Finite Group Databasebeta

← C3×#25634 #10644241

Subgroups — C3×#25634 #10644241

Order 768.

1082 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic
2C2×C3×#494C211yesmaximal, characteristic
3C2×C3×#462C211yesmaximal, characteristic
4C3×#2568C211yesmaximal, characteristic
5C3×#2016C214yesmaximal
6C3×#1984C211yesmaximal, characteristic
7C3×#1981C211yesmaximal, characteristic
8C3×#1878C214yesmaximal
9C3×C4×#103C211yesmaximal, characteristic
10#25634C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×Q16C2214yes—
12C22×C3×Q16C2212yes—
13C3×#557—24no—
14C3×#557—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125—21no—
19C2×C3×#125—22no—
20C2×C3×C4×Q8—21no—
21C2×C3×C4×Q8C2212yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C3×#494—24no—
25C3×#462—24no—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2214yes—
30C2×C3×C4×C8C2211yescharacteristic
31C3×#400—22no—
32C3×#391—21no—
33C3×#391C2214yes—
34C3×#382—21no—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×#340C2212yes—
37C3×#336C2214yes—
38C2×#557C2×C311yescharacteristic
39C2×#494C2×C311yescharacteristic
40C2×#462C2×C311yescharacteristic
41#2568C2×C311yescharacteristic
42#2016C2×C314yes—
43#1984C2×C311yescharacteristic
44#1981C2×C311yescharacteristic
45#1878C2×C314yes—
46C4×#103C2×C311yescharacteristic
47C22×C3×Q8D411yescharacteristic
48C22×C3×Q8C2312yes—
49C22×C3×Q8C2311yescharacteristic
50C22×C3×Q8C2312yes—
51C2×C3×Q16—24no—
52C2×C3×Q16—28no—
53C2×C3×Q16—28no—
54C2×C3×Q16—44no—
55C2×C3×Q16—44no—
56C22×C3×C8C2312yes—
57C22×C3×C8C2312yes—
58C3×C4⋊Q8—24no—
59C3×C4⋊Q8—24no—
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8D414yes—
64C3×#125—216no—
65C3×#125—28no—
66C3×C4×Q8—28no—
67C3×C4×Q8—24no—
68C3×C4×Q8—24no—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8D414yes—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4D412yes—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42—22no—
86C2×C3×C42—21no—
87C2×C3×C42—21no—
88C2×C3×C42C2312yes—
89C2×C3×C42C2311yescharacteristic
90C3×C4⋊C8—22no—
91C3×#103—24no—
92C3×#103—24no—
93C3×#103—44no—
94C3×C4×C8—22no—
95C3×#95—21no—
96C3×#95—24no—
97C3×#95—24no—
98C22×Q16C22×C314yes—
99C22×Q16C22×C312yes—
100#557—24no—
101#557—24no—
102C2×C4⋊Q8C22×C311yescharacteristic
103C2×C4⋊Q8C22×C312yes—
104C2×C4⋊Q8C22×C311yescharacteristic
105C2×#125—21no—
106C2×#125—22no—
107C2×C4×Q8—21no—
108C2×C4×Q8C22×C312yes—
109C2×C4×Q8C22×C311yescharacteristic
110C2×C4×Q8C22×C311yescharacteristic
111#494—24no—
112#462—24no—
113C2×C4⋊C8C22×C311yescharacteristic
114C2×#103C22×C312yes—
115C2×#103C22×C312yes—
116C2×#103C22×C314yes—
117C2×C4×C8C22×C311yescharacteristic
118#400—22no—
119#391—21no—
120#391C22×C314yes—
121#382—21no—
122C4×C4⋊C4C22×C311yescharacteristic
123#340C22×C312yes—
124#336C22×C314yes—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—21no—
129C2×C3×Q8—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—24no—
133C2×C3×Q8—24no—
134C2×C3×Q8—24no—
135C2×C3×Q8—22no—
136C2×C3×Q8—24no—
137C2×C3×Q8Q1614yescomplemented
138C2×C3×Q8C2×D414yes—
139C2×C3×Q8C2×D412yes—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2411yescharacteristic
150C3×Q16—416no—
151C3×Q16—48no—
152C2×C3×C8—22no—
153C2×C3×C8—22no—
154C2×C3×C8—22no—
155C2×C3×C8—24no—
156C2×C3×C8#4114yes—
157C3×C4⋊C4—28no—
158C3×C4⋊C4—28no—
159C3×C4⋊C4—28no—
160C3×C4⋊C4—24no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—24no—
163C3×C4⋊C4—24no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—42no—
169C3×C4⋊C4—42no—
170C3×C4⋊C4C2×D412yes—
171C3×C42—28no—
172C3×C42—24no—
173C3×C42—22no—
174C3×C42—22no—
175C3×C42—42no—
176C3×C42C2×D412yes—
177C3×C42C2×D412yes—
178C22×Q8C3×D411yescharacteristic
179C22×Q8C23×C312yes—
180C22×Q8C23×C311yescharacteristic
181C22×Q8C23×C312yes—
182C2×Q16—24no—
183C2×Q16—28no—
184C2×Q16—28no—
185C2×Q16—44no—
186C2×Q16—44no—
