Finite Group Databasebeta

← C3×#29686 #10648293

Subgroups — C3×#29686 #10648293

Order 768.

1312 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3459C211yesmaximal, characteristic, complemented
2C3×#2534C213yesmaximal, complemented
3C3×#2030C211yesmaximal, characteristic, complemented
4C3×#2029C211yesmaximal, characteristic, complemented
5C3×#2022C213yesmaximal, complemented
6C3×#2021C213yesmaximal, complemented
7C3×#1799C213yesmaximal, complemented
8#29686C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#543C2211yescharacteristic
10C3×#539—23no—
11C3×#538—23no—
12C3×#534C2213yes—
13C3×#529—23no—
14C3×#525—23no—
15C2×C3×#123C2213yes—
16C2×C3×#120C2213yes—
17C2×C3×C4×D4C2213yes—
18C22×C3×C22⋊C4C2211yescharacteristic
19C2×C3×#99C2216yes—
20C3×#396—23no—
21C3×#394—23no—
22C3×#388—23no—
23C3×#386—23no—
24C3×#381C2213yes—
25C3×#380C2213yes—
26C3×#379C2213yes—
27C3×#342C2213yes—
28C3×#342C2213yes—
29#3459C2×C311yescharacteristic, complemented
30#2534C2×C313yescomplemented
31#2030C2×C311yescharacteristic, complemented
32#2029C2×C311yescharacteristic, complemented
33#2022C2×C313yescomplemented
34#2021C2×C313yescomplemented
35#1799C2×C313yescomplemented
36C25×C3C2311yescharacteristic
37C2×C3×#41—21no—
38C22×C3×D4—23no—
39C22×C3×D4C2313yes—
40C23×C3×C4D413yescomplemented
41C23×C3×C4C2311yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C3×#126—26no—
44C3×#125—23no—
45C3×#124—23no—
46C3×#123—26no—
47C3×#123—26no—
48C3×#123—46no—
49C3×C22⋊Q8—23no—
50C3×C22⋊Q8—23no—
51C3×C22⋊Q8—23no—
52C3×#121—23no—
53C3×#121—23no—
54C3×#121—23no—
55C3×#120—26no—
56C3×#120—46no—
57C3×C4×Q8—21no—
58C3×C4×D4—23no—
59C3×C4×D4—26no—
60C3×C4×D4—26no—
61C3×#117—23no—
62C2×C3×C4⋊C4—23no—
63C2×C3×C4⋊C4—23no—
64C2×C3×C4⋊C4D413yescomplemented
65C2×C3×C22⋊C4—23no—
66C2×C3×C22⋊C4—23no—
67C2×C3×C22⋊C4—23no—
68C2×C3×C22⋊C4—23no—
69C2×C3×C22⋊C4—23no—
70C2×C3×C22⋊C4—23no—
71C2×C3×C22⋊C4D413yescomplemented
72C2×C3×C22⋊C4C2313yes—
73C2×C3×C22⋊C4C2313yes—
74C2×C3×C22⋊C4C2313yes—
75C2×C3×C42D413yescomplemented
76C3×#99—46no—
77C3×#95—23no—
78C3×#95—23no—
79C3×#95—23no—
80#543C22×C311yescharacteristic
81#539—23no—
82#538—23no—
83#534C22×C313yes—
84#529—23no—
85#525—23no—
86C2×#123C22×C313yes—
87C2×#120C22×C313yes—
88C2×C4×D4C22×C313yes—
89C22×C22⋊C4C22×C311yescharacteristic
90C2×#99C22×C316yes—
91#396—23no—
92#394—23no—
93#388—23no—
94#386—23no—
95#381C22×C313yes—
96#380C22×C313yes—
97#379C22×C313yes—
98#342C22×C313yes—
99#342C22×C313yes—
100C24×C3—23no—
101C24×C3—23no—
102C24×C3—21no—
103C24×C3—42no—
104C24×C3—81no—
105C24×C3C2×D413yes—
106C24×C3C2411yescharacteristic
107C3×#41—82no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×D4—26no—
111C2×C3×D4—23no—
112C2×C3×D4—26no—
113C2×C3×D4—23no—
114C2×C3×D4—23no—
115C2×C3×D4—23no—
116C2×C3×D4—46no—
117C2×C3×D4—46no—
118C2×C3×D4—83no—
119C22×C3×C4—23no—
120C22×C3×C4—23no—
121C22×C3×C4—23no—
122C22×C3×C4—23no—
123C22×C3×C4—23no—
124C22×C3×C4—23no—
125C22×C3×C4—21no—
