Finite Group Databasebeta

← C3×#69165 #11796489

Subgroups — C3×#69165 #11796489

Order 1152.

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

normal subgroups only

This group has 532 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×#5668C211yesmaximal, characteristic
2C3×#4751C212yesmaximal, complemented
3C32×#355C211yesmaximal, characteristic, complemented
4C3×#4542C211yesmaximal, characteristic, complemented
5C3×#4478C212yesmaximal, complemented
6C3×#2132—31nomaximal
7#69165C311yesmaximal, characteristic, direct product factor
8C2×C3×#239C2211yescharacteristic
9C2×C3×#237C411yescharacteristic
10C32×#124C2211yescharacteristic
11C3×#944C2211yescharacteristic
12C3×#940C411yescharacteristic
13C3×D3×M16—22no—
14C3×#884—22no—
15C3×#67⋊C4—22no—
16C3×#875C411yescharacteristic
17C3×C42×D3C411yescharacteristic
18C32×#101C2212yes—
19C3×#836C2212yes—
20C3×#823—22no—
21C3×#532—31no—
22#5668C2×C311yescharacteristic
23C3×#413—32no—
24#4751C2×C312yescomplemented
25C3×#356—32no—
26C3×#355—21no—
27C3×#355—31no—
28C3×#355D311yescharacteristic, complemented
29C3×#355C2×C311yescharacteristic, complemented
30#4542C2×C311yescharacteristic, complemented
31#4478C2×C312yescomplemented
32C2×C32×Q8C2311yescharacteristic
33C2×C32×D4C2×C411yescharacteristic
34C2×C3×#67—22no—
35C2×C3×#67—21no—
36C22×C3×Dic3—21no—
37C3×#239—41no—
38C3×#239—41no—
39C3×#237—41no—
40C3×#237—41no—
41C2×C3×D12C2×C411yescharacteristic
42C2×C3×D12D412yes—
43C2×C3×C4×D3—21no—
44C2×C3×C4×D3C2×C411yescharacteristic
45C2×C3×C4×D3D412yes—
46C2×C3×Dic6C2×C411yescharacteristic
47C32×M16—22no—
48C32×C22⋊C4—22no—
49C32×C42C2×C411yescharacteristic
50C3×(C2×D3)⋊C4—22no—
51C3×Dic3⋊C4—22no—
52C3×C4×Dic3—21no—
53C3×C4×Dic3C2×C411yescharacteristic
54C3×#208—22no—
55C3×ΓL(1,𝔽25)—42no—
56C3×D3×C8—42no—
57#2132—31noSylow
58C2×C3×#41—31no—
59C2×C3×#41—31no—
60C2×#239C22×C311yescharacteristic
61C2×#237C3×C411yescharacteristic
62C2×C3×M16—62no—
63C3×C4⋊Q8—31no—
64C3×#127—31no—
65C3×#124—21no—
66C3×#124—31no—
67C3×#124C2×D311yescharacteristic
68C3×#124C22×C311yescharacteristic
69C3×#121—62no—
70C3×C4×D4—62no—
71C2×C3×C42—31no—
72#944C22×C311yescharacteristic
73#940C3×C411yescharacteristic
74D3×M16—22no—
75#884—22no—
76#67⋊C4—22no—
77#875C3×C411yescharacteristic
78C42×D3C3×C411yescharacteristic
79C3×#101—22no—
80C3×#101—32no—
81C3×#101C2×D312yes—
82C3×#101C22×C312yes—
83C3×#100—62no—
84#836C22×C312yes—
85#823—22no—
86C23×C32—21no—
87C22×C3×D3C22⋊C412yes—
88C22×C3×D3C22⋊C411yescharacteristic
89C32×Q8—41no—
90C32×D4—41no—
91C2×C32×C4—21no—
92C2×C32×C4C22×C411yescharacteristic
93C2×C32×C4C2×D412yes—
94C3×#67—44no—
95C3×#67—42no—
96C2×C3×Dic3—21no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—41no—
99C2×C3×Dic3—41no—
100C2×C3×Dic3C22⋊C411yescharacteristic
101C3×D12—41no—
102C3×D12—42no—
103C3×C4×D3—22no—
104C3×C4×D3—22no—
105C3×C4×D3—21no—
