Finite Group Databasebeta

← C3×#32927 #10651534

Subgroups — C3×#32927 #10651534

Order 768.

1482 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 1482 conjugacy classes of subgroups — too many to draw. Only its 298 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×C4×H(5,𝔽2)C211yesmaximal, characteristic
2C2×C3×#458C212yesmaximal
3C3×#2486C213yesmaximal, complemented
4C3×#1890C216yesmaximal, complemented
5C3×#1799C213yesmaximal, complemented
6#32927C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×H(5,𝔽2)C2211yescharacteristic
8C2×C3×#123C2216yes—
9C3×#520—23no—
10C3×#518—23no—
11C3×#518C2213yes—
12C3×C4×#41—23no—
13C2×C3×C4×D4—23no—
14C2×C3×C4×D4C2213yes—
15C22×C3×C4⋊C4C2211yescharacteristic, complemented
16C3×#458—28no—
17C2×C3×#99C2216yes—
18C3×#398—23no—
19C3×#397—23no—
20C3×#389—23no—
21C3×#381C2216yescomplemented
22C3×#380—23no—
23C3×#380C2213yescomplemented
24C3×#342C2216yescomplemented
25C4×H(5,𝔽2)C2×C311yescharacteristic
26C2×#458C2×C312yes—
27#2486C2×C313yescomplemented
28#1890C2×C316yescomplemented
29#1799C2×C313yescomplemented
30C3×H(5,𝔽2)—26no—
31C3×H(5,𝔽2)D414yes—
32C2×C3×#41—23no—
33C22×C3×D4—23no—
34C22×C3×D4C2313yes—
35C23×C3×C4—21no—
36C23×C3×C4D413yes—
37C23×C3×C4C2312yes—
38C23×C3×C4C2311yescharacteristic
39C3×#123—212no—
40C3×#123—212no—
41C3×C4×Q8—21no—
42C3×C4×Q8—23no—
43C3×C4×D4—23no—
44C3×C4×D4—26no—
45C3×C4×D4—23no—
46C3×C4×D4—26no—
47C3×C4×D4—26no—
48C3×C4×D4—26no—
49C3×C4×D4—26no—
50C3×#117—26no—
51C3×#117—23no—
52C2×C3×C4⋊C4—23no—
53C2×C3×C4⋊C4—23no—
54C2×C3×C4⋊C4—23no—
55C2×C3×C4⋊C4—26no—
56C2×C3×C4⋊C4—23no—
57C2×C3×C4⋊C4D413yes—
58C2×C3×C22⋊C4—23no—
59C2×C3×C22⋊C4—23no—
60C2×C3×C22⋊C4—26no—
61C2×C3×C22⋊C4D413yes—
62C2×C3×C22⋊C4C2316yes—
63C2×C3×C22⋊C4C2313yes—
64C2×C3×C42—23no—
65C2×C3×C42D413yes—
66C3×#99—212no—
67C3×#95—26no—
68C3×#95—23no—
69C3×#95—26no—
70C2×H(5,𝔽2)C22×C311yescharacteristic
71C2×#123C22×C316yes—
72#520—23no—
73#518—23no—
74#518C22×C313yes—
75C4×#41—23no—
76C2×C4×D4—23no—
77C2×C4×D4C22×C313yes—
78C22×C4⋊C4C22×C311yescharacteristic, complemented
79#458—28no—
80C2×#99C22×C316yes—
81#398—23no—
82#397—23no—
83#389—23no—
84#381C22×C316yescomplemented
85#380—23no—
86#380C22×C313yescomplemented
87#342C22×C316yescomplemented
88C24×C3—21no—
89C24×C3C2×D413yes—
90C24×C3C2411yescharacteristic
91C3×#41—212no—
92C3×#41—212no—
93C2×C3×Q8—21no—
94C2×C3×D4—23no—
95C2×C3×D4—212no—
96C2×C3×D4—212no—
97C2×C3×D4—26no—
98C2×C3×D4—26no—
99C2×C3×D4C2×D416yes—
100C2×C3×D4#4116yes—
101C22×C3×C4—23no—
102C22×C3×C4—23no—
103C22×C3×C4—23no—
104C22×C3×C4—23no—
105C22×C3×C4—23no—
