Finite Group Databasebeta

← C3×#29272 #10647879

Subgroups — C3×#29272 #10647879

Order 768.

736 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C4⋊Q8C211yesmaximal, characteristic
2C3×C4×C4⋊Q8C211yesmaximal, characteristic
3C3×#2099C212yesmaximal
4C3×#2057C212yesmaximal
5C3×#1988C212yesmaximal
6C3×#1947C212yesmaximal
7C3×#1918C214yesmaximal
8C3×#1831C211yesmaximal, characteristic
9#29272C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4×Q8—21no—
13C2×C3×C4×Q8—21no—
14C2×C3×#107C2214yes—
15C2×C3×C4⋊C8C2212yes—
16C3×#391—21no—
17C3×#391—21no—
18C3×#389—21no—
19C3×#389—21no—
20C3×#389C2212yes—
21C3×#389C2212yes—
22C3×#384C2211yescharacteristic
23C3×#384C2211yescharacteristic
24C3×C4×C4⋊C4—21no—
25C3×C4×C4⋊C4—21no—
26C3×C4×C4⋊C4C2212yes—
27C3×C4×C4⋊C4C2212yes—
28C3×C43C2211yescharacteristic
29C3×#340C2214yes—
30C3×#340C2214yes—
31C3×#340C2214yes—
32C3×#336C2212yes—
33C3×#336C2212yes—
34C4×C4⋊Q8C2×C311yescharacteristic
35C4×C4⋊Q8C2×C311yescharacteristic
36#2099C2×C312yes—
37#2057C2×C312yes—
38#1988C2×C312yes—
39#1947C2×C312yes—
40#1918C2×C314yes—
41#1831C2×C311yescharacteristic
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×C8C2314yes—
45C3×C4⋊Q8—24no—
46C3×C4⋊Q8—22no—
47C3×C4⋊Q8Q814yescomplemented
48C3×C4×Q8—44no—
49C3×C4×Q8—44no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C42—22no—
63C2×C3×C42—22no—
64C2×C3×C42D411yescharacteristic
65C2×C3×C42D411yescharacteristic
66C2×C3×C42D411yescharacteristic
67C2×C3×C42D411yescharacteristic
68C2×C3×C42C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C3×#107—28no—
72C3×C4⋊C8—24no—
73C3×#95—22no—
74C3×#95—22no—
75C3×#95—22no—
76C3×#95—22no—
77C2×C4⋊Q8C22×C311yescharacteristic
78C2×C4⋊Q8C22×C311yescharacteristic
79C2×C4×Q8—21no—
80C2×C4×Q8—21no—
81C2×#107C22×C314yes—
82C2×C4⋊C8C22×C312yes—
83#391—21no—
84#391—21no—
85#389—21no—
86#389—21no—
87#389C22×C312yes—
88#389C22×C312yes—
89#384C22×C311yescharacteristic
90#384C22×C311yescharacteristic
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4C22×C312yes—
94C4×C4⋊C4C22×C312yes—
95C43C22×C311yescharacteristic
96#340C22×C314yes—
97#340C22×C314yes—
98#340C22×C314yes—
99#336C22×C312yes—
100#336C22×C312yes—
101C2×C3×Q8—24no—
102C2×C3×Q8—24no—
103C2×C3×Q8—42no—
104C2×C3×Q8—42no—
105C2×C3×Q8—42no—
106C2×C3×Q8—42no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2411yescharacteristic
118C2×C3×C8—24no—
119C2×C3×C8—24no—
120C2×C3×C8—24no—
121C3×C4⋊C4—24no—
122C3×C4⋊C4—24no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—24no—
126C3×C4⋊C4—24no—
127C3×C4⋊C4—42no—
128C3×C4⋊C4—44no—
129C3×C4⋊C4—44no—
130C3×C4⋊C4—42no—
131C3×C4⋊C4—42no—
132C3×C4⋊C4—42no—
133C3×C4⋊C4C2×Q814yes—
134C3×C4⋊C4#4114yes—
135C3×C42—22no—
136C3×C42—22no—
137C3×C42—22no—
138C3×C42—22no—
139C3×C42—41no—
140C3×C42—44no—
141C3×C42—44no—
142C3×C42D814yes—
143C3×C42C2×Q812yes—
144C3×C42#4112yes—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×C8C23×C314yes—
148C4⋊Q8—24no—
149C4⋊Q8—22no—
150C4⋊Q8C3×Q814yescomplemented
151C4×Q8—44no—
152C4×Q8—44no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4C23×C312yes—
162C2×C4⋊C4C23×C312yes—
163C2×C4⋊C4C23×C312yes—
164C2×C4⋊C4C23×C312yes—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42C3×D411yescharacteristic
168C2×C42C3×D411yescharacteristic
169C2×C42C3×D411yescharacteristic
170C2×C42C3×D411yescharacteristic
171C2×C42C23×C311yescharacteristic
