Finite Group Databasebeta

← C3×#69150 #11796382

Subgroups — C3×#69150 #11796382

Order 1152.

615 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 615 conjugacy classes of subgroups — too many to draw. Only its 110 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×#5715C211yesmaximal, characteristic
2C3×D3×#99C212yesmaximal, complemented
3C32×#353C211yesmaximal, characteristic, complemented
4C3×#4545C211yesmaximal, characteristic, complemented
5C3×#4473C212yesmaximal, complemented
6C3×#2127—31nomaximal
7#69150C311yesmaximal, characteristic, direct product factor
8C2×C3×#237C411yescharacteristic, complemented
9C2×C3×D3×D4C2211yescharacteristic
10C32×#127C2211yescharacteristic
11C3×#940C411yescharacteristic, complemented
12C3×#939—22no—
13C3×#937—22no—
14C3×#936C2211yescharacteristic
15C3×D3×C22⋊C4—22no—
16C3×#875C411yescharacteristic, complemented
17C3×#872C411yescharacteristic, complemented
18C32×#99C2212yes—
19C3×#834C2212yes—
20C3×#806—22no—
21C3×#535—31no—
22#5715C2×C311yescharacteristic
23C2×C3×#99—32no—
24D3×#99C2×C312yescomplemented
25C3×#354—32no—
26C3×#353—21no—
27C3×#353—31no—
28C3×#353D311yescharacteristic, complemented
29C3×#353C2×C311yescharacteristic, complemented
30#4545C2×C311yescharacteristic, complemented
31#4473C2×C312yescomplemented
32C23×C3×D3D412yes—
33C2×C32×D4—22no—
34C2×C32×D4C2×C411yescharacteristic, complemented
35C2×C32×D4C2311yescharacteristic
36C2×C3×#67—21no—
37C2×C3×#67—22no—
38C2×C3×#67D412yes—
39C22×C3×Dic3—21no—
40C3×#237—42no—
41C3×D3×D4—42no—
42C2×C3×D12C2×C411yescharacteristic
43C2×C3×C4×D3—42no—
44C2×C3×C4×D3C2×C411yescharacteristic
45C2×C3×Dic6C2×C411yescharacteristic, complemented
46C32×C22⋊C4—22no—
47C32×C42C2×C411yescharacteristic, complemented
48C3×#217—22no—
49C3×#217—22no—
50C3×(C2×D3)⋊C4—21no—
51C3×(C2×D3)⋊C4—22no—
52C3×(C2×D3)⋊C4—42no—
53C3×Dic3⋊C4—21no—
54C3×C4×Dic3C2×C411yescharacteristic, complemented
55#2127—31noSylow
56C2×C3×#41—31no—
57C22×C3×D4—31no—
58C2×#237C3×C411yescharacteristic, complemented
59C2×D3×D4C22×C311yescharacteristic
60C3×#127—21no—
61C3×#127—31no—
62C3×#127C2×D311yescharacteristic
63C3×#127C22×C311yescharacteristic
64C3×#124—31no—
65C3×#124—31no—
66C3×#121—62no—
67C3×#120—62no—
68C3×#117—31no—
69C2×C3×C22⋊C4—62no—
70#940C3×C411yescharacteristic, complemented
71#939—22no—
72#937—22no—
73#936C22×C311yescharacteristic
74D3×C22⋊C4—22no—
75#875C3×C411yescharacteristic, complemented
76#872C3×C411yescharacteristic, complemented
77C3×#99—22no—
78C3×#99—32no—
79C3×#99—62no—
80C3×#99C2×D312yes—
81C3×#99C22×C312yes—
82#834C22×C312yes—
83#806—22no—
84C23×C32—21no—
85C23×C32C2×D412yes—
86C22×C3×D3—22no—
87C22×C3×D3—22no—
88C22×C3×D3—82no—
89C22×C3×D3C22⋊C412yes—
90C22×C3×D3C22⋊C411yescharacteristic
91C32×D4—41no—
92C32×D4—41no—
93C32×D4—82no—
94C2×C32×C4—21no—
95C2×C32×C4—42no—
96C2×C32×C4C22×C411yescharacteristic
97C3×#67—42no—
98C3×#67—42no—
99C3×#67—82no—
100C2×C3×Dic3—21no—
101C2×C3×Dic3—21no—
102C2×C3×Dic3—42no—
103C2×C3×Dic3—42no—
104C2×C3×Dic3C22⋊C411yescharacteristic
105C3×D12—41no—
106C3×C4×D3—22no—
107C3×C4×D3—84no—
108C3×Dic6—41no—
109#535—31no—
110C2×#99—32no—
111#354—32no—
112#353—31no—
113#353C3×D311yescharacteristic, complemented
114C24×C3—32no—
115C23×D3C3×D412yes—
116C3×#41—122no—
117C2×C3×Q8—31no—
118C2×C3×D4—22no—
119C2×C3×D4—22no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—32no—
123C2×C3×D4—31no—
124C2×C3×D4—42no—
125C2×C3×D4—61no—
126C2×C3×D4—62no—
127C2×C3×D4—122no—
128C2×C3×D4C4×D311yescharacteristic, complemented
129C2×C3×D4C2×C3×C411yescharacteristic, complemented
130C2×C3×D4C22×D311yescharacteristic
131C2×C3×D4C23×C311yescharacteristic
132C22×C3×C4—31no—
133C22×C3×C4—61no—
134C22×C3×C4—122no—
135C2×#67—21no—
136C2×#67—22no—
137C2×#67C3×D412yes—
138C22×Dic3—21no—
139#237—42no—
140D3×D4—42no—
141C2×D12C2×C3×C411yescharacteristic
142C2×C4×D3—42no—
143C2×C4×D3C2×C3×C411yescharacteristic
144C2×Dic6C2×C3×C411yescharacteristic, complemented
145C3×C4⋊C4—61no—
146C3×C22⋊C4—22no—
147C3×C22⋊C4—22no—
148C3×C22⋊C4—42no—
149C3×C22⋊C4—61no—
150C3×C22⋊C4—62no—
151C3×C22⋊C4—62no—
152C3×C22⋊C4—62no—
153C3×C22⋊C4—122no—
154C3×C42—21no—
155C3×C42—31no—
156C3×C42C4×D311yescharacteristic, complemented
157C3×C42C2×C3×C411yescharacteristic, complemented
158#217—22no—
159#217—22no—
160(C2×D3)⋊C4—21no—
161(C2×D3)⋊C4—22no—
162(C2×D3)⋊C4—42no—
163Dic3⋊C4—21no—
