Finite Group Databasebeta

← C3×#12702 #10631309

Subgroups — C3×#12702 #10631309

Order 768.

962 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×C3×#536C211yesmaximal, characteristic
2C2×C3×#356C212yesmaximal, complemented
3C2×C3×#355C212yesmaximal, complemented
4C3×#1784C212yesmaximal, complemented
5#12702C311yesmaximal, characteristic, direct product factor, Sylow
6C22×C3×#41C411yescharacteristic
7C23×C3×Q8C2211yescharacteristic
8C3×#536—24no—
9C3×#536—24no—
10C2×C3×C4⋊Q8C411yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×#124C411yescharacteristic
13C2×C3×#124C2211yescharacteristic
14C2×C3×#123—22no—
15C2×C3×C22⋊Q8—22no—
16C2×C3×#117C411yescharacteristic
17C2×C3×#101—22no—
18C2×C3×#101C2214yes—
19C3×#407—22no—
20C3×#356—24no—
21C3×#355—24no—
22C2×#536C2×C311yescharacteristic
23C2×#356C2×C312yescomplemented
24C2×#355C2×C312yescomplemented
25#1784C2×C312yescomplemented
26C2×C3×#41—22no—
27C2×C3×#41—22no—
28C2×C3×#41—42no—
29C2×C3×#41—42no—
30C22×C3×Q8—22no—
31C22×C3×Q8—22no—
32C22×C3×Q8—42no—
33C22×C3×Q8—42no—
34C22×C3×Q8C2×C411yescharacteristic
35C22×C3×Q8C2×C411yescharacteristic
36C22×C3×Q8D412yes—
37C22×C3×Q8C2311yescharacteristic
38C22×C3×D4—21no—
39C22×C3×D4C2×C411yescharacteristic
40C23×C3×C4—21no—
41C23×C3×C4C2×C411yescharacteristic
42C23×C3×C4D412yes—
43C2×C3×M16—24no—
44C3×C4⋊Q8—22no—
45C3×C4⋊Q8—22no—
46C3×C4⋊Q8—22no—
47C3×C4⋊Q8D414yes—
48C3×#124—22no—
49C3×#124—22no—
50C3×#124—22no—
51C3×#124D414yes—
52C3×#123—216no—
53C3×C22⋊Q8—216no—
54C3×#117—22no—
55C3×#117—22no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—22no—
62C2×C3×C42C2×C411yescharacteristic
63C2×C3×C42C2×C411yescharacteristic
64C3×#101—24no—
65C3×#101—24no—
66C3×#101—44no—
67C3×C22⋊C8—44no—
68C22×#41C3×C411yescharacteristic
69C23×Q8C22×C311yescharacteristic
70#536—24no—
71#536—24no—
72C2×C4⋊Q8C3×C411yescharacteristic
73C2×C4⋊Q8C22×C311yescharacteristic
74C2×#124C3×C411yescharacteristic
75C2×#124C22×C311yescharacteristic
76C2×#123—22no—
77C2×C22⋊Q8—22no—
78C2×#117C3×C411yescharacteristic
79C2×#101—22no—
80C2×#101C22×C314yes—
81#407—22no—
82#356—24no—
83#355—24no—
84C24×C3—21no—
85C24×C3C22⋊C411yescharacteristic
86C3×#41—48no—
87C3×#41—48no—
88C2×C3×Q8—24no—
89C2×C3×Q8—21no—
90C2×C3×Q8—21no—
91C2×C3×Q8—21no—
92C2×C3×Q8—24no—
93C2×C3×Q8—41no—
94C2×C3×Q8—41no—
95C2×C3×Q8—42no—
96C2×C3×Q8—41no—
97C2×C3×Q8—41no—
98C2×C3×Q8—48no—
99C2×C3×Q8—48no—
100C2×C3×Q8—41no—
101C2×C3×Q8—41no—
102C2×C3×Q8C22⋊C412yes—
103C2×C3×Q8C2×D412yes—
104C2×C3×Q8C2×D412yes—
105C2×C3×D4—24no—
106C2×C3×D4—21no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C2×C3×D4—42no—
110C2×C3×D4—42no—
111C2×C3×D4C22⋊C412yes—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4—24no—
117C22×C3×C4—24no—
118C22×C3×C4—21no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—24no—
122C22×C3×C4—42no—
123C22×C3×C4—42no—
124C22×C3×C4—42no—
125C22×C3×C4C22⋊C412yes—
126C22×C3×C4C22⋊C411yescharacteristic
127C22×C3×C4C22×C411yescharacteristic
128C22×C3×C4C2×D412yes—
129C3×M16—44no—
130C3×M16—44no—
131C2×C3×C8—44no—
132C3×C4⋊C4—24no—
133C3×C4⋊C4—28no—
134C3×C4⋊C4—28no—
135C3×C22⋊C4—24no—
136C3×C22⋊C4—28no—
137C3×C22⋊C4—28no—
138C3×C42—21no—
139C3×C42—21no—
140C3×C42C22⋊C412yes—
141C3×C42C22⋊C412yes—
142C2×#41—22no—
143C2×#41—22no—
144C2×#41—42no—
145C2×#41—42no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8C2×C3×C411yescharacteristic
151C22×Q8C2×C3×C411yescharacteristic
152C22×Q8C3×D412yes—
153C22×Q8C23×C311yescharacteristic
154C22×D4—21no—
155C22×D4C2×C3×C411yescharacteristic
156C23×C4—21no—
157C23×C4C2×C3×C411yescharacteristic
158C23×C4C3×D412yes—
159C2×M16—24no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8C3×D414yes—
164#124—22no—
165#124—22no—
166#124—22no—
167#124C3×D414yes—
168#123—216no—
169C22⋊Q8—216no—
170#117—22no—
171#117—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C42C2×C3×C411yescharacteristic
