Finite Group Databasebeta

← C3×D3×#355 #11796449

Subgroups — C3×D3×#355 #11796449

Order 1152.

552 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 114 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×D3×#124C211yesmaximal, characteristic
2C3×D3×#101C212yesmaximal, complemented
3C32×#355C211yesmaximal, characteristic, complemented
4C3×#4542C211yesmaximal, characteristic, complemented
5C3×#4477C212yesmaximal, complemented
6C2×C3×#355—31nomaximal
7D3×#355C311yesmaximal, characteristic, direct product factor
8C2×C3×D3×Q8C2211yescharacteristic
9C2×C3×D3×D4C411yescharacteristic
10C32×#124C2211yescharacteristic
11C3×#947C2211yescharacteristic
12C3×#936C411yescharacteristic
13C3×D3×M16—22no—
14C3×#885—22no—
15C3×D3×C22⋊C4—22no—
16C3×#875C411yescharacteristic
17C3×C42×D3C411yescharacteristic
18C32×#101C2212yescomplemented
19C3×#836C2212yescomplemented
20C3×#824—22no—
21C2×C3×#124—31no—
22D3×#124C2×C311yescharacteristic
23C2×C3×#101—32no—
24D3×#101C2×C312yescomplemented
25C3×#355—21no—
26C3×#355—31no—
27C3×#355—32no—
28C3×#355D311yescharacteristic, direct product factor
29C3×#355C2×C311yescharacteristic, complemented
30#4542C2×C311yescharacteristic, complemented
31#4477C2×C312yescomplemented
32C23×C3×D3—21no—
33C2×C32×Q8C2311yescharacteristic
34C2×C32×D4C2×C411yescharacteristic
35C2×C3×#67—21no—
36C3×D3×Q8—41no—
37C3×D3×Q8—41no—
38C3×D3×D4—41no—
39C3×D3×D4—41no—
40C2×C3×D12C2×C411yescharacteristic
41C2×C3×C4×D3—21no—
42C2×C3×C4×D3C2×C411yescharacteristic
43C2×C3×C4×D3D412yes—
44C2×C3×Dic6C2×C411yescharacteristic
45C2×C3×Dic6D412yes—
46C32×M16—22no—
47C32×C22⋊C4—22no—
48C32×C42C2×C411yescharacteristic
49C3×#217—22no—
50C3×(C2×D3)⋊C4—22no—
51C3×(C2×D3)⋊C4—22no—
52C3×C4×Dic3—21no—
53C3×C4×Dic3C2×C411yescharacteristic
54C3×#208—22no—
55C3×ΓL(1,𝔽25)—42no—
56C3×D3×C8—42no—
57C2×#355—31noSylow
58C22×C3×Q8—31no—
59C22×C3×D4—31no—
60C2×D3×Q8C22×C311yescharacteristic
61C2×D3×D4C3×C411yescharacteristic
62C2×C3×M16—62no—
63C3×#124—21no—
64C3×#124—31no—
65C3×#124—31no—
66C3×#124—31no—
67C3×#124—62no—
68C3×#124C2×D311yescharacteristic
69C3×#124C22×C311yescharacteristic
70C2×C3×C22⋊C4—62no—
71C2×C3×C42—31no—
72#947C22×C311yescharacteristic
73#936C3×C411yescharacteristic
74D3×M16—22no—
75#885—22no—
76D3×C22⋊C4—22no—
77#875C3×C411yescharacteristic
78C42×D3C3×C411yescharacteristic
79C3×#101—22no—
80C3×#101—32no—
81C3×#101—62no—
82C3×#101C2×D312yescomplemented
83C3×#101C22×C312yescomplemented
84#836C22×C312yescomplemented
85#824—22no—
86C23×C32—21no—
87C22×C3×D3—21no—
88C22×C3×D3—44no—
89C22×C3×D3—41no—
90C22×C3×D3—41no—
91C22×C3×D3C22⋊C411yescharacteristic
92C32×Q8—41no—
93C32×D4—41no—
94C2×C32×C4—21no—
95C2×C32×C4C22×C411yescharacteristic
96C2×C32×C4C2×D412yes—
97C3×#67—42no—
98C2×C3×Dic3—21no—
99C2×C3×Dic3C22⋊C411yescharacteristic
100C2×C3×Dic3C22⋊C412yes—
101C3×D12—41no—
102C3×C4×D3—22no—
103C3×C4×D3—22no—
104C3×C4×D3—21no—
105C3×C4×D3—21no—
106C3×C4×D3—42no—
107C3×Dic6—41no—
108C3×Dic6—42no—
109C32×C8—42no—
110C3×C3⋊C8—42no—
111C2×#124—31no—
112C2×#101—32no—
113#355—31no—
114#355—32no—
115#355C3×D311yescharacteristic, direct product factor
116C24×C3—61no—
