D4—41no—
298D4—61no—
299D4—122no—
300D4—121no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—62no—
305C2×C4—62no—
306C2×C4—62no—
307C2×C4—121no—
308C2×C4—122no—
309C2×C4C2×C3×D1211yescharacteristic
310C8—121no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—41no—
315C2×C3—42no—
316C2×C3—42no—
317C2×C3—82no—
318C2×C3—241no—
319C2×C3#474111yescharacteristic
320C2×C3C3×B(4,𝔽2)11yescharacteristic
321D3—81no—
322C22—22no—
323C22—42no—
324C22—121no—
325C22—242no—
326C22C3×#88211yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—82no—
331C4—122no—
332C4—121no—
333C4—122no—
334C3—21no—
335C3#6915511yescharacteristic, direct product factor
336C3C3×#220111yescharacteristic, complemented
337C2—21no—
338C2—42no—
339C2—241no—
340C2C3×#474111yescharacteristic
subgroupG/H
centerC2×C3#4741
commutator, cosocleC2×C3×D4C23×C3
FrattiniC2×D4C22×C3×D3
FittingC32×#354C2
socleC2×C32B(4,𝔽2)
prime radicalC3×#1004C22
hypercenterC3×#354D3
hypocenterC3C3×#2201
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#354

Complements

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

NH ≅ G/Ntype
1. C3×#5672C2semidirect
4. C32×#354C2semidirect
5. C3×#4472C2semidirect
7. #69155C3direct (N × H)
22. #5672C2×C3semidirect
28. C3×#354D3semidirect
29. C3×#354C2×C3semidirect
31. #4472C2×C3semidirect
107. #354C3×D3semidirect
291. C32#2201semidirect
335. C3#69155direct (N × H)
336. C3C3×#2201semidirect

Maximal subgroups

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

subgroupindex
C3×#56722
C3×#50612
C3×#47422
C32×#3542
C3×#44722
C3×#22013
#691553