Finite Group Databasebeta

← C3×#32976 #10651583

Subgroups — C3×#32976 #10651583

Order 768.

1382 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 1382 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
2C3×#2487C213yesmaximal, complemented
3C3×#1867C216yesmaximal, complemented
4C3×#1799C213yesmaximal, complemented
5C3×#1790C212yesmaximal
6#32976C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×H(5,𝔽2)C2211yescharacteristic
8C3×#520—23no—
9C3×#518—23no—
10C3×#518C2213yes—
11C3×C4×#41—23no—
12C2×C3×C4×D4—23no—
13C2×C3×C4×D4C2213yes—
14C22×C3×C4⋊C4C2211yescharacteristic, complemented
15C2×C3×#99C2216yes—
16C3×#400—23no—
17C3×#399—23no—
18C3×#389—23no—
19C3×#386C2216yes—
20C3×#385C2216yescomplemented
21C3×#380—23no—
22C3×#380C2213yescomplemented
23C3×#342C2216yescomplemented
24C4×H(5,𝔽2)C2×C311yescharacteristic
25#2487C2×C313yescomplemented
26#1867C2×C316yescomplemented
27#1799C2×C313yescomplemented
28#1790C2×C312yes—
29C3×H(5,𝔽2)—26no—
30C3×H(5,𝔽2)Q814yes—
31C2×C3×#41—23no—
32C22×C3×D4—23no—
33C22×C3×D4C2313yes—
34C23×C3×C4—21no—
35C23×C3×C4D413yes—
36C23×C3×C4C2312yes—
37C23×C3×C4C2311yescharacteristic
38C3×C4×Q8—21no—
39C3×C4×Q8—23no—
40C3×C4×D4—23no—
41C3×C4×D4—26no—
42C3×C4×D4—23no—
43C3×C4×D4—26no—
44C3×C4×D4—26no—
45C3×C4×D4—26no—
46C3×C4×D4—26no—
47C3×#117—26no—
48C3×#117—23no—
49C2×C3×C4⋊C4—23no—
50C2×C3×C4⋊C4—23no—
51C2×C3×C4⋊C4—23no—
52C2×C3×C4⋊C4—23no—
53C2×C3×C4⋊C4—26no—
54C2×C3×C4⋊C4D413yes—
55C2×C3×C22⋊C4—23no—
56C2×C3×C22⋊C4—23no—
57C2×C3×C22⋊C4—26no—
58C2×C3×C22⋊C4D413yes—
59C2×C3×C22⋊C4C2316yes—
60C2×C3×C22⋊C4C2313yes—
61C2×C3×C42—23no—
62C2×C3×C42D413yes—
63C3×#99—212no—
64C3×#95—26no—
65C3×#95—23no—
66C3×#95—26no—
67C2×H(5,𝔽2)C22×C311yescharacteristic
68#520—23no—
69#518—23no—
70#518C22×C313yes—
71C4×#41—23no—
72C2×C4×D4—23no—
73C2×C4×D4C22×C313yes—
74C22×C4⋊C4C22×C311yescharacteristic, complemented
75C2×#99C22×C316yes—
76#400—23no—
77#399—23no—
78#389—23no—
79#386C22×C316yes—
80#385C22×C316yescomplemented
81#380—23no—
82#380C22×C313yescomplemented
83#342C22×C316yescomplemented
84C24×C3—21no—
85C24×C3C2×D413yes—
86C24×C3C2411yescharacteristic
87C3×#41—212no—
88C3×#41—212no—
89C2×C3×Q8—21no—
90C2×C3×D4—23no—
91C2×C3×D4—212no—
92C2×C3×D4—212no—
93C2×C3×D4—26no—
94C2×C3×D4—26no—
95C2×C3×D4C2×Q816yes—
96C2×C3×D4#4116yes—
97C22×C3×C4—23no—
98C22×C3×C4—23no—
99C22×C3×C4—23no—
100C22×C3×C4—23no—
101C22×C3×C4—23no—
102C22×C3×C4—26no—
103C22×C3×C4—26no—