117C23×D3—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—31no—
120C2×C3×Q8—32no—
121C2×C3×Q8—121no—
122C2×C3×Q8—121no—
123C2×C3×Q8C22×D311yescharacteristic
124C2×C3×Q8C23×C311yescharacteristic
125C2×C3×D4—21no—
126C2×C3×D4—31no—
127C2×C3×D4—61no—
128C2×C3×D4—121no—
129C2×C3×D4—121no—
130C2×C3×D4C4×D311yescharacteristic
131C2×C3×D4C2×C3×C411yescharacteristic
132C22×C3×C4—31no—
133C22×C3×C4—32no—
134C22×C3×C4—61no—
135C2×#67—21no—
136D3×Q8—41no—
137D3×Q8—41no—
138D3×D4—41no—
139D3×D4—41no—
140C2×D12C2×C3×C411yescharacteristic
141C2×C4×D3—21no—
142C2×C4×D3C2×C3×C411yescharacteristic
143C2×C4×D3C3×D412yes—
144C2×Dic6C2×C3×C411yescharacteristic
145C2×Dic6C3×D412yes—
146C3×M16—22no—
147C3×M16—22no—
148C3×M16—42no—
149C3×M16—62no—
150C3×M16—122no—
151C2×C3×C8—122no—
152C3×C22⋊C4—22no—
153C3×C22⋊C4—22no—
154C3×C22⋊C4—42no—
155C3×C22⋊C4—62no—
156C3×C22⋊C4—62no—
157C3×C22⋊C4—62no—
158C3×C42—21no—
159C3×C42—31no—
160C3×C42—61no—
161C3×C42C4×D311yescharacteristic
162C3×C42C2×C3×C411yescharacteristic
163#217—22no—
164(C2×D3)⋊C4—22no—
165(C2×D3)⋊C4—22no—
166C4×Dic3—21no—
167C4×Dic3C2×C3×C411yescharacteristic
168#208—22no—
169ΓL(1,𝔽25)—42no—
170D3×C8—42no—
171C22×C32—42no—
172C22×C32—41no—
173C22×C32C2×C22⋊C411yescharacteristic
174C2×C3×D3—22no—
175C2×C3×D3—42no—
176C2×C3×D3—41no—
177C2×C3×D3—82no—
178C2×C3×D3—82no—
179C2×C3×D3#9911yescharacteristic
180C2×C3×D3#9911yescharacteristic
181C32×C4—22no—
182C32×C4—21no—
183C32×C4—41no—
184C3×Dic3—22no—
185C3×Dic3—21no—
186C3×Dic3—41no—
187C22×Q8—31no—
188C22×D4—31no—
189C2×M16—62no—
190#124—31no—
191#124—31no—
192#124—31no—
193#124—62no—
194#124C2×C3×D311yescharacteristic
195C2×C22⋊C4—62no—
196C2×C42—31no—
197#101—32no—
198#101—62no—
199#101C2×C3×D312yescomplemented
200C23×C3—21no—
201C23×C3—21no—
202C23×C3—31no—
203C23×C3—41no—
204C23×C3—61no—
205C23×C3—121no—
206C23×C3—121no—
207C23×C3—124no—
208C22×D3—21no—
209C22×D3—44no—
210C22×D3—41no—
211C22×D3—41no—
212C22×D3C3×C22⋊C411yescharacteristic
213C3×Q8—41no—
214C3×Q8—41no—
215C3×Q8—81no—
216C3×Q8—122no—
217C3×Q8—121no—
218C3×D4—41no—
219C3×D4—41no—
220C3×D4—81no—
221C3×D4—121no—
222C3×D4—122no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—22no—
227C2×C3×C4—31no—
228C2×C3×C4—32no—
229C2×C3×C4—41no—
230C2×C3×C4—62no—
231C2×C3×C4—62no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—122no—
236C2×C3×C4C2×C4×D311yescharacteristic
237C2×C3×C4D3×D412yes—
238C2×C3×C4C22×C3×C411yescharacteristic
239C2×C3×C4C2×C3×D412yes—
240#67—42no—
241C2×Dic3—21no—
242C2×Dic3C3×C22⋊C411yescharacteristic
243C2×Dic3C3×C22⋊C412yes—
244D12—41no—
245C4×D3—22no—
246C4×D3—22no—
247C4×D3—21no—
248C4×D3—21no—
249C4×D3—42no—
250Dic6—41no—
251Dic6—42no—
252C3×C8—42no—
253C3×C8—42no—
254C3×C8—82no—
255C3×C8—122no—
256C3⋊C8—42no—
257C2×C32—21no—
258C2×C32—81no—
259C2×C32C2×#9911yescharacteristic
260C3×D3—21no—
261C3×D3—81no—
262C3×D3#35512yesdirect product factor
263C24—61no—
264C2×Q8—31no—
265C2×Q8—32no—
266C2×Q8—121no—
267C2×Q8—121no—
268C2×Q8C22×C3×D311yescharacteristic
269C2×D4—31no—
270C2×D4—61no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4C3×C4×D311yescharacteristic
274C22×C4—31no—
275C22×C4—32no—
276C22×C4—61no—
277M16—22no—
