Finite Group Databasebeta

← C3×#12232 #10630839

Subgroups — C3×#12232 #10630839

Order 768.

360 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2834C211yesmaximal, characteristic
2C3×#2062C211yesmaximal, characteristic
3C3×#1614C211yesmaximal, characteristic
4C3×#1608C211yesmaximal, characteristic
5C3×#1352C211yesmaximal, characteristic
6C3×#1344C211yesmaximal, characteristic
7C3×#1326C211yesmaximal, characteristic
8#12232C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#543C2211yescharacteristic
10C3×#484—21no—
11C3×#483—21no—
12C3×#401—21no—
13C3×#399C2211yescharacteristic
14C3×#398—21no—
15C3×#398—21no—
16C3×#398—21no—
17C3×#397—21no—
18C3×#397C2211yescharacteristic
19C3×#385—21no—
20C3×#385—21no—
21C3×#385C2211yescharacteristic
22C3×#342—21no—
23C3×#342C2211yescharacteristic
24C3×#328—21no—
25C3×#328—21no—
26C3×#323C2211yescharacteristic
27C3×#323C2211yescharacteristic
28#2834C2×C311yescharacteristic
29#2062C2×C311yescharacteristic
30#1614C2×C311yescharacteristic
31#1608C2×C311yescharacteristic
32#1352C2×C311yescharacteristic
33#1344C2×C311yescharacteristic
34#1326C2×C311yescharacteristic
35C23×C3×C4D411yescharacteristic
36C3×C22⋊Q8—21no—
37C3×C22⋊Q8—21no—
38C3×C22⋊Q8—41no—
39C3×C22⋊Q8—41no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—41no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4D411yescharacteristic
53C2×C3×C22⋊C4C2311yescharacteristic
54C3×#107—41no—
55C3×#106—41no—
56C3×#103—41no—
57C3×#103—41no—
58C3×C22⋊C8—21no—
59C3×C22⋊C8—21no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95—21no—
63C3×#95—21no—
64C3×#95—21no—
65C3×#95—21no—
66C3×#95—21no—
67C3×#95—41no—
68C3×#95D411yescharacteristic
69C3×#95D411yescharacteristic
70#543C22×C311yescharacteristic
71#484—21no—
72#483—21no—
73#401—21no—
74#399C22×C311yescharacteristic
75#398—21no—
76#398—21no—
77#398—21no—
78#397—21no—
79#397C22×C311yescharacteristic
80#385—21no—
81#385—21no—
82#385C22×C311yescharacteristic
83#342—21no—
84#342C22×C311yescharacteristic
85#328—21no—
86#328—21no—
87#323C22×C311yescharacteristic
88#323C22×C311yescharacteristic
89C24×C3C2×D411yescharacteristic
90C2×C3×Q8—21no—
91C2×C3×Q8—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—41no—
101C22×C3×C4—41no—
102C22×C3×C4—81no—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C2×C3×C8—41no—
106C2×C3×C8—41no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—41no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—41no—
112C3×C4⋊C4—41no—
113C3×C4⋊C4—81no—
114C3×C4⋊C4—81no—
115C3×C4⋊C4—81no—
116C3×C4⋊C4—81no—
117C3×C4⋊C4—81no—
118C3×C4⋊C4—82no—
119C3×C22⋊C4—22no—
120C3×C22⋊C4—21no—
121C3×C22⋊C4—21no—
122C3×C22⋊C4—41no—
123C3×C22⋊C4—41no—
124C3×C22⋊C4—41no—
125C3×C22⋊C4—41no—
126C3×C22⋊C4—41no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—81no—
129C23×C4C3×D411yescharacteristic
130C22⋊Q8—21no—
131C22⋊Q8—21no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4C3×D411yescharacteristic
141C2×C4⋊C4C3×D411yescharacteristic
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4C3×D411yescharacteristic
147C2×C22⋊C4C23×C311yescharacteristic
148#107—41no—
149#106—41no—
150#103—41no—
151#103—41no—
152C22⋊C8—21no—
153C22⋊C8—21no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95—41no—
162#95C3×D411yescharacteristic
163#95C3×D411yescharacteristic
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—41no—
167C23×C3—41no—
168C23×C3#12011yescharacteristic
169C23×C3#13511yescharacteristic
170C23×C3#13711yescharacteristic
171C3×Q8—81no—
172C3×Q8—81no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—42no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—42no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—81no—
200C2×C3×C4—81no—
201C2×C3×C4—81no—
202C2×C3×C4—81no—
203C2×C3×C4—81no—
204C3×C8—81no—
205C3×C8—81no—
206C24C2×C3×D411yescharacteristic
207C2×Q8—21no—
208C2×Q8—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—81no—
220C22×C4C2×C3×D411yescharacteristic
221C22×C4C2×C3×D411yescharacteristic
222C2×C8—41no—
223C2×C8—41no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—81no—
231C4⋊C4—81no—
232C4⋊C4—81no—
233C4⋊C4—81no—
234C4⋊C4—81no—
235C4⋊C4—82no—
236C22⋊C4—22no—
237C22⋊C4—21no—
238C22⋊C4—21no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—81no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—81no—
255C22×C3#45211yescharacteristic
256C22×C3#45611yescharacteristic
257C22×C3B(4,𝔽2)11yescharacteristic
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—81no—
261C3×C4—81no—
262C3×C4—81no—
263C3×C4—82no—
264C3×C4—81no—
265C3×C4—81no—
266C3×C4—81no—
267C3×C4—81no—
268C3×C4—81no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23C3×#12011yescharacteristic
274C23C3×#13511yescharacteristic
275C23C3×#13711yescharacteristic
276Q8—81no—
277Q8—81no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C8—81no—
310C8—81no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3#160311yescharacteristic
316C2×C3#220011yescharacteristic
317C2×C3#220111yescharacteristic
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22C3×#45211yescharacteristic
328C22C3×#45611yescharacteristic
329C22C3×B(4,𝔽2)11yescharacteristic
330C4—41no—
331C4—41no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4—82no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—81no—
341C3#1223211yescharacteristic, direct product factor, Sylow
342C2—21no—
343C2—21no—
344C2—41no—
345C2—41no—
346C2C3×#160311yescharacteristic
347C2C3×#220011yescharacteristic
348C2C3×#220111yescharacteristic
subgroupG/H
center, socleC22×C3B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C3
prime radicalC24×C3C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C2×C3×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
8. #12232C3direct (N × H)
341. C3#12232direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28342
C3×#20622
C3×#16142
C3×#16082
C3×#13522
C3×#13442
C3×#13262
#122323