106C22×C3×C4—26no—
107C22×C3×C4—26no—
108C22×C3×C4—22no—
109C22×C3×C4—26no—
110C22×C3×C4—42no—
111C22×C3×C4—46no—
112C22×C3×C4—82no—
113C22×C3×C4C2×D413yes—
114C22×C3×C4C2×D413yes—
115C22×C3×C4C2×D413yes—
116C22×C3×C4C2×D413yes—
117C22×C3×C4C2×D413yes—
118C3×C4⋊C4—23no—
119C3×C4⋊C4—23no—
120C3×C4⋊C4—26no—
121C3×C4⋊C4—46no—
122C3×C4⋊C4—43no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—412no—
125C3×C22⋊C4—26no—
126C3×C22⋊C4—26no—
127C3×C22⋊C4—26no—
128C3×C22⋊C4—26no—
129C3×C22⋊C4—26no—
130C3×C22⋊C4—412no—
131C3×C42—26no—
132C3×C42—23no—
133C3×C42—23no—
134H(5,𝔽2)—26no—
135H(5,𝔽2)C3×D414yes—
136C2×#41—23no—
137C22×D4—23no—
138C22×D4C23×C313yes—
139C23×C4—21no—
140C23×C4C3×D413yes—
141C23×C4C23×C312yes—
142C23×C4C23×C311yescharacteristic
143#123—212no—
144#123—212no—
145C4×Q8—21no—
146C4×Q8—23no—
147C4×D4—23no—
148C4×D4—26no—
149C4×D4—23no—
150C4×D4—26no—
151C4×D4—26no—
152C4×D4—26no—
153C4×D4—26no—
154#117—26no—
155#117—23no—
156C2×C4⋊C4—23no—
157C2×C4⋊C4—23no—
158C2×C4⋊C4—23no—
159C2×C4⋊C4—26no—
160C2×C4⋊C4—23no—
161C2×C4⋊C4C3×D413yes—
162C2×C22⋊C4—23no—
163C2×C22⋊C4—23no—
164C2×C22⋊C4—26no—
165C2×C22⋊C4C3×D413yes—
166C2×C22⋊C4C23×C316yes—
167C2×C22⋊C4C23×C313yes—
168C2×C42—23no—
169C2×C42C3×D413yes—
170#99—212no—
171#95—26no—
172#95—23no—
173#95—26no—
174C23×C3—24no—
175C23×C3—23no—
176C23×C3—23no—
177C23×C3—41no—
178C23×C3—41no—
179C23×C3—43no—
180C23×C3—43no—
181C23×C3—81no—
182C23×C3—81no—
183C23×C3#12116yes—
184C23×C3C22×D413yes—
185C23×C3C22×D411yescharacteristic
186C23×C3C2×#4113yes—
187C3×Q8—24no—
188C3×D4—212no—
189C3×D4—212no—
190C3×D4—212no—
191C2×C3×C4—212no—
192C2×C3×C4—23no—
193C2×C3×C4—23no—
194C2×C3×C4—26no—
195C2×C3×C4—23no—
196C2×C3×C4—23no—
197C2×C3×C4—43no—
198C2×C3×C4—46no—
199C2×C3×C4—43no—
200C2×C3×C4—43no—
201C2×C3×C4—46no—
202C2×C3×C4—46no—
203C2×C3×C4—42no—
204C2×C3×C4—46no—
205C2×C3×C4—46no—
206C2×C3×C4—46no—
207C2×C3×C4—83no—
208C2×C3×C4—83no—
209C2×C3×C4—83no—
210C2×C3×C4#12013yes—
211C2×C3×C4#12011yescharacteristic
212C2×C3×C4#12116yes—
213C24—21no—
214C24C2×C3×D413yes—
215C24C24×C311yescharacteristic
216#41—212no—
217#41—212no—
218C2×Q8—21no—
219C2×D4—23no—
220C2×D4—212no—
221C2×D4—212no—
222C2×D4—26no—
223C2×D4—26no—
224C2×D4C2×C3×D416yes—
225C2×D4C3×#4116yes—
226C22×C4—23no—
227C22×C4—23no—
228C22×C4—23no—
229C22×C4—23no—
230C22×C4—23no—
231C22×C4—26no—
232C22×C4—26no—
233C22×C4—22no—
234C22×C4—26no—
235C22×C4—42no—
236C22×C4—46no—
237C22×C4—82no—
238C22×C4C2×C3×D413yes—
239C22×C4C2×C3×D413yes—
240C22×C4C2×C3×D413yes—
241C22×C4C2×C3×D413yes—
242C22×C4C2×C3×D413yes—