278M16—62no—
279M16—122no—
280C2×C8—122no—
281C22⋊C4—22no—
282C22⋊C4—62no—
283C22⋊C4—62no—
284C22⋊C4—62no—
285C42—31no—
286C42—61no—
287C42C3×C4×D311yescharacteristic
288C22×C3—21no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—42no—
292C22×C3—42no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—62no—
296C22×C3—81no—
297C22×C3—82no—
298C22×C3—122no—
299C22×C3—121no—
300C22×C3—242no—
301C22×C3—242no—
302C22×C3D3×C22⋊C411yescharacteristic
303C22×C3C2×C3×C22⋊C411yescharacteristic
304C2×D3—22no—
305C2×D3—42no—
306C2×D3—41no—
307C2×D3—82no—
308C2×D3—82no—
309C2×D3C3×#9911yescharacteristic
310C2×D3C3×#9911yescharacteristic
311C3×C4—22no—
312C3×C4—21no—
313C3×C4—22no—
314C3×C4—21no—
315C3×C4—42no—
316C3×C4—41no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—61no—
320C3×C4—62no—
321C3×C4—81no—
322C3×C4—121no—
323Dic3—21no—
324Dic3—22no—
325Dic3—41no—
326C32C2×#35511yescharacteristic, complemented, Sylow
327C23—21no—
328C23—31no—
329C23—61no—
330C23—121no—
331C23—121no—
332C23—124no—
333Q8—41no—
334Q8—122no—
335Q8—121no—
336D4—41no—
337D4—121no—
338D4—122no—
339C2×C4—21no—
340C2×C4—31no—
341C2×C4—32no—
342C2×C4—62no—
343C2×C4—62no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—122no—
348C2×C4C2×C3×C4×D311yescharacteristic
349C2×C4C3×D3×D412yes—
350C8—42no—
351C8—122no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—32no—
356C2×C3—41no—
357C2×C3—61no—
358C2×C3—81no—
359C2×C3—81no—
360C2×C3—161no—
361C2×C3—241no—
362C2×C3D3×#9911yescharacteristic
363C2×C3C2×C3×#9911yescharacteristic
364D3—21no—
365D3—81no—
366D3C3×#35512yesdirect product factor
367C22—31no—
368C22—31no—
369C22—42no—
370C22—41no—
371C22—62no—
372C22—122no—
373C22—121no—
374C22—242no—
375C22—242no—
376C22C3×D3×C22⋊C411yescharacteristic
377C4—22no—
378C4—21no—
379C4—41no—
380C4—61no—
381C4—62no—
382C4—121no—
383C3—21no—
384C3D3×#35511yescharacteristic, direct product factor
385C3C2×C3×#35511yescharacteristic, complemented
386C2—21no—
387C2—32no—
388C2—61no—
389C2—81no—
390C2—241no—
391C2C3×D3×#9911yescharacteristic
subgroupG/H
centerC2×C3D3×#99
commutatorC2×C3×C4C22×C3×C4
FrattiniC2×Q8C22×C3×D3
FittingC32×#355C2
socleC2×C32C2×#99
prime radicalC2×C3×D3×D4C4
cosocleC2×C3×Q8C23×C3
hypercenterC3×#355D3
hypocenterC3C2×C3×#355
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×Q8 ◁ C3×#355

Complements

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

NH ≅ G/Ntype
2. C3×D3×#101C2semidirect
3. C32×#355C2semidirect
4. C3×#4542C2semidirect
5. C3×#4477C2semidirect
7. D3×#355C3direct (N × H)
18. C32×#101C22semidirect
19. C3×#836C22semidirect
24. D3×#101C2×C3semidirect
28. C3×#355D3direct (N × H)
29. C3×#355C2×C3semidirect
30. #4542C2×C3semidirect
31. #4477C2×C3semidirect
82. C3×#101C2×D3semidirect
83. C3×#101C22×C3semidirect
84. #836C22×C3semidirect
115. #355C3×D3direct (N × H)
199. #101C2×C3×D3semidirect
262. C3×D3#355direct (N × H)
326. C32C2×#355semidirect
366. D3C3×#355direct (N × H)
384. C3D3×#355direct (N × H)
385. C3C2×C3×#355semidirect

Maximal subgroups

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

subgroupindex
C3×D3×#1242
C3×D3×#1012
C32×#3552
C3×#45422
C3×#44772
C2×C3×#3553
D3×#3553