Finite Group Databasebeta

← C3×#32926 #10651533

Subgroups — C3×#32926 #10651533

Order 768.

1922 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 1922 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, complemented
2C2×C3×B(4,𝔽2)C212yesmaximal, complemented
3C3×#2408C213yesmaximal, complemented
4C3×#1889C216yesmaximal, complemented
5C3×#1799C213yesmaximal, complemented
6#32926C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×H(5,𝔽2)C2211yescharacteristic
8C23×C3×D4C2211yescharacteristic, complemented
9C2×C3×#121—23no—
10C2×C3×#120C2216yes—
11C3×#520—23no—
12C3×#518—23no—
13C3×#518C2213yes—
14C3×C4×#41—23no—
15C2×C3×C4×D4—23no—
16C2×C3×C4×D4C2213yes—
17C3×B(4,𝔽2)—28no—
18C2×C3×#99C2216yes—
19C3×#394—23no—
20C3×#392—23no—
21C3×#390—23no—
22C3×#380C2213yescomplemented
23C3×#379C2216yescomplemented
24C3×#342C2216yescomplemented
25C4×H(5,𝔽2)C2×C311yescharacteristic, complemented
26C2×B(4,𝔽2)C2×C312yescomplemented
27#2408C2×C313yescomplemented
28#1889C2×C316yescomplemented
29#1799C2×C313yescomplemented
30C25×C3C2312yes—
31C3×H(5,𝔽2)—26no—
32C3×H(5,𝔽2)D414yescomplemented
33C2×C3×#41—23no—
34C22×C3×D4—23no—
35C22×C3×D4—26no—
36C22×C3×D4—23no—
37C22×C3×D4—23no—
38C22×C3×D4—44no—
39C22×C3×D4C2313yes—
40C23×C3×C4—21no—
41C23×C3×C4D413yescomplemented
42C23×C3×C4C2311yescharacteristic
43C3×#121—412no—
44C3×#120—212no—
45C3×#120—212no—
46C3×C4×Q8—21no—
47C3×C4×Q8—23no—
48C3×C4×D4—23no—
49C3×C4×D4—26no—
50C3×C4×D4—23no—
51C3×C4×D4—26no—
52C3×C4×D4—26no—
53C3×C4×D4—26no—
54C3×C4×D4—26no—
55C3×#117—26no—
56C3×#117—23no—
57C2×C3×C4⋊C4—23no—
58C2×C3×C4⋊C4—23no—
59C2×C3×C4⋊C4D413yescomplemented
60C2×C3×C22⋊C4—23no—
61C2×C3×C22⋊C4—23no—
62C2×C3×C22⋊C4—26no—
63C2×C3×C22⋊C4—26no—
64C2×C3×C22⋊C4—26no—
65C2×C3×C22⋊C4D413yescomplemented
66C2×C3×C22⋊C4C2316yes—
67C2×C3×C22⋊C4C2313yes—
68C2×C3×C42—23no—
69C2×C3×C42D413yescomplemented
70C3×#99—212no—
71C3×#95—23no—
72C2×H(5,𝔽2)C22×C311yescharacteristic
73C23×D4C22×C311yescharacteristic, complemented
74C2×#121—23no—
75C2×#120C22×C316yes—
76#520—23no—
77#518—23no—
78#518C22×C313yes—
79C4×#41—23no—
80C2×C4×D4—23no—
81C2×C4×D4C22×C313yes—
82B(4,𝔽2)—28no—
83C2×#99C22×C316yes—
84#394—23no—
85#392—23no—
86#390—23no—
87#380C22×C313yescomplemented
88#379C22×C316yescomplemented
89#342C22×C316yescomplemented
90C24×C3—21no—
91C24×C3—26no—
92C24×C3—22no—
93C24×C3—26no—
94C24×C3—44no—
95C24×C3—82no—
96C24×C3C2×D413yes—
97C24×C3C2411yescharacteristic
98C3×#41—212no—
99C3×#41—212no—
100C2×C3×Q8—21no—
101C2×C3×D4—23no—
102C2×C3×D4—212no—
103C2×C3×D4—212no—
104C2×C3×D4—26no—
105C2×C3×D4—26no—
106C2×C3×D4—212no—
107C2×C3×D4—46no—
108C2×C3×D4—412no—
109C2×C3×D4—412no—
110C2×C3×D4—412no—
111C2×C3×D4—43no—
112C2×C3×D4—41no—
113C2×C3×D4—86no—
114C2×C3×D4C2×D416yes—
115C2×C3×D4#4116yes—
116C22×C3×C4—23no—
117C22×C3×C4—23no—
118C22×C3×C4—23no—
119C22×C3×C4—23no—
120C22×C3×C4—23no—
121C22×C3×C4—26no—
122C22×C3×C4—42no—
123C22×C3×C4—46no—
124C22×C3×C4—82no—
125C22×C3×C4C2×D413yes—
126C22×C3×C4C2×D413yes—
127C22×C3×C4C2×D413yes—
128C22×C3×C4C2×D413yes—