164C4×Dic3C2×C3×C411yescharacteristic, complemented
165C22×C32—22no—
166C22×C32—42no—
167C22×C32—42no—
168C22×C32—41no—
169C22×C32C2×C22⋊C411yescharacteristic
170C2×C3×D3—22no—
171C2×C3×D3—41no—
172C2×C3×D3—44no—
173C2×C3×D3—42no—
174C2×C3×D3—82no—
175C2×C3×D3#9912yes—
176C32×C4—21no—
177C32×C4—41no—
178C32×C4—82no—
179C3×Dic3—21no—
180C3×Dic3—41no—
181C3×Dic3—41no—
182C3×Dic3—82no—
183C2×#41—31no—
184C22×D4—31no—
185#127—31no—
186#127C2×C3×D311yescharacteristic
187#124—31no—
188#124—31no—
189#121—62no—
190#120—62no—
191#117—31no—
192C2×C22⋊C4—62no—
193#99—32no—
194#99—62no—
195#99C2×C3×D312yes—
196C23×C3—21no—
197C23×C3—21no—
198C23×C3—22no—
199C23×C3—31no—
200C23×C3—32no—
201C23×C3—41no—
202C23×C3—62no—
203C23×C3—62no—
204C23×C3—242no—
205C23×C3D3×D412yes—
206C23×C3C2×C3×D412yes—
207C22×D3—22no—
208C22×D3—22no—
209C22×D3—82no—
210C22×D3C3×C22⋊C412yes—
211C22×D3C3×C22⋊C411yescharacteristic
212C3×Q8—121no—
213C3×D4—41no—
214C3×D4—41no—
215C3×D4—41no—
216C3×D4—41no—
217C3×D4—81no—
218C3×D4—81no—
219C3×D4—82no—
220C3×D4—82no—
221C3×D4—84no—
222C3×D4—121no—
223C3×D4—122no—
224C3×D4—122no—
225C3×D4—242no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—31no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—62no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—82no—
237C2×C3×C4—122no—
238C2×C3×C4—122no—
239C2×C3×C4—244no—
240C2×C3×C4C2×C4×D311yescharacteristic
241C2×C3×C4C22×C3×C411yescharacteristic
242#67—42no—
243#67—42no—
244#67—82no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3C3×C22⋊C411yescharacteristic
250D12—41no—
251C4×D3—22no—
252C4×D3—84no—
253Dic6—41no—
254C2×C32—21no—
255C2×C32—42no—
256C2×C32—81no—
257C2×C32C2×#9911yescharacteristic
258C3×D3—22no—
259C3×D3—42no—
260C24—32no—
261#41—122no—
262C2×Q8—31no—
263C2×D4—22no—
264C2×D4—32no—
265C2×D4—31no—
266C2×D4—61no—
267C2×D4—62no—
268C2×D4—122no—
269C2×D4C3×C4×D311yescharacteristic, complemented
270C2×D4C22×C3×D311yescharacteristic
271C22×C4—31no—
272C22×C4—61no—
273C22×C4—122no—
274C4⋊C4—61no—
275C22⋊C4—22no—
276C22⋊C4—61no—
277C22⋊C4—62no—
278C22⋊C4—62no—
279C22⋊C4—62no—
280C22⋊C4—122no—
281C42—31no—
282C42C3×C4×D311yescharacteristic, complemented
283C22×C3—22no—
284C22×C3—22no—
285C22×C3—21no—
286C22×C3—32no—
287C22×C3—42no—
288C22×C3—42no—
289C22×C3—42no—
290C22×C3—42no—
291C22×C3—42no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—62no—
295C22×C3—81no—
296C22×C3—82no—
297C22×C3—82no—
298C22×C3—121no—
299C22×C3—122no—
300C22×C3—124no—
301C22×C3—242no—
302C22×C3D3×C22⋊C411yescharacteristic
303C22×C3C2×C3×C22⋊C411yescharacteristic
304C2×D3—22no—
305C2×D3—41no—
306C2×D3—44no—
307C2×D3—42no—
308C2×D3—82no—
309C2×D3C3×#9912yes—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—41no—
313C3×C4—41no—
314C3×C4—41no—
315C3×C4—42no—
316C3×C4—61no—
317C3×C4—82no—
318C3×C4—82no—
319C3×C4—121no—
320C3×C4—121no—
321C3×C4—162no—
322C3×C4—242no—
323Dic3—21no—
324Dic3—41no—
325Dic3—41no—
326Dic3—82no—
327C32#212711yescharacteristic, complemented, Sylow
328C23—21no—
329C23—31no—
330C23—32no—
331C23—62no—
332C23—62no—
333C23—242no—
334C23C3×D3×D412yes—
335Q8—121no—
336D4—41no—
337D4—41no—
338D4—82no—
339D4—121no—
340D4—122no—
341D4—122no—
342D4—242no—
343C2×C4—21no—
344C2×C4—31no—
345C2×C4—42no—
346C2×C4—62no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—244no—
352C2×C4C2×C3×C4×D311yescharacteristic
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—41no—
357C2×C3—42no—
358C2×C3—42no—
359C2×C3—62no—
360C2×C3—82no—
361C2×C3—81no—
362C2×C3—81no—
363C2×C3—122no—
364C2×C3—161no—
365C2×C3D3×#9911yescharacteristic
366C2×C3C2×C3×#9911yescharacteristic
367D3—22no—
368D3—42no—
369C22—22no—
370C22—32no—
371C22—42no—
372C22—42no—
373C22—41no—
374C22—62no—
375C22—121no—
376C22—122no—
377C22—124no—
378C22—242no—
379C22C3×D3×C22⋊C411yescharacteristic
380C4—21no—
381C4—41no—
382C4—61no—
383C4—82no—
384C4—121no—
385C4—121no—
386C4—242no—
387C3—21no—
388C3#6915011yescharacteristic, direct product factor
389C3C3×#212711yescharacteristic, complemented
390C2—21no—
391C2—42no—
392C2—62no—
393C2—81no—
394C2—122no—
395C2C3×D3×#9911yescharacteristic
subgroupG/H
centerC2×C3D3×#99
commutatorC2×C3×C4C22×C3×C4
FrattiniC2×D4C22×C3×D3
FittingC32×#353C2
socleC2×C32C2×#99
prime radicalC3×#5715C2
cosocleC2×C3×D4C23×C3
hypercenterC3×#353D3
hypocenterC3C3×#2127
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#353