179C2×C42C2×C3×C411yescharacteristic
180#101—24no—
181#101—24no—
182#101—44no—
183C22⋊C8—44no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—22no—
187C23×C3—42no—
188C23×C3—44no—
189C23×C3—41no—
190C23×C3—41no—
191C23×C3#9912yes—
192C23×C3C2×C22⋊C411yescharacteristic
193C3×Q8—42no—
194C3×Q8—42no—
195C3×Q8—48no—
196C3×Q8—42no—
197C3×Q8—42no—
198C3×Q8—42no—
199C3×Q8—42no—
200C3×D4—42no—
201C3×D4—42no—
202C3×D4—48no—
203C2×C3×C4—24no—
204C2×C3×C4—24no—
205C2×C3×C4—21no—
206C2×C3×C4—22no—
207C2×C3×C4—21no—
208C2×C3×C4—22no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—48no—
218C2×C3×C4—48no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—41no—
222C2×C3×C4—41no—
223C2×C3×C4#9912yes—
224C2×C3×C4C2×C22⋊C411yescharacteristic
225C2×C3×C4C2×C22⋊C411yescharacteristic
226C2×C3×C4#12014yes—
227C3×C8—84no—
228C24—21no—
229C24C3×C22⋊C411yescharacteristic
230#41—48no—
231#41—48no—
232C2×Q8—24no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—24no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—42no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—48no—
243C2×Q8—48no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8C3×C22⋊C412yes—
247C2×Q8C2×C3×D412yes—
248C2×Q8C2×C3×D412yes—
249C2×D4—24no—
250C2×D4—21no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4C3×C22⋊C412yes—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—24no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4C3×C22⋊C412yes—
270C22×C4C3×C22⋊C411yescharacteristic
271C22×C4C22×C3×C411yescharacteristic
272C22×C4C2×C3×D412yes—
273M16—44no—
274M16—44no—
275C2×C8—44no—
276C4⋊C4—24no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C22⋊C4—24no—
280C22⋊C4—28no—
281C22⋊C4—28no—
282C42—21no—
283C42—21no—
284C42C3×C22⋊C412yes—
285C42C3×C22⋊C412yes—
286C22×C3—22no—
287C22×C3—24no—
288C22×C3—22no—
289C22×C3—42no—
290C22×C3—42no—
291C22×C3—44no—
292C22×C3—42no—
293C22×C3—82no—
294C22×C3—82no—
295C22×C3#37911yescharacteristic
296C22×C3C2×#9911yescharacteristic
297C22×C3C2×#9911yescharacteristic
298C3×C4—24no—
299C3×C4—22no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—42no—
303C3×C4—42no—
304C3×C4—42no—
305C3×C4—42no—
306C23—21no—
307C23—21no—
308C23—22no—
309C23—42no—
310C23—44no—
311C23—41no—
312C23—41no—
313C23C3×#9912yes—
314C23C2×C3×C22⋊C411yescharacteristic
315Q8—42no—
316Q8—42no—
317Q8—48no—
318Q8—42no—
319Q8—42no—
320Q8—42no—
321Q8—42no—
322D4—42no—
323D4—42no—
324D4—48no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—48no—
340C2×C4—48no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4C3×#9912yes—
346C2×C4C2×C3×C22⋊C411yescharacteristic
347C2×C4C2×C3×C22⋊C411yescharacteristic
348C2×C4C3×#12014yes—
349C8—84no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—42no—
353C2×C3—82no—
354C2×C3#178111yescharacteristic
355C2×C3#213312yes—
356C22—22no—
357C22—22no—
358C22—24no—
359C22—42no—
360C22—42no—
361C22—44no—
362C22—42no—
363C22—82no—
364C22—82no—
365C22C3×#37911yescharacteristic
366C22C2×C3×#9911yescharacteristic
367C22C2×C3×#9911yescharacteristic
368C4—24no—
369C4—22no—
370C4—21no—
371C4—21no—
372C4—42no—
373C4—42no—
374C4—42no—
375C4—42no—
376C3#1270211yescharacteristic, direct product factor, Sylow
377C2—21no—
378C2—21no—
379C2—42no—
380C2—82no—
381C2C3×#178111yescharacteristic
382C2C3×#213312yes—
subgroupG/H
center, socleC22×C3C2×#99
commutatorC22×C4C22×C3×C4
Frattini, cosocleC22×Q8C23×C3
prime radicalC22×C3×#41C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C23×C3×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#356C2semidirect
3. C2×C3×#355C2semidirect
4. C3×#1784C2semidirect
5. #12702C3direct (N × H)
23. C2×#356C2×C3semidirect
24. C2×#355C2×C3semidirect
25. #1784C2×C3semidirect
376. C3#12702direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×C3×#3562
C2×C3×#3552
C3×#17842
#127023