Finite Group Databasebeta

← C3×#33632 #10652239

Subgroups — C3×#33632 #10652239

Order 768.

520 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3164C211yesmaximal, characteristic
2C3×#2950C211yesmaximal, characteristic
3C3×#2229C212yesmaximal, complemented
4C3×#2229C212yesmaximal, complemented
5C3×#2227C212yesmaximal, complemented
6C3×#2225C212yesmaximal, complemented
7C3×#2159C211yesmaximal, characteristic
8C3×#2158C211yesmaximal, characteristic
9C3×#2151C211yesmaximal, characteristic
10C3×#2141C211yesmaximal, characteristic
11C2×C3×#358C211yesmaximal, characteristic
12#33632C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2211yescharacteristic
14C3×#544—21no—
15C3×#519—21no—
16C2×C3×M32C2211yescharacteristic
17C22×C3×C16C2211yescharacteristic
18C3×#501—22no—
19C3×#500C2212yes—
20C3×#499C2212yes—
21C3×#441—22no—
22C3×C4×Q16C2212yes—
23C3×C4×Q16C2212yes—
24C3×#428—21no—
25C3×#427C2211yescharacteristic
26C3×#427C2211yescharacteristic
27C2×C3×#107C2211yescharacteristic
28C3×#419—21no—
29C3×#405C2211yescharacteristic
30C3×#367C2211yescharacteristic
31C3×#367C2211yescharacteristic
32C3×#367C2212yes—
33C3×#366C2211yescharacteristic
34C3×#366C2211yescharacteristic
35C3×#366C2212yes—
36C3×C4⋊C16C2212yes—
37C3×C4⋊C16C2212yes—
38C3×#358C2212yes—
39C3×#358C2212yes—
40C3×#358C2212yes—
41C3×#358C2212yes—
42#3164C2×C311yescharacteristic
43#2950C2×C311yescharacteristic
44#2229C2×C312yescomplemented
45#2229C2×C312yescomplemented
46#2227C2×C312yescomplemented
47#2225C2×C312yescomplemented
48#2159C2×C311yescharacteristic
49#2158C2×C311yescharacteristic
50#2151C2×C311yescharacteristic
51#2141C2×C311yescharacteristic
52C2×#358C2×C311yescharacteristic
53C22×C3×Q8—21no—
54C2×C3×Q16—21no—
55C2×C3×Q16—22no—
56C2×C3×Q16Q814yescomplemented
57C2×C3×Q16C2312yes—
58C2×C3×M16D411yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8—41no—
62C3×#125—22no—
63C3×#125—41no—
64C3×C22⋊Q8—42no—
65C3×C4×Q8—22no—
66C3×C4×Q8—22no—
67C3×C4×Q8—41no—
68C3×C4×Q8—41no—
69C3×#117—21no—
70C3×#117—21no—
71C3×#117—42no—
72C3×#117D411yescharacteristic
73C2×C3×C4⋊C4—21no—
74C3×M32—22no—
75C2×C3×C16—22no—
76C2×C3×C16C2311yescharacteristic
77C2×C3×C16C2311yescharacteristic
78C2×C3×C16C2312yes—
79C3×#107—21no—
80C3×#107—22no—
81C3×#107C2311yescharacteristic
82C3×#107C2311yescharacteristic
83C3×#107C2312yes—
84C3×#107C2311yescharacteristic
85C3×#107C2311yescharacteristic
86C3×#106—21no—
87C3×#106—21no—
88C3×#106—22no—
89C3×#103—22no—
90C3×#103—22no—
91C3×#97D412yes—
92C3×C4×C8C2312yes—
93C22×Q16C22×C311yescharacteristic
94#544—21no—
95#519—21no—
96C2×M32C22×C311yescharacteristic
97C22×C16C22×C311yescharacteristic
98#501—22no—
99#500C22×C312yes—
100#499C22×C312yes—
101#441—22no—
102C4×Q16C22×C312yes—
103C4×Q16C22×C312yes—
104#428—21no—
105#427C22×C311yescharacteristic
106#427C22×C311yescharacteristic
107C2×#107C22×C311yescharacteristic
108#419—21no—
109#405C22×C311yescharacteristic
110#367C22×C311yescharacteristic
111#367C22×C311yescharacteristic
112#367C22×C312yes—
113#366C22×C311yescharacteristic
114#366C22×C311yescharacteristic
115#366C22×C312yes—
116C4⋊C16C22×C312yes—
117C4⋊C16C22×C312yes—
118#358C22×C312yes—
119#358C22×C312yes—
120#358C22×C312yes—
121#358C22×C312yes—
122C2×C3×Q8—24no—
123C2×C3×Q8—22no—
124C2×C3×Q8—41no—
125C2×C3×Q8—42no—
126C2×C3×Q8—42no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4C2×D411yescharacteristic
130C3×Q16—24no—
131C3×Q16—22no—
132C3×Q16C2×Q814yes—
133C3×M16—22no—
134C2×C3×C8C2×D412yes—
135C2×C3×C8C2×D411yescharacteristic
136C2×C3×C8C2×Q812yes—
137C2×C3×C8#4112yes—
138C2×C3×C8C2411yescharacteristic
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—42no—
146C3×C4⋊C4—42no—
147C3×C4⋊C4—42no—
148C3×C4⋊C4—42no—
149C3×C4⋊C4—42no—
150C3×C4⋊C4D811yescharacteristic
151C3×C4⋊C4D811yescharacteristic
152C3×C22⋊C4—41no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4D812yes—
155C3×C42—41no—
156C3×C42—41no—
157C3×C42—42no—
158C3×C42—42no—
159C3×C42C2×D412yes—
160C3×C16—21no—
161C3×C16—21no—