126C22×C3×C4—81no—
127C22×C3×C4—83no—
128C22×C3×C4C2×D413yes—
129C22×C3×C4C2×D413yes—
130C22×C3×C4C2×D413yes—
131C22×C3×C4C2×D413yes—
132C22×C3×C4C2×D413yes—
133C3×C4⋊C4—23no—
134C3×C4⋊C4—23no—
135C3×C4⋊C4—23no—
136C3×C4⋊C4—23no—
137C3×C4⋊C4—26no—
138C3×C4⋊C4—46no—
139C3×C4⋊C4—83no—
140C3×C22⋊C4—26no—
141C3×C22⋊C4—26no—
142C3×C22⋊C4—26no—
143C3×C22⋊C4—26no—
144C3×C22⋊C4—26no—
145C3×C22⋊C4—26no—
146C3×C22⋊C4—46no—
147C3×C22⋊C4—46no—
148C3×C22⋊C4—46no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—43no—
151C3×C22⋊C4—83no—
152C3×C42—23no—
153C3×C42—23no—
154C25C23×C311yescharacteristic
155C2×#41—21no—
156C22×D4—23no—
157C22×D4C23×C313yes—
158C23×C4C3×D413yescomplemented
159C23×C4C23×C311yescharacteristic
160C23×C4C23×C311yescharacteristic
161#126—26no—
162#125—23no—
163#124—23no—
164#123—26no—
165#123—26no—
166#123—46no—
167C22⋊Q8—23no—
168C22⋊Q8—23no—
169C22⋊Q8—23no—
170#121—23no—
171#121—23no—
172#121—23no—
173#120—26no—
174#120—46no—
175C4×Q8—21no—
176C4×D4—23no—
177C4×D4—26no—
178C4×D4—26no—
179#117—23no—
180C2×C4⋊C4—23no—
181C2×C4⋊C4—23no—
182C2×C4⋊C4C3×D413yescomplemented
183C2×C22⋊C4—23no—
184C2×C22⋊C4—23no—
185C2×C22⋊C4—23no—
186C2×C22⋊C4—23no—
187C2×C22⋊C4—23no—
188C2×C22⋊C4—23no—
189C2×C22⋊C4C3×D413yescomplemented
190C2×C22⋊C4C23×C313yes—
191C2×C22⋊C4C23×C313yes—
192C2×C22⋊C4C23×C313yes—
193C2×C42C3×D413yescomplemented
194#99—46no—
195#95—23no—
196#95—23no—
197#95—23no—
198C23×C3—26no—
199C23×C3—23no—
200C23×C3—23no—
201C23×C3—23no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—43no—
205C23×C3—43no—
206C23×C3—43no—
207C23×C3—43no—
208C23×C3—43no—
209C23×C3—41no—
210C23×C3—43no—
211C23×C3—43no—
212C23×C3—86no—
213C23×C3—82no—
214C23×C3C22×D413yes—
215C23×C3H(5,𝔽2)13yes—
216C23×C32−1+411yescharacteristic
217C3×Q8—81no—
218C3×Q8—81no—
219C3×D4—46no—
220C3×D4—46no—
221C3×D4—83no—
222C3×D4—83no—
223C3×D4—812no—
224C2×C3×C4—23no—
225C2×C3×C4—23no—
226C2×C3×C4—23no—
227C2×C3×C4—23no—
228C2×C3×C4—23no—
229C2×C3×C4—43no—
230C2×C3×C4—43no—
231C2×C3×C4—46no—
232C2×C3×C4—43no—
233C2×C3×C4—43no—
234C2×C3×C4—43no—
235C2×C3×C4—43no—
236C2×C3×C4—43no—
237C2×C3×C4—43no—
238C2×C3×C4—41no—
239C2×C3×C4—43no—
240C2×C3×C4—43no—
241C2×C3×C4—43no—
242C2×C3×C4—43no—
243C2×C3×C4—83no—
244C2×C3×C4#12011yescharacteristic
245C2×C3×C4#12013yes—
246C24—23no—
247C24—23no—
248C24—21no—
249C24—42no—
250C24—81no—
251C24C2×C3×D413yes—
252C24C24×C311yescharacteristic
253#41—82no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×D4—26no—
257C2×D4—23no—
258C2×D4—26no—
259C2×D4—23no—
260C2×D4—23no—
261C2×D4—23no—
262C2×D4—46no—
263C2×D4—46no—
264C2×D4—83no—
265C22×C4—23no—
266C22×C4—23no—
267C22×C4—23no—
268C22×C4—23no—
269C22×C4—23no—
270C22×C4—23no—
271C22×C4—21no—
272C22×C4—81no—
273C22×C4—83no—
274C22×C4C2×C3×D413yes—
275C22×C4C2×C3×D413yes—
276C22×C4C2×C3×D413yes—
277C22×C4C2×C3×D413yes—
278C22×C4C2×C3×D413yes—
279C4⋊C4—23no—
280C4⋊C4—23no—
281C4⋊C4—23no—
282C4⋊C4—23no—
283C4⋊C4—26no—
284C4⋊C4—46no—
285C4⋊C4—83no—