187C22×C8C23×C312yes—
188C22×C8C23×C312yes—
189C4⋊Q8—24no—
190C4⋊Q8—24no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8C3×D414yes—
195#125—216no—
196#125—28no—
197C4×Q8—28no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8C3×D414yes—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C3×D411yescharacteristic
213C2×C4⋊C4C3×D412yes—
214C2×C4⋊C4C23×C312yes—
215C2×C4⋊C4C23×C311yescharacteristic
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42C23×C312yes—
220C2×C42C23×C311yescharacteristic
221C4⋊C8—22no—
222#103—24no—
223#103—24no—
224#103—44no—
225C4×C8—22no—
226#95—21no—
227#95—24no—
228#95—24no—
229C23×C3C22×D411yescharacteristic
230C3×Q8—28no—
231C3×Q8—22no—
232C3×Q8—22no—
233C3×Q8—28no—
234C3×Q8—42no—
235C3×Q8—42no—
236C3×Q8—42no—
237C3×Q8—42no—
238C3×Q8—42no—
239C3×Q8—44no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—22no—
247C2×C3×C4—24no—
248C2×C3×C4—42no—
249C2×C3×C4—44no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—44no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4#12012yes—
257C2×C3×C4#12012yes—
258C2×C3×C4C2×Q1614yes—
259C2×C3×C4C2×Q1612yes—
260C2×C3×C4#13712yes—
261C2×C3×C4C22×D411yescharacteristic
262C2×C3×C4C22×D411yescharacteristic
263C2×C3×C4C2×#4112yes—
264C2×C3×C4H(5,𝔽2)12yes—
265C3×C8—24no—
266C3×C8—42no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8C3×Q1614yescomplemented
280C2×Q8C2×C3×D414yes—
281C2×Q8C2×C3×D412yes—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4C2×C3×D412yes—
288C22×C4C2×C3×D411yescharacteristic
289C22×C4C2×C3×D412yes—
290C22×C4C2×C3×D411yescharacteristic
291C22×C4C24×C311yescharacteristic
292Q16—416no—
293Q16—48no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—24no—
298C2×C8C3×#4114yes—
299C4⋊C4—28no—
300C4⋊C4—28no—
301C4⋊C4—28no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4C2×C3×D412yes—
313C42—28no—
314C42—24no—
315C42—22no—
316C42—22no—
317C42—42no—
318C42C2×C3×D412yes—
319C42C2×C3×D412yes—
320C22×C3C2×#12011yescharacteristic
321C22×C3#55012yes—
322C22×C3C22×Q1611yescharacteristic
323C22×C3C2×#13711yescharacteristic
324C22×C3#57812yes—
325C3×C4—24no—
326C3×C4—22no—
327C3×C4—42no—
328C3×C4—44no—
329C3×C4—42no—
330C3×C4—42no—
331C3×C4—82no—
332C3×C4—82no—
333C3×C4#45114yes—
334C3×C4#53211yescharacteristic
335C3×C4#53511yescharacteristic
336C3×C4#54612yes—
337C23C22×C3×D411yescharacteristic
338Q8—28no—
339Q8—22no—
340Q8—22no—
341Q8—28no—
342Q8—42no—
343Q8—42no—
344Q8—42no—
345Q8—42no—
346Q8—42no—
347Q8—44no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4C3×#12012yes—
365C2×C4C3×#12012yes—
366C2×C4C2×C3×Q1614yes—
367C2×C4C2×C3×Q1612yes—
368C2×C4C3×#13712yes—
369C2×C4C22×C3×D411yescharacteristic
370C2×C4C22×C3×D411yescharacteristic
371C2×C4C2×C3×#4112yes—
372C2×C4C3×H(5,𝔽2)12yes—
373C8—24no—
374C8—42no—
375C2×C3#243111yescharacteristic
376C2×C3C2×#45111yescharacteristic
377C2×C3#301111yescharacteristic
378C2×C3#339512yes—
379C2×C3#339812yes—
380C22C2×C3×#12011yescharacteristic
381C22C3×#55012yes—
382C22C22×C3×Q1611yescharacteristic
383C22C2×C3×#13711yescharacteristic
384C22C3×#57812yes—
385C4—24no—
386C4—22no—
387C4—42no—
388C4—44no—
389C4—42no—
390C4—42no—
391C4—82no—
392C4—82no—
393C4C3×#45114yes—
394C4C3×#53211yescharacteristic
395C4C3×#53511yescharacteristic
396C4C3×#54612yes—
397C3#2563411yescharacteristic, direct product factor, Sylow
398C2C3×#243111yescharacteristic
399C2C2×C3×#45111yescharacteristic
400C2C3×#301111yescharacteristic
401C2C3×#339512yes—
402C2C3×#339812yes—
subgroupG/H
center, socle, prime radicalC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
10. #25634C3direct (N × H)
137. C2×C3×Q8Q16semidirect
279. C2×Q8C3×Q16semidirect
397. C3#25634direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×C3×#4942
C2×C3×#4622
C3×#25682
C3×#20162
C3×#19842
C3×#19812
C3×#18782
C3×C4×#1032
#256343