104C22×C3×C4—42no—
105C22×C3×C4—46no—
106C22×C3×C4—82no—
107C22×C3×C4—82no—
108C22×C3×C4C2×D413yes—
109C22×C3×C4C2×D413yes—
110C22×C3×C4C2×D413yes—
111C22×C3×C4C2×D413yes—
112C22×C3×C4C2×D413yes—
113C3×C4⋊C4—23no—
114C3×C4⋊C4—23no—
115C3×C4⋊C4—26no—
116C3×C4⋊C4—46no—
117C3×C4⋊C4—43no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—86no—
120C3×C22⋊C4—26no—
121C3×C22⋊C4—26no—
122C3×C22⋊C4—26no—
123C3×C22⋊C4—212no—
124C3×C22⋊C4—86no—
125C3×C42—26no—
126C3×C42—23no—
127C3×C42—23no—
128H(5,𝔽2)—26no—
129H(5,𝔽2)C3×Q814yes—
130C2×#41—23no—
131C22×D4—23no—
132C22×D4C23×C313yes—
133C23×C4—21no—
134C23×C4C3×D413yes—
135C23×C4C23×C312yes—
136C23×C4C23×C311yescharacteristic
137C4×Q8—21no—
138C4×Q8—23no—
139C4×D4—23no—
140C4×D4—26no—
141C4×D4—23no—
142C4×D4—26no—
143C4×D4—26no—
144C4×D4—26no—
145C4×D4—26no—
146#117—26no—
147#117—23no—
148C2×C4⋊C4—23no—
149C2×C4⋊C4—23no—
150C2×C4⋊C4—23no—
151C2×C4⋊C4—23no—
152C2×C4⋊C4—26no—
153C2×C4⋊C4C3×D413yes—
154C2×C22⋊C4—23no—
155C2×C22⋊C4—23no—
156C2×C22⋊C4—26no—
157C2×C22⋊C4C3×D413yes—
158C2×C22⋊C4C23×C316yes—
159C2×C22⋊C4C23×C313yes—
160C2×C42—23no—
161C2×C42C3×D413yes—
162#99—212no—
163#95—26no—
164#95—23no—
165#95—26no—
166C23×C3—24no—
167C23×C3—23no—
168C23×C3—23no—
169C23×C3—41no—
170C23×C3—41no—
171C23×C3—43no—
172C23×C3—43no—
173C23×C3—81no—
174C23×C3—81no—
175C23×C3C22⋊Q816yes—
176C23×C3C22×D413yes—
177C23×C3C22×Q811yescharacteristic
178C23×C3C2×#4113yes—
179C3×Q8—24no—
180C3×D4—212no—
181C3×D4—212no—
182C3×D4—212no—
183C2×C3×C4—212no—
184C2×C3×C4—23no—
185C2×C3×C4—23no—
186C2×C3×C4—26no—
187C2×C3×C4—23no—
188C2×C3×C4—23no—
189C2×C3×C4—43no—
190C2×C3×C4—46no—
191C2×C3×C4—43no—
192C2×C3×C4—43no—
193C2×C3×C4—46no—
194C2×C3×C4—46no—
195C2×C3×C4—46no—
196C2×C3×C4—46no—
197C2×C3×C4—42no—
198C2×C3×C4—46no—
199C2×C3×C4—83no—
200C2×C3×C4—83no—
201C2×C3×C4—83no—
202C2×C3×C4#12013yes—
203C2×C3×C4#12011yescharacteristic
204C2×C3×C4C22⋊Q816yes—
205C24—21no—
206C24C2×C3×D413yes—
207C24C24×C311yescharacteristic
208#41—212no—
209#41—212no—
210C2×Q8—21no—
211C2×D4—23no—
212C2×D4—212no—
213C2×D4—212no—
214C2×D4—26no—
215C2×D4—26no—
216C2×D4C2×C3×Q816yes—
217C2×D4C3×#4116yes—
218C22×C4—23no—
219C22×C4—23no—
220C22×C4—23no—
221C22×C4—23no—
222C22×C4—23no—
223C22×C4—26no—
224C22×C4—26no—
225C22×C4—42no—
226C22×C4—46no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4C2×C3×D413yes—
230C22×C4C2×C3×D413yes—
231C22×C4C2×C3×D413yes—
232C22×C4C2×C3×D413yes—
233C22×C4C2×C3×D413yes—