129C22×C3×C4C2×D413yes—
130C3×C4⋊C4—23no—
131C3×C4⋊C4—23no—
132C3×C4⋊C4—26no—
133C3×C4⋊C4—46no—
134C3×C22⋊C4—26no—
135C3×C22⋊C4—26no—
136C3×C22⋊C4—26no—
137C3×C22⋊C4—26no—
138C3×C22⋊C4—26no—
139C3×C22⋊C4—412no—
140C3×C22⋊C4—412no—
141C3×C22⋊C4—86no—
142C3×C42—26no—
143C3×C42—23no—
144C3×C42—23no—
145C25C23×C312yes—
146H(5,𝔽2)—26no—
147H(5,𝔽2)C3×D414yescomplemented
148C2×#41—23no—
149C22×D4—23no—
150C22×D4—26no—
151C22×D4—23no—
152C22×D4—23no—
153C22×D4—44no—
154C22×D4C23×C313yes—
155C23×C4—21no—
156C23×C4C3×D413yescomplemented
157C23×C4C23×C311yescharacteristic
158#121—412no—
159#120—212no—
160#120—212no—
161C4×Q8—21no—
162C4×Q8—23no—
163C4×D4—23no—
164C4×D4—26no—
165C4×D4—23no—
166C4×D4—26no—
167C4×D4—26no—
168C4×D4—26no—
169C4×D4—26no—
170#117—26no—
171#117—23no—
172C2×C4⋊C4—23no—
173C2×C4⋊C4—23no—
174C2×C4⋊C4C3×D413yescomplemented
175C2×C22⋊C4—23no—
176C2×C22⋊C4—23no—
177C2×C22⋊C4—26no—
178C2×C22⋊C4—26no—
179C2×C22⋊C4—26no—
180C2×C22⋊C4C3×D413yescomplemented
181C2×C22⋊C4C23×C316yes—
182C2×C22⋊C4C23×C313yes—
183C2×C42—23no—
184C2×C42C3×D413yescomplemented
185#99—212no—
186#95—23no—
187C23×C3—24no—
188C23×C3—23no—
189C23×C3—23no—
190C23×C3—212no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3—43no—
194C23×C3—43no—
195C23×C3—46no—
196C23×C3—46no—
197C23×C3—46no—
198C23×C3—42no—
199C23×C3—46no—
200C23×C3—46no—
201C23×C3—84no—
202C23×C3—812no—
203C23×C3—81no—
204C23×C3—81no—
205C23×C3#12116yescomplemented
206C23×C3C22×D413yes—
207C23×C3C22×D411yescharacteristic
208C23×C3C2×#4113yes—
209C3×Q8—24no—
210C3×D4—212no—
211C3×D4—212no—
212C3×D4—212no—
213C3×D4—412no—
214C3×D4—44no—
215C3×D4—824no—
216C2×C3×C4—212no—
217C2×C3×C4—23no—
218C2×C3×C4—23no—
219C2×C3×C4—26no—
220C2×C3×C4—23no—
221C2×C3×C4—23no—
222C2×C3×C4—43no—
223C2×C3×C4—46no—
224C2×C3×C4—43no—
225C2×C3×C4—43no—
226C2×C3×C4—46no—
227C2×C3×C4—46no—
228C2×C3×C4—46no—
229C2×C3×C4—83no—
230C2×C3×C4—83no—
231C2×C3×C4—83no—
232C2×C3×C4#12013yes—
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12116yescomplemented
235C24—21no—
236C24—26no—
237C24—22no—
238C24—26no—
239C24—44no—
240C24—82no—
241C24C2×C3×D413yes—
242C24C24×C311yescharacteristic
243#41—212no—
244#41—212no—
245C2×Q8—21no—
246C2×D4—23no—
247C2×D4—212no—
248C2×D4—212no—
249C2×D4—26no—
250C2×D4—26no—
251C2×D4—212no—
252C2×D4—46no—
253C2×D4—412no—
254C2×D4—412no—
255C2×D4—412no—
256C2×D4—43no—
257C2×D4—41no—
258C2×D4—86no—
259C2×D4C2×C3×D416yes—
260C2×D4C3×#4116yes—
261C22×C4—23no—
262C22×C4—23no—
263C22×C4—23no—
264C22×C4—23no—
265C22×C4—23no—
266C22×C4—26no—
267C22×C4—42no—
268C22×C4—46no—
269C22×C4—82no—
270C22×C4C2×C3×D413yes—
271C22×C4C2×C3×D413yes—
272C22×C4C2×C3×D413yes—
273C22×C4C2×C3×D413yes—
274C22×C4C2×C3×D413yes—
275C4⋊C4—23no—
276C4⋊C4—23no—
277C4⋊C4—26no—
278C4⋊C4—46no—
279C22⋊C4—26no—
280C22⋊C4—26no—
281C22⋊C4—26no—
282C22⋊C4—26no—
283C22⋊C4—26no—
284C22⋊C4—412no—
285C22⋊C4—412no—
286C22⋊C4—86no—
287C42—26no—
288C42—23no—
289C42—23no—
290C22×C3—23no—
291C22×C3—23no—
292C22×C3—26no—
293C22×C3—41no—
294C22×C3—43no—
295C22×C3—46no—
296C22×C3—412no—
297C22×C3—412no—