106C3×C4×D3—21no—
107C3×C4×D3—42no—
108C3×Dic6—41no—
109C32×C8—42no—
110C3×C3⋊C8—42no—
111#532—31no—
112#413—32no—
113#356—32no—
114#355—31no—
115#355C3×D311yescharacteristic, complemented
116C3×#41—121no—
117C3×#41—121no—
118C3×#41—121no—
119C3×#41—121no—
120C2×C3×Q8—21no—
121C2×C3×Q8—31no—
122C2×C3×Q8C22×D311yescharacteristic
123C2×C3×Q8C23×C311yescharacteristic
124C2×C3×D4—21no—
125C2×C3×D4—31no—
126C2×C3×D4—32no—
127C2×C3×D4—62no—
128C2×C3×D4—61no—
129C2×C3×D4C4×D311yescharacteristic
130C2×C3×D4C2×C3×C411yescharacteristic
131C22×C3×C4—31no—
132C22×C3×C4—32no—
133C22×C3×C4—61no—
134C22×C3×C4—61no—
135C2×#67—22no—
136C2×#67—21no—
137C22×Dic3—21no—
138#239—41no—
139#239—41no—
140#237—41no—
141#237—41no—
142C2×D12C2×C3×C411yescharacteristic
143C2×D12C3×D412yes—
144C2×C4×D3—21no—
145C2×C4×D3C2×C3×C411yescharacteristic
146C2×C4×D3C3×D412yes—
147C2×Dic6C2×C3×C411yescharacteristic
148C3×M16—22no—
149C3×M16—22no—
150C3×M16—42no—
151C3×M16—62no—
152C3×M16—122no—
153C2×C3×C8—122no—
154C3×C4⋊C4—62no—
155C3×C22⋊C4—22no—
156C3×C22⋊C4—22no—
157C3×C22⋊C4—42no—
158C3×C22⋊C4—62no—
159C3×C42—21no—
160C3×C42—31no—
161C3×C42—61no—
162C3×C42C4×D311yescharacteristic
163C3×C42C2×C3×C411yescharacteristic
164(C2×D3)⋊C4—22no—
165Dic3⋊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—21no—
175C2×C3×D3—22no—
176C2×C3×D3—41no—
177C2×C3×D3—42no—
178C32×C4—22no—
179C32×C4—21no—
180C32×C4—41no—
181C3×Dic3—22no—
182C3×Dic3—41no—
183C3×Dic3#9911yescharacteristic
184C3×Dic3#9911yescharacteristic
185C2×#41—31no—
186C2×#41—31no—
187C2×M16—62no—
188C4⋊Q8—31no—
189#127—31no—
190#124—31no—
191#124C2×C3×D311yescharacteristic
192#121—62no—
193C4×D4—62no—
194C2×C42—31no—
195#101—32no—
196#101C2×C3×D312yes—
197#100—62no—
198C23×C3—21no—
199C23×C3—21no—
200C23×C3—32no—
201C23×C3—31no—
202C23×C3—41no—
203C22×D3C3×C22⋊C412yes—
204C22×D3C3×C22⋊C411yescharacteristic
205C3×Q8—41no—
206C3×Q8—41no—
207C3×Q8—81no—
208C3×Q8—121no—
209C3×D4—41no—
210C3×D4—41no—
211C3×D4—81no—
212C3×D4—122no—
213C3×D4—121no—
214C3×D4—124no—
215C3×D4—122no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—22no—
220C2×C3×C4—31no—
221C2×C3×C4—41no—
222C2×C3×C4—62no—
223C2×C3×C4—62no—
224C2×C3×C4—61no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—121no—
229C2×C3×C4—121no—
230C2×C3×C4—122no—
231C2×C3×C4C2×C4×D311yescharacteristic
232C2×C3×C4D3×D412yes—
233C2×C3×C4C22×C3×C411yescharacteristic
234C2×C3×C4C2×C3×D412yes—
235#67—44no—
236#67—42no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3C3×C22⋊C411yescharacteristic
242D12—41no—
243D12—42no—
244C4×D3—22no—
245C4×D3—22no—
246C4×D3—21no—
247C4×D3—21no—
248C4×D3—42no—
249Dic6—41no—
250C3×C8—42no—
251C3×C8—42no—
252C3×C8—82no—
253C3×C8—122no—
254C3⋊C8—42no—
255C2×C32—21no—
256C2×C32—81no—
257C2×C32C2×#9911yescharacteristic
258C3×D3—41no—
259C3×D3—42no—
260#41—121no—
261#41—121no—