243C4⋊C4—23no—
244C4⋊C4—23no—
245C4⋊C4—26no—
246C4⋊C4—46no—
247C4⋊C4—43no—
248C4⋊C4—41no—
249C4⋊C4—412no—
250C22⋊C4—26no—
251C22⋊C4—26no—
252C22⋊C4—26no—
253C22⋊C4—26no—
254C22⋊C4—26no—
255C22⋊C4—412no—
256C42—26no—
257C42—23no—
258C42—23no—
259C22×C3—23no—
260C22×C3—23no—
261C22×C3—26no—
262C22×C3—41no—
263C22×C3—43no—
264C22×C3—46no—
265C22×C3—43no—
266C22×C3—43no—
267C22×C3—83no—
268C22×C3—84no—
269C22×C3C2×#12011yescharacteristic
270C22×C3C2×#12113yes—
271C22×C3#52513yes—
272C3×C4—26no—
273C3×C4—23no—
274C3×C4—43no—
275C3×C4—43no—
276C3×C4—86no—
277C3×C4—82no—
278C3×C4#45812yescomplemented
279C23—24no—
280C23—23no—
281C23—23no—
282C23—41no—
283C23—41no—
284C23—43no—
285C23—43no—
286C23—81no—
287C23—81no—
288C23C3×#12116yes—
289C23C22×C3×D413yes—
290C23C22×C3×D411yescharacteristic
291C23C2×C3×#4113yes—
292Q8—24no—
293D4—212no—
294D4—212no—
295D4—212no—
296C2×C4—212no—
297C2×C4—23no—
298C2×C4—23no—
299C2×C4—26no—
300C2×C4—23no—
301C2×C4—23no—
302C2×C4—43no—
303C2×C4—46no—
304C2×C4—43no—
305C2×C4—43no—
306C2×C4—46no—
307C2×C4—46no—
308C2×C4—42no—
309C2×C4—46no—
310C2×C4—46no—
311C2×C4—46no—
312C2×C4—83no—
313C2×C4—83no—
314C2×C4—83no—
315C2×C4C3×#12013yes—
316C2×C4C3×#12011yescharacteristic
317C2×C4C3×#12116yes—
318C2×C3—23no—
319C2×C3—23no—
320C2×C3—46no—
321C2×C3#284711yescharacteristic
322C2×C3C2×#45811yescharacteristic
323C2×C3#302811yescharacteristic
324C22—23no—
325C22—23no—
326C22—26no—
327C22—41no—
328C22—43no—
329C22—46no—
330C22—43no—
331C22—43no—
332C22—83no—
333C22—84no—
334C22C2×C3×#12011yescharacteristic
335C22C2×C3×#12113yes—
336C22C3×#52513yes—
337C4—23no—
338C4—26no—
339C4—43no—
340C4—43no—
341C4—86no—
342C4—82no—
343C4C3×#45812yescomplemented
344C3#3292711yescharacteristic, direct product factor, Sylow
345C2—23no—
346C2—23no—
347C2—46no—
348C2C3×#284711yescharacteristic
349C2C2×C3×#45811yescharacteristic
350C2C3×#302811yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC2×C3×H(5,𝔽2)C22
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
3. C3×#2486C2semidirect
4. C3×#1890C2semidirect
5. C3×#1799C2semidirect
6. #32927C3direct (N × H)
15. C22×C3×C4⋊C4C22semidirect
21. C3×#381C22semidirect
23. C3×#380C22semidirect
24. C3×#342C22semidirect
27. #2486C2×C3semidirect
28. #1890C2×C3semidirect
29. #1799C2×C3semidirect
78. C22×C4⋊C4C22×C3semidirect
84. #381C22×C3semidirect
86. #380C22×C3semidirect
87. #342C22×C3semidirect
278. C3×C4#458semidirect
343. C4C3×#458semidirect
344. C3#32927direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×H(5,𝔽2)2
C2×C3×#4582
C3×#24862
C3×#18902
C3×#17992
#329273