Finite Group Databasebeta

← C3×#32975 #10651582

Subgroups — C3×#32975 #10651582

Order 768.

1550 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 1550 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×#2409C213yesmaximal, complemented
3C3×#1867C216yesmaximal, complemented
4C3×#1799C213yesmaximal, complemented
5C3×#1790C212yesmaximal
6#32975C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×H(5,𝔽2)C2211yescharacteristic
8C23×C3×Q8C2211yescharacteristic, complemented
9C2×C3×C22⋊Q8—23no—
10C3×#520—23no—
11C3×#518—23no—
12C3×#518C2213yes—
13C3×C4×#41—23no—
14C2×C3×C4×D4—23no—
15C2×C3×C4×D4C2213yes—
16C2×C3×#99C2216yes—
17C3×#395—23no—
18C3×#393—23no—
19C3×#391—23no—
20C3×#386C2216yes—
21C3×#385C2216yescomplemented
22C3×#380C2213yescomplemented
23C3×#342C2216yescomplemented
24C4×H(5,𝔽2)C2×C311yescharacteristic
25#2409C2×C313yescomplemented
26#1867C2×C316yescomplemented
27#1799C2×C313yescomplemented
28#1790C2×C312yes—
29C3×H(5,𝔽2)—26no—
30C3×H(5,𝔽2)Q814yescomplemented
31C2×C3×#41—23no—
32C22×C3×Q8—23no—
33C22×C3×Q8—23no—
34C22×C3×Q8—44no—
35C22×C3×D4—23no—
36C22×C3×D4C2313yes—
37C23×C3×C4—21no—
38C23×C3×C4D413yes—
39C23×C3×C4C2312yes—
40C23×C3×C4C2311yescharacteristic
41C3×C22⋊Q8—412no—
42C3×C4×Q8—21no—
43C3×C4×Q8—23no—
44C3×C4×D4—23no—
45C3×C4×D4—26no—
46C3×C4×D4—23no—
47C3×C4×D4—26no—
48C3×C4×D4—26no—
49C3×C4×D4—26no—
50C3×C4×D4—26no—
51C3×#117—26no—
52C3×#117—23no—
53C2×C3×C4⋊C4—23no—
54C2×C3×C4⋊C4—23no—
55C2×C3×C4⋊C4—26no—
56C2×C3×C4⋊C4D413yes—
57C2×C3×C22⋊C4—23no—
58C2×C3×C22⋊C4—23no—
59C2×C3×C22⋊C4—26no—
60C2×C3×C22⋊C4D413yes—
61C2×C3×C22⋊C4C2316yes—
62C2×C3×C22⋊C4C2313yes—
63C2×C3×C42—23no—
64C2×C3×C42D413yes—
65C3×#99—212no—
66C3×#95—23no—
67C3×#95—26no—
68C3×#95—26no—
69C2×H(5,𝔽2)C22×C311yescharacteristic
70C23×Q8C22×C311yescharacteristic, complemented
71C2×C22⋊Q8—23no—
72#520—23no—
73#518—23no—
74#518C22×C313yes—
75C4×#41—23no—
76C2×C4×D4—23no—
77C2×C4×D4C22×C313yes—
78C2×#99C22×C316yes—
79#395—23no—
80#393—23no—
81#391—23no—
82#386C22×C316yes—
83#385C22×C316yescomplemented
84#380C22×C313yescomplemented
85#342C22×C316yescomplemented
86C24×C3—21no—
87C24×C3C2×D413yes—
88C24×C3C2411yescharacteristic
89C3×#41—212no—
90C3×#41—212no—
91C2×C3×Q8—21no—
92C2×C3×Q8—212no—
93C2×C3×Q8—46no—
94C2×C3×Q8—412no—
95C2×C3×Q8—43no—
96C2×C3×Q8—41no—
97C2×C3×D4—23no—
98C2×C3×D4—212no—
99C2×C3×D4—212no—
100C2×C3×D4—26no—
101C2×C3×D4—26no—
102C2×C3×D4C2×Q816yes—
103C2×C3×D4#4116yes—
104C22×C3×C4—23no—
105C22×C3×C4—26no—
106C22×C3×C4—23no—
107C22×C3×C4—23no—
108C22×C3×C4—23no—
109C22×C3×C4—23no—
110C22×C3×C4—26no—
111C22×C3×C4—44no—