172C2×C42C23×C311yescharacteristic
173C2×C42C23×C311yescharacteristic
174#107—28no—
175C4⋊C8—24no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180C23×C3C22×D411yescharacteristic
181C3×Q8—48no—
182C3×Q8—48no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—24no—
192C2×C3×C4—24no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4C22⋊Q812yes—
207C2×C3×C4C22⋊Q812yes—
208C2×C3×C4C2×D812yes—
209C2×C3×C4C2×D812yes—
210C2×C3×C4C2×D812yes—
211C2×C3×C4#13712yes—
212C2×C3×C4#13712yes—
213C2×C3×C4C22×Q811yescharacteristic
214C2×C3×C4C2×#4111yescharacteristic
215C2×C3×C4C2×#4112yes—
216C2×C3×C42−1+412yes—
217C3×C8—44no—
218C2×Q8—24no—
219C2×Q8—24no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4C2×C3×D411yescharacteristic
229C22×C4C2×C3×D411yescharacteristic
230C22×C4C2×C3×D411yescharacteristic
231C22×C4C2×C3×D411yescharacteristic
232C22×C4C2×C3×D411yescharacteristic
233C22×C4C2×C3×D411yescharacteristic
234C22×C4C24×C311yescharacteristic
235C2×C8—24no—
236C2×C8—24no—
237C2×C8—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4C2×C3×Q814yes—
251C4⋊C4C3×#4114yes—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—41no—
257C42—44no—
258C42—44no—
259C42C3×D814yes—
260C42C2×C3×Q812yes—
261C42C3×#4112yes—
262C22×C3C2×C22⋊Q811yescharacteristic
263C22×C3#52511yescharacteristic
264C22×C3#53611yescharacteristic
265C22×C3C22×D811yescharacteristic
266C22×C3C2×#13711yescharacteristic
267C22×C3C2×#13711yescharacteristic
268C22×C3#57511yescharacteristic
269C3×C4—22no—
270C3×C4—22no—
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—44no—
274C3×C4—41no—
275C3×C4—44no—
276C3×C4—82no—
277C3×C4—82no—
278C3×C4#47414yes—
279C3×C4#55612yes—
280C3×C4#55712yes—
281C23C22×C3×D411yescharacteristic
282Q8—48no—
283Q8—48no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4C3×C22⋊Q812yes—
308C2×C4C3×C22⋊Q812yes—
309C2×C4C2×C3×D812yes—
310C2×C4C2×C3×D812yes—
311C2×C4C2×C3×D812yes—
312C2×C4C3×#13712yes—
313C2×C4C3×#13712yes—
314C2×C4C22×C3×Q811yescharacteristic
315C2×C4C2×C3×#4111yescharacteristic
316C2×C4C2×C3×#4112yes—
317C2×C4C3×2−1+412yes—
318C8—44no—
319C2×C3#258111yescharacteristic
320C2×C3C2×#47411yescharacteristic
321C2×C3#307811yescharacteristic
322C2×C3#310011yescharacteristic
323C2×C3#311311yescharacteristic
324C2×C3#322611yescharacteristic
325C2×C3#323711yescharacteristic
326C22C2×C3×C22⋊Q811yescharacteristic
327C22C3×#52511yescharacteristic
328C22C3×#53611yescharacteristic
329C22C22×C3×D811yescharacteristic
330C22C2×C3×#13711yescharacteristic
331C22C2×C3×#13711yescharacteristic
332C22C3×#57511yescharacteristic
333C4—22no—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—41no—
338C4—44no—
339C4—44no—
340C4—82no—
341C4—82no—
342C4C3×#47414yes—
343C4C3×#55612yes—
344C4C3×#55712yes—
345C3#2927211yescharacteristic, direct product factor, Sylow
346C2C3×#258111yescharacteristic
347C2C2×C3×#47411yescharacteristic
348C2C3×#307811yescharacteristic
349C2C3×#310011yescharacteristic
350C2C3×#311311yescharacteristic
351C2C3×#322611yescharacteristic
352C2C3×#323711yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C43 ◁ G

Complements

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

NH ≅ G/Ntype
9. #29272C3direct (N × H)
47. C3×C4⋊Q8Q8semidirect
150. C4⋊Q8C3×Q8semidirect
345. C3#29272direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊Q82
C3×#20992
C3×#20572
C3×#19882
C3×#19472
C3×#19182
C3×#18312
#292723