286C22⋊C4—26no—
287C22⋊C4—26no—
288C22⋊C4—26no—
289C22⋊C4—26no—
290C22⋊C4—26no—
291C22⋊C4—26no—
292C22⋊C4—46no—
293C22⋊C4—46no—
294C22⋊C4—46no—
295C22⋊C4—41no—
296C22⋊C4—43no—
297C22⋊C4—83no—
298C42—23no—
299C42—23no—
300C22×C3—23no—
301C22×C3—23no—
302C22×C3—43no—
303C22×C3—41no—
304C22×C3—46no—
305C22×C3—46no—
306C22×C3—46no—
307C22×C3—43no—
308C22×C3—43no—
309C22×C3—42no—
310C22×C3—43no—
311C22×C3—83no—
312C22×C3—83no—
313C22×C3—83no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3C2×#12011yescharacteristic
317C22×C3#53513yes—
318C22×C3#53713yes—
319C3×C4—21no—
320C3×C4—23no—
321C3×C4—43no—
322C3×C4—43no—
323C3×C4—43no—
324C3×C4—83no—
325C3×C4—83no—
326C3×C4—81no—
327C23—26no—
328C23—23no—
329C23—23no—
330C23—23no—
331C23—41no—
332C23—41no—
333C23—43no—
334C23—43no—
335C23—43no—
336C23—43no—
337C23—43no—
338C23—41no—
339C23—43no—
340C23—43no—
341C23—86no—
342C23—82no—
343C23C22×C3×D413yes—
344C23C3×H(5,𝔽2)13yes—
345C23C3×2−1+411yescharacteristic
346Q8—81no—
347Q8—81no—
348D4—46no—
349D4—46no—
350D4—83no—
351D4—83no—
352D4—812no—
353C2×C4—23no—
354C2×C4—23no—
355C2×C4—23no—
356C2×C4—23no—
357C2×C4—23no—
358C2×C4—43no—
359C2×C4—43no—
360C2×C4—46no—
361C2×C4—43no—
362C2×C4—43no—
363C2×C4—43no—
364C2×C4—43no—
365C2×C4—43no—
366C2×C4—43no—
367C2×C4—41no—
368C2×C4—43no—
369C2×C4—43no—
370C2×C4—43no—
371C2×C4—43no—
372C2×C4—83no—
373C2×C4C3×#12011yescharacteristic
374C2×C4C3×#12013yes—
375C2×C3—23no—
376C2×C3—23no—
377C2×C3—82no—
378C2×C3—83no—
379C2×C3#261511yescharacteristic
380C2×C3#302511yescharacteristic
381C2×C3#302611yescharacteristic
382C22—23no—
383C22—23no—
384C22—43no—
385C22—41no—
386C22—46no—
387C22—46no—
388C22—46no—
389C22—43no—
390C22—42no—
391C22—43no—
392C22—43no—
393C22—83no—
394C22—83no—
395C22—83no—
396C22—81no—
397C22—81no—
398C22C2×C3×#12011yescharacteristic
399C22C3×#53513yes—
400C22C3×#53713yes—
401C4—21no—
402C4—23no—
403C4—43no—
404C4—43no—
405C4—43no—
406C4—83no—
407C4—83no—
408C4—81no—
409C3#2968611yescharacteristic, direct product factor, Sylow
410C2—23no—
411C2—23no—
412C2—82no—
413C2—83no—
414C2C3×#261511yescharacteristic
415C2C3×#302511yescharacteristic
416C2C3×#302611yescharacteristic
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×#3459C2semidirect
2. C3×#2534C2semidirect
3. C3×#2030C2semidirect
4. C3×#2029C2semidirect
5. C3×#2022C2semidirect
6. C3×#2021C2semidirect
7. C3×#1799C2semidirect
8. #29686C3direct (N × H)
29. #3459C2×C3semidirect
30. #2534C2×C3semidirect
31. #2030C2×C3semidirect
32. #2029C2×C3semidirect
33. #2022C2×C3semidirect
34. #2021C2×C3semidirect
35. #1799C2×C3semidirect
40. C23×C3×C4D4semidirect
64. C2×C3×C4⋊C4D4semidirect
71. C2×C3×C22⋊C4D4semidirect
75. C2×C3×C42D4semidirect
158. C23×C4C3×D4semidirect
182. C2×C4⋊C4C3×D4semidirect
189. C2×C22⋊C4C3×D4semidirect
193. C2×C42C3×D4semidirect
409. C3#29686direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34592
C3×#25342
C3×#20302
C3×#20292
C3×#20222
C3×#20212
C3×#17992
#296863