112C22×C3×C4—42no—
113C22×C3×C4—46no—
114C22×C3×C4—82no—
115C22×C3×C4C2×D413yes—
116C22×C3×C4C2×D413yes—
117C22×C3×C4C2×D413yes—
118C22×C3×C4C2×D413yes—
119C22×C3×C4C2×D413yes—
120C3×C4⋊C4—23no—
121C3×C4⋊C4—23no—
122C3×C4⋊C4—26no—
123C3×C4⋊C4—46no—
124C3×C4⋊C4—412no—
125C3×C22⋊C4—26no—
126C3×C22⋊C4—26no—
127C3×C22⋊C4—26no—
128C3×C22⋊C4—212no—
129C3×C22⋊C4—412no—
130C3×C42—26no—
131C3×C42—23no—
132C3×C42—23no—
133H(5,𝔽2)—26no—
134H(5,𝔽2)C3×Q814yescomplemented
135C2×#41—23no—
136C22×Q8—23no—
137C22×Q8—23no—
138C22×Q8—44no—
139C22×D4—23no—
140C22×D4C23×C313yes—
141C23×C4—21no—
142C23×C4C3×D413yes—
143C23×C4C23×C312yes—
144C23×C4C23×C311yescharacteristic
145C22⋊Q8—412no—
146C4×Q8—21no—
147C4×Q8—23no—
148C4×D4—23no—
149C4×D4—26no—
150C4×D4—23no—
151C4×D4—26no—
152C4×D4—26no—
153C4×D4—26no—
154C4×D4—26no—
155#117—26no—
156#117—23no—
157C2×C4⋊C4—23no—
158C2×C4⋊C4—23no—
159C2×C4⋊C4—26no—
160C2×C4⋊C4C3×D413yes—
161C2×C22⋊C4—23no—
162C2×C22⋊C4—23no—
163C2×C22⋊C4—26no—
164C2×C22⋊C4C3×D413yes—
165C2×C22⋊C4C23×C316yes—
166C2×C22⋊C4C23×C313yes—
167C2×C42—23no—
168C2×C42C3×D413yes—
169#99—212no—
170#95—23no—
171#95—26no—
172#95—26no—
173C23×C3—24no—
174C23×C3—23no—
175C23×C3—23no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3—43no—
179C23×C3—43no—
180C23×C3—81no—
181C23×C3—81no—
182C23×C3C22⋊Q816yescomplemented
183C23×C3C22×D413yes—
184C23×C3C22×Q811yescharacteristic
185C23×C3C2×#4113yes—
186C3×Q8—24no—
187C3×Q8—412no—
188C3×Q8—44no—
189C3×D4—212no—
190C3×D4—212no—
191C3×D4—212no—
192C2×C3×C4—212no—
193C2×C3×C4—23no—
194C2×C3×C4—23no—
195C2×C3×C4—26no—
196C2×C3×C4—212no—
197C2×C3×C4—23no—
198C2×C3×C4—23no—
199C2×C3×C4—43no—
200C2×C3×C4—46no—
201C2×C3×C4—42no—
202C2×C3×C4—46no—
203C2×C3×C4—43no—
204C2×C3×C4—43no—
205C2×C3×C4—46no—
206C2×C3×C4—46no—
207C2×C3×C4—46no—
208C2×C3×C4—83no—
209C2×C3×C4—83no—
210C2×C3×C4—83no—
211C2×C3×C4#12013yes—
212C2×C3×C4#12011yescharacteristic
213C2×C3×C4C22⋊Q816yescomplemented
214C24—21no—
215C24C2×C3×D413yes—
216C24C24×C311yescharacteristic
217#41—212no—
218#41—212no—
219C2×Q8—21no—
220C2×Q8—212no—
221C2×Q8—46no—
222C2×Q8—412no—
223C2×Q8—43no—
224C2×Q8—41no—
225C2×D4—23no—
226C2×D4—212no—
227C2×D4—212no—
228C2×D4—26no—
229C2×D4—26no—
230C2×D4C2×C3×Q816yes—
231C2×D4C3×#4116yes—
232C22×C4—23no—
233C22×C4—26no—
234C22×C4—23no—
235C22×C4—23no—
236C22×C4—23no—
237C22×C4—23no—
238C22×C4—26no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—46no—
242C22×C4—82no—
243C22×C4C2×C3×D413yes—
244C22×C4C2×C3×D413yes—
245C22×C4C2×C3×D413yes—
246C22×C4C2×C3×D413yes—
247C22×C4C2×C3×D413yes—
248C4⋊C4—23no—