234C4⋊C4—23no—
235C4⋊C4—23no—
236C4⋊C4—26no—
237C4⋊C4—46no—
238C4⋊C4—43no—
239C4⋊C4—41no—
240C4⋊C4—86no—
241C22⋊C4—26no—
242C22⋊C4—26no—
243C22⋊C4—26no—
244C22⋊C4—212no—
245C22⋊C4—86no—
246C42—26no—
247C42—23no—
248C42—23no—
249C22×C3—23no—
250C22×C3—23no—
251C22×C3—26no—
252C22×C3—41no—
253C22×C3—43no—
254C22×C3—46no—
255C22×C3—43no—
256C22×C3—43no—
257C22×C3—83no—
258C22×C3—84no—
259C22×C3C2×#12011yescharacteristic
260C22×C3C2×C22⋊Q813yes—
261C22×C3#52513yes—
262C3×C4—26no—
263C3×C4—23no—
264C3×C4—43no—
265C3×C4—43no—
266C3×C4—86no—
267C3×C4—82no—
268C3×C4#45812yes—
269C23—24no—
270C23—23no—
271C23—23no—
272C23—41no—
273C23—41no—
274C23—43no—
275C23—43no—
276C23—81no—
277C23—81no—
278C23C3×C22⋊Q816yes—
279C23C22×C3×D413yes—
280C23C22×C3×Q811yescharacteristic
281C23C2×C3×#4113yes—
282Q8—24no—
283D4—212no—
284D4—212no—
285D4—212no—
286C2×C4—212no—
287C2×C4—23no—
288C2×C4—23no—
289C2×C4—26no—
290C2×C4—23no—
291C2×C4—23no—
292C2×C4—43no—
293C2×C4—46no—
294C2×C4—43no—
295C2×C4—43no—
296C2×C4—46no—
297C2×C4—46no—
298C2×C4—46no—
299C2×C4—46no—
300C2×C4—42no—
301C2×C4—46no—
302C2×C4—83no—
303C2×C4—83no—
304C2×C4—83no—
305C2×C4C3×#12013yes—
306C2×C4C3×#12011yescharacteristic
307C2×C4C3×C22⋊Q816yes—
308C2×C3—23no—
309C2×C3—23no—
310C2×C3—46no—
311C2×C3#284811yescharacteristic
312C2×C3C2×#45811yescharacteristic
313C2×C3#302711yescharacteristic
314C22—23no—
315C22—23no—
316C22—26no—
317C22—41no—
318C22—43no—
319C22—46no—
320C22—43no—
321C22—43no—
322C22—83no—
323C22—84no—
324C22C2×C3×#12011yescharacteristic
325C22C2×C3×C22⋊Q813yes—
326C22C3×#52513yes—
327C4—23no—
328C4—26no—
329C4—43no—
330C4—43no—
331C4—86no—
332C4—82no—
333C4C3×#45812yes—
334C3#3297611yescharacteristic, direct product factor, Sylow
335C2—23no—
336C2—23no—
337C2—46no—
338C2C3×#284811yescharacteristic
339C2C2×C3×#45811yescharacteristic
340C2C3×#302711yescharacteristic
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
2. C3×#2487C2semidirect
3. C3×#1867C2semidirect
4. C3×#1799C2semidirect
6. #32976C3direct (N × H)
14. C22×C3×C4⋊C4C22semidirect
20. C3×#385C22semidirect
22. C3×#380C22semidirect
23. C3×#342C22semidirect
25. #2487C2×C3semidirect
26. #1867C2×C3semidirect
27. #1799C2×C3semidirect
74. C22×C4⋊C4C22×C3semidirect
80. #385C22×C3semidirect
82. #380C22×C3semidirect
83. #342C22×C3semidirect
334. C3#32976direct (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
C3×#24872
C3×#18672
C3×#17992
C3×#17902
#329763