162C3×C16—22no—
163C22×Q8—21no—
164C2×Q16—21no—
165C2×Q16—22no—
166C2×Q16C3×Q814yescomplemented
167C2×Q16C23×C312yes—
168C2×M16C3×D411yescharacteristic
169C22×C8C23×C311yescharacteristic
170C4⋊Q8—22no—
171C4⋊Q8—41no—
172#125—22no—
173#125—41no—
174C22⋊Q8—42no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—41no—
178C4×Q8—41no—
179#117—21no—
180#117—21no—
181#117—42no—
182#117C3×D411yescharacteristic
183C2×C4⋊C4—21no—
184M32—22no—
185C2×C16—22no—
186C2×C16C23×C311yescharacteristic
187C2×C16C23×C312yes—
188C2×C16C23×C311yescharacteristic
189#107—21no—
190#107—22no—
191#107C23×C312yes—
192#107C23×C311yescharacteristic
193#107C23×C311yescharacteristic
194#107C23×C311yescharacteristic
195#107C23×C311yescharacteristic
196#106—21no—
197#106—21no—
198#106—22no—
199#103—22no—
200#103—22no—
201#97C3×D412yes—
202C4×C8C23×C312yes—
203C23×C3C2×D811yescharacteristic
204C3×Q8—24no—
205C3×Q8—42no—
206C3×Q8—44no—
207C3×Q8—162no—
208C2×C3×C4—42no—
209C2×C3×C4—44no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—161no—
216C2×C3×C4C22⋊Q812yes—
217C2×C3×C4C2×D812yes—
218C2×C3×C4C2×D812yes—
219C2×C3×C4C2×D811yescharacteristic
220C2×C3×C4#13712yes—
221C2×C3×C4C22×D411yescharacteristic
222C3×C8—22no—
223C3×C8C22⋊Q812yes—
224C3×C8C22×Q811yescharacteristic
225C3×C8C2×#4111yescharacteristic
226C2×Q8—24no—
227C2×Q8—22no—
228C2×Q8—41no—
229C2×Q8—42no—
230C2×Q8—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4C2×C3×D411yescharacteristic
234Q16—24no—
235Q16—22no—
236Q16C2×C3×Q814yes—
237M16—22no—
238C2×C8C2×C3×D412yes—
239C2×C8C2×C3×D411yescharacteristic
240C2×C8C2×C3×Q812yes—
241C2×C8C3×#4112yes—
242C2×C8C24×C311yescharacteristic
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—22no—
247C4⋊C4—21no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4C3×D811yescharacteristic
255C4⋊C4C3×D811yescharacteristic
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4C3×D812yes—
259C42—41no—
260C42—41no—
261C42—42no—
262C42—42no—
263C42C2×C3×D412yes—
264C16—21no—
265C16—21no—
266C16—22no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3#47412yes—
270C22×C3C22×D811yescharacteristic
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—44no—
274C3×C4—81no—
275C3×C4—82no—
276C3×C4—81no—
277C3×C4—81no—
278C3×C4—81no—
279C3×C4#47412yes—
280C3×C4C2×C22⋊Q811yescharacteristic
281C3×C4C2×#13711yescharacteristic
282C23C2×C3×D811yescharacteristic
283Q8—24no—
284Q8—42no—
285Q8—44no—
286Q8—162no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—161no—
295C2×C4C3×C22⋊Q812yes—
296C2×C4C2×C3×D812yes—
297C2×C4C2×C3×D812yes—
298C2×C4C2×C3×D811yescharacteristic
299C2×C4C3×#13712yes—
300C2×C4C22×C3×D411yescharacteristic
301C8—22no—
302C8C3×C22⋊Q812yes—
303C8C22×C3×Q811yescharacteristic
304C8C2×C3×#4111yescharacteristic
305C2×C3—22no—
306C2×C3C2×#47411yescharacteristic
307C2×C3#341611yescharacteristic
308C2×C3#341711yescharacteristic
309C22—21no—
310C22—21no—
311C22C3×#47412yes—
312C22C22×C3×D811yescharacteristic
313C4—22no—
314C4—22no—
315C4—44no—
316C4—82no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4C3×#47412yes—
322C4C2×C3×C22⋊Q811yescharacteristic
323C4C2×C3×#13711yescharacteristic
324C3#3363211yescharacteristic, direct product factor, Sylow
325C2—22no—
326C2C2×C3×#47411yescharacteristic
327C2C3×#341611yescharacteristic
328C2C3×#341711yescharacteristic
subgroupG/H
center, socleC22×C3C22×D8
commutator, Frattini, cosocleC2×C8C24×C3
prime radicalC23×C3C2×D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×#117 ◁ C3×#405 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2229C2semidirect
4. C3×#2229C2semidirect
5. C3×#2227C2semidirect
6. C3×#2225C2semidirect
12. #33632C3direct (N × H)
44. #2229C2×C3semidirect
45. #2229C2×C3semidirect
46. #2227C2×C3semidirect
47. #2225C2×C3semidirect
56. C2×C3×Q16Q8semidirect
166. C2×Q16C3×Q8semidirect
324. C3#33632direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#31642
C3×#29502
C3×#22292
C3×#22272
C3×#22252
C3×#21592
C3×#21582
C3×#21512
C3×#21412
C2×C3×#3582
#336323