249C4⋊C4—23no—
250C4⋊C4—26no—
251C4⋊C4—46no—
252C4⋊C4—412no—
253C22⋊C4—26no—
254C22⋊C4—26no—
255C22⋊C4—26no—
256C22⋊C4—212no—
257C22⋊C4—412no—
258C42—26no—
259C42—23no—
260C42—23no—
261C22×C3—23no—
262C22×C3—23no—
263C22×C3—26no—
264C22×C3—41no—
265C22×C3—43no—
266C22×C3—46no—
267C22×C3—43no—
268C22×C3—43no—
269C22×C3—83no—
270C22×C3—84no—
271C22×C3C2×#12011yescharacteristic
272C22×C3C2×C22⋊Q813yes—
273C22×C3#52513yes—
274C3×C4—26no—
275C3×C4—23no—
276C3×C4—44no—
277C3×C4—43no—
278C3×C4—43no—
279C3×C4—86no—
280C3×C4B(4,𝔽2)12yes—
281C23—24no—
282C23—23no—
283C23—23no—
284C23—41no—
285C23—41no—
286C23—43no—
287C23—43no—
288C23—81no—
289C23—81no—
290C23C3×C22⋊Q816yescomplemented
291C23C22×C3×D413yes—
292C23C22×C3×Q811yescharacteristic
293C23C2×C3×#4113yes—
294Q8—24no—
295Q8—412no—
296Q8—44no—
297D4—212no—
298D4—212no—
299D4—212no—
300C2×C4—212no—
301C2×C4—23no—
302C2×C4—23no—
303C2×C4—26no—
304C2×C4—212no—
305C2×C4—23no—
306C2×C4—23no—
307C2×C4—43no—
308C2×C4—46no—
309C2×C4—42no—
310C2×C4—46no—
311C2×C4—43no—
312C2×C4—43no—
313C2×C4—46no—
314C2×C4—46no—
315C2×C4—46no—
316C2×C4—83no—
317C2×C4—83no—
318C2×C4—83no—
319C2×C4C3×#12013yes—
320C2×C4C3×#12011yescharacteristic
321C2×C4C3×C22⋊Q816yescomplemented
322C2×C3—23no—
323C2×C3—23no—
324C2×C3—46no—
325C2×C3#284811yescharacteristic
326C2×C3C2×B(4,𝔽2)11yescharacteristic
327C2×C3#302811yescharacteristic
328C22—23no—
329C22—23no—
330C22—26no—
331C22—41no—
332C22—43no—
333C22—46no—
334C22—43no—
335C22—43no—
336C22—83no—
337C22—84no—
338C22C2×C3×#12011yescharacteristic
339C22C2×C3×C22⋊Q813yes—
340C22C3×#52513yes—
341C4—23no—
342C4—26no—
343C4—44no—
344C4—43no—
345C4—43no—
346C4—86no—
347C4C3×B(4,𝔽2)12yes—
348C3#3297511yescharacteristic, direct product factor, Sylow
349C2—23no—
350C2—23no—
351C2—46no—
352C2C3×#284811yescharacteristic
353C2C2×C3×B(4,𝔽2)11yescharacteristic
354C2C3×#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
2. C3×#2409C2semidirect
3. C3×#1867C2semidirect
4. C3×#1799C2semidirect
6. #32975C3direct (N × H)
8. C23×C3×Q8C22semidirect
21. C3×#385C22semidirect
22. C3×#380C22semidirect
23. C3×#342C22semidirect
25. #2409C2×C3semidirect
26. #1867C2×C3semidirect
27. #1799C2×C3semidirect
30. C3×H(5,𝔽2)Q8semidirect
70. C23×Q8C22×C3semidirect
83. #385C22×C3semidirect
84. #380C22×C3semidirect
85. #342C22×C3semidirect
134. H(5,𝔽2)C3×Q8semidirect
182. C23×C3C22⋊Q8semidirect
213. C2×C3×C4C22⋊Q8semidirect
290. C23C3×C22⋊Q8semidirect
321. C2×C4C3×C22⋊Q8semidirect
348. C3#32975direct (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×#24092
C3×#18672
C3×#17992
C3×#17902
#329753