262#41—121no—
263#41—121no—
264C2×Q8—31no—
265C2×Q8C22×C3×D311yescharacteristic
266C2×D4—31no—
267C2×D4—32no—
268C2×D4—62no—
269C2×D4—61no—
270C2×D4C3×C4×D311yescharacteristic
271C22×C4—31no—
272C22×C4—32no—
273C22×C4—61no—
274C22×C4—61no—
275M16—22no—
276M16—62no—
277M16—122no—
278C2×C8—122no—
279C4⋊C4—62no—
280C22⋊C4—22no—
281C22⋊C4—62no—
282C42—31no—
283C42—61no—
284C42C3×C4×D311yescharacteristic
285C22×C3—21no—
286C22×C3—42no—
287C22×C3—42no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—61no—
291C22×C3—62no—
292C22×C3—81no—
293C22×C3—82no—
294C22×C3—121no—
295C22×C3—122no—
296C22×C3D3×C22⋊C411yescharacteristic
297C22×C3C2×C3×C22⋊C411yescharacteristic
298C2×D3—21no—
299C2×D3—22no—
300C2×D3—41no—
301C2×D3—42no—
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—22no—
305C3×C4—21no—
306C3×C4—31no—
307C3×C4—31no—
308C3×C4—42no—
309C3×C4—41no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—62no—
313C3×C4—81no—
314C3×C4—121no—
315Dic3—22no—
316Dic3—41no—
317Dic3C3×#9911yescharacteristic
318Dic3C3×#9911yescharacteristic
319C32#213211yescharacteristic, complemented, Sylow
320C23—21no—
321C23—32no—
322C23—31no—
323Q8—41no—
324Q8—121no—
325D4—41no—
326D4—122no—
327D4—121no—
328D4—124no—
329D4—122no—
330C2×C4—21no—
331C2×C4—31no—
332C2×C4—62no—
333C2×C4—62no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—122no—
341C2×C4C2×C3×C4×D311yescharacteristic
342C2×C4C3×D3×D412yes—
343C8—42no—
344C8—122no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—41no—
349C2×C3—81no—
350C2×C3—81no—
351C2×C3—121no—
352C2×C3—122no—
353C2×C3—161no—
354C2×C3D3×#9911yescharacteristic
355C2×C3C2×C3×#9911yescharacteristic
356D3—41no—
357D3—42no—
358C22—42no—
359C22—41no—
360C22—61no—
361C22—62no—
362C22—121no—
363C22—122no—
364C22C3×D3×C22⋊C411yescharacteristic
365C4—22no—
366C4—21no—
367C4—31no—
368C4—31no—
369C4—41no—
370C4—62no—
371C4—121no—
372C3—21no—
373C3#6916511yescharacteristic, direct product factor
374C3C3×#213211yescharacteristic, complemented
375C2—21no—
376C2—81no—
377C2—121no—
378C2—122no—
379C2C3×D3×#9911yescharacteristic
subgroupG/H
centerC2×C3D3×#99
commutatorC2×C3×C4C22×C3×C4
FrattiniC2×Q8C22×C3×D3
FittingC32×#355C2
socleC2×C32C2×#99
prime radicalC3×#5668C2
cosocleC2×C3×Q8C23×C3
hypercenterC3×#355D3
hypocenterC3C3×#2132
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×#4751C2semidirect
3. C32×#355C2semidirect
4. C3×#4542C2semidirect
5. C3×#4478C2semidirect
7. #69165C3direct (N × H)
24. #4751C2×C3semidirect
28. C3×#355D3semidirect
29. C3×#355C2×C3semidirect
30. #4542C2×C3semidirect
31. #4478C2×C3semidirect
115. #355C3×D3semidirect
319. C32#2132semidirect
373. C3#69165direct (N × H)
374. C3C3×#2132semidirect

Maximal subgroups

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

subgroupindex
C3×#56682
C3×#47512
C32×#3552
C3×#45422
C3×#44782
C3×#21323
#691653