Complements

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

NH ≅ G/Ntype
2. C3×D3×#99C2semidirect
3. C32×#353C2semidirect
4. C3×#4545C2semidirect
5. C3×#4473C2semidirect
7. #69150C3direct (N × H)
8. C2×C3×#237C4semidirect
11. C3×#940C4semidirect
16. C3×#875C4semidirect
17. C3×#872C4semidirect
24. D3×#99C2×C3semidirect
28. C3×#353D3semidirect
29. C3×#353C2×C3semidirect
30. #4545C2×C3semidirect
31. #4473C2×C3semidirect
34. C2×C32×D4C2×C4semidirect
45. C2×C3×Dic6C2×C4semidirect
47. C32×C42C2×C4semidirect
54. C3×C4×Dic3C2×C4semidirect
58. C2×#237C3×C4semidirect
70. #940C3×C4semidirect
75. #875C3×C4semidirect
76. #872C3×C4semidirect
113. #353C3×D3semidirect
128. C2×C3×D4C4×D3semidirect
129. C2×C3×D4C2×C3×C4semidirect
144. C2×Dic6C2×C3×C4semidirect
156. C3×C42C4×D3semidirect
157. C3×C42C2×C3×C4semidirect
164. C4×Dic3C2×C3×C4semidirect
269. C2×D4C3×C4×D3semidirect
282. C42C3×C4×D3semidirect
327. C32#2127semidirect
388. C3#69150direct (N × H)
389. C3C3×#2127semidirect

Maximal subgroups

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

subgroupindex
C3×#57152
C3×D3×#992
C32×#3532
C3×#45452
C3×#44732
C3×#21273
#691503