298C22×C3—43no—
299C22×C3—43no—
300C22×C3—44no—
301C22×C3—83no—
302C22×C3—86no—
303C22×C3—86no—
304C22×C3—84no—
305C22×C3—82no—
306C22×C3—82no—
307C22×C3C2×#12011yescharacteristic
308C22×C3C2×#12113yes—
309C22×C3#52513yes—
310C3×C4—26no—
311C3×C4—23no—
312C3×C4—43no—
313C3×C4—43no—
314C3×C4—86no—
315C3×C4B(4,𝔽2)12yescomplemented
316C23—24no—
317C23—23no—
318C23—23no—
319C23—212no—
320C23—41no—
321C23—41no—
322C23—43no—
323C23—43no—
324C23—46no—
325C23—46no—
326C23—46no—
327C23—42no—
328C23—46no—
329C23—46no—
330C23—84no—
331C23—812no—
332C23—81no—
333C23—81no—
334C23C3×#12116yescomplemented
335C23C22×C3×D413yes—
336C23C22×C3×D411yescharacteristic
337C23C2×C3×#4113yes—
338Q8—24no—
339D4—212no—
340D4—212no—
341D4—212no—
342D4—412no—
343D4—44no—
344D4—824no—
345C2×C4—212no—
346C2×C4—23no—
347C2×C4—23no—
348C2×C4—26no—
349C2×C4—23no—
350C2×C4—23no—
351C2×C4—43no—
352C2×C4—46no—
353C2×C4—43no—
354C2×C4—43no—
355C2×C4—46no—
356C2×C4—46no—
357C2×C4—46no—
358C2×C4—83no—
359C2×C4—83no—
360C2×C4—83no—
361C2×C4C3×#12013yes—
362C2×C4C3×#12011yescharacteristic
363C2×C4C3×#12116yescomplemented
364C2×C3—23no—
365C2×C3—23no—
366C2×C3—46no—
367C2×C3—84no—
368C2×C3#284711yescharacteristic
369C2×C3C2×B(4,𝔽2)11yescharacteristic
370C2×C3#302711yescharacteristic
371C22—23no—
372C22—23no—
373C22—26no—
374C22—41no—
375C22—43no—
376C22—46no—
377C22—412no—
378C22—412no—
379C22—43no—
380C22—43no—
381C22—44no—
382C22—83no—
383C22—86no—
384C22—86no—
385C22—84no—
386C22—82no—
387C22—82no—
388C22C2×C3×#12011yescharacteristic
389C22C2×C3×#12113yes—
390C22C3×#52513yes—
391C4—23no—
392C4—26no—
393C4—43no—
394C4—43no—
395C4—86no—
396C4C3×B(4,𝔽2)12yescomplemented
397C3#3292611yescharacteristic, direct product factor, Sylow
398C2—23no—
399C2—23no—
400C2—46no—
401C2—84no—
402C2C3×#284711yescharacteristic
403C2C2×C3×B(4,𝔽2)11yescharacteristic
404C2C3×#302711yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
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
1. C3×C4×H(5,𝔽2)C2semidirect
2. C2×C3×B(4,𝔽2)C2semidirect
3. C3×#2408C2semidirect
4. C3×#1889C2semidirect
5. C3×#1799C2semidirect
6. #32926C3direct (N × H)
8. C23×C3×D4C22semidirect
22. C3×#380C22semidirect
23. C3×#379C22semidirect
24. C3×#342C22semidirect
25. C4×H(5,𝔽2)C2×C3semidirect
26. C2×B(4,𝔽2)C2×C3semidirect
27. #2408C2×C3semidirect
28. #1889C2×C3semidirect
29. #1799C2×C3semidirect
32. C3×H(5,𝔽2)D4semidirect
41. C23×C3×C4D4semidirect
59. C2×C3×C4⋊C4D4semidirect
65. C2×C3×C22⋊C4D4semidirect
69. C2×C3×C42D4semidirect
73. C23×D4C22×C3semidirect
87. #380C22×C3semidirect
88. #379C22×C3semidirect
89. #342C22×C3semidirect
147. H(5,𝔽2)C3×D4semidirect
156. C23×C4C3×D4semidirect
174. C2×C4⋊C4C3×D4semidirect
180. C2×C22⋊C4C3×D4semidirect
184. C2×C42C3×D4semidirect
205. C23×C3#121semidirect
234. C2×C3×C4#121semidirect
315. C3×C4B(4,𝔽2)semidirect
334. C23C3×#121semidirect
363. C2×C4C3×#121semidirect
396. C4C3×B(4,𝔽2)semidirect
397. C3#32926direct (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
C2×C3×B(4,𝔽2)2
C3×#24082
C3×#18892
C3×#17992
#329263