Finite Group Databasebeta

← C3×#25836 #10644443

Subgroups — C3×#25836 #10644443

Order 768.

986 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic
2C2×C3×#498C211yesmaximal, characteristic
3C2×C3×#477C211yesmaximal, characteristic
4C3×#2567C211yesmaximal, characteristic
5C3×#2059C214yesmaximal
6C3×#1988C211yesmaximal, characteristic
7C3×#1984C211yesmaximal, characteristic
8C3×#1865C214yesmaximal
9C3×C4×#103C211yesmaximal, characteristic
10#25836C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#557—24no—
12C3×#557—24no—
13C2×C3×C4⋊Q8—21no—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#125—22no—
17C2×C3×#125C2212yes—
18C2×C3×C4×Q8—21no—
19C2×C3×C4×Q8C2212yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C3×#498—24no—
23C3×#477—24no—
24C2×C3×#107C2212yes—
25C2×C3×#106C2214yes—
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×C4×C8C2211yescharacteristic
30C3×#395—22no—
31C3×#391—21no—
32C3×#389C2214yes—
33C3×#384—21no—
34C3×C4×C4⋊C4C2211yescharacteristic
35C3×#340C2212yes—
36C3×#340C2214yes—
37C3×#336C2214yes—
38C2×#557C2×C311yescharacteristic
39C2×#498C2×C311yescharacteristic
40C2×#477C2×C311yescharacteristic
41#2567C2×C311yescharacteristic
42#2059C2×C314yes—
43#1988C2×C311yescharacteristic
44#1984C2×C311yescharacteristic
45#1865C2×C314yes—
46C4×#103C2×C311yescharacteristic
47C22×C3×Q8—21no—
48C22×C3×Q8—21no—
49C22×C3×Q8D411yescharacteristic
50C22×C3×Q8C2311yescharacteristic
51C22×C3×C8C2312yes—
52C22×C3×C8C2312yes—
53C3×C4⋊Q8—28no—
54C3×C4⋊Q8—22no—
55C3×C4⋊Q8—22no—
56C3×C4⋊Q8—22no—
57C3×C4⋊Q8D414yes—
58C3×#125—216no—
59C3×#125—24no—
60C3×#125—24no—
61C3×C4×Q8—28no—
62C3×C4×Q8—24no—
63C3×C4×Q8—24no—
64C3×C4×Q8—22no—
65C3×C4×Q8—22no—
66C3×C4×Q8—22no—
67C3×C4×Q8D414yes—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C4⋊C4D412yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C42—22no—
82C2×C3×C42—21no—
83C2×C3×C42—21no—
84C2×C3×C42C2312yes—
85C2×C3×C42C2311yescharacteristic
86C3×#107—24no—
87C3×#106—28no—
88C3×C4⋊C8—22no—
89C3×#103—24no—
90C3×#103Q818yescomplemented
91C3×C4×C8—22no—
92C3×#95—21no—
93C3×#95—24no—
94#557—24no—
95#557—24no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C22×C311yescharacteristic
98C2×C4⋊Q8C22×C311yescharacteristic
99C2×#125—22no—
100C2×#125C22×C312yes—
101C2×C4×Q8—21no—
102C2×C4×Q8C22×C312yes—
103C2×C4×Q8C22×C311yescharacteristic
104C2×C4×Q8C22×C311yescharacteristic
105#498—24no—
106#477—24no—
107C2×#107C22×C312yes—
108C2×#106C22×C314yes—
109C2×C4⋊C8C22×C311yescharacteristic
110C2×#103C22×C312yes—
111C2×#103C22×C312yes—
112C2×C4×C8C22×C311yescharacteristic
113#395—22no—
114#391—21no—
115#389C22×C314yes—
116#384—21no—
117C4×C4⋊C4C22×C311yescharacteristic
118#340C22×C312yes—
119#340C22×C314yes—
120#336C22×C314yes—
121C2×C3×Q8—22no—
122C2×C3×Q8—24no—
123C2×C3×Q8—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8—21no—
126C2×C3×Q8—22no—
127C2×C3×Q8—42no—
128C2×C3×Q8—42no—
129C2×C3×Q8—42no—
130C2×C3×Q8—82no—
131C2×C3×Q8C2×D412yes—
132C2×C3×Q8C2×Q814yes—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2411yescharacteristic
143C2×C3×C8—22no—
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C2×C3×C8—24no—
147C2×C3×C8C2×Q814yes—
148C3×C4⋊C4—28no—
149C3×C4⋊C4—28no—
150C3×C4⋊C4—28no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—24no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—42no—
161C3×C4⋊C4—84no—
162C3×C4⋊C4C2×D414yes—
163C3×C4⋊C4C2×D412yes—
164C3×C4⋊C4C2×Q814yes—
165C3×C42—28no—
166C3×C42—24no—
167C3×C42—22no—
168C3×C42—22no—
169C3×C42—42no—
170C3×C42C2×D412yes—
171C3×C42C2×D412yes—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8C3×D411yescharacteristic
175C22×Q8C23×C311yescharacteristic
176C22×C8C23×C312yes—
177C22×C8C23×C312yes—
178C4⋊Q8—28no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—22no—
182C4⋊Q8C3×D414yes—
183#125—216no—
184#125—24no—
185#125—24no—
186C4×Q8—28no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8C3×D414yes—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4C3×D411yescharacteristic
201C2×C4⋊C4C3×D412yes—
202C2×C4⋊C4C23×C312yes—
203C2×C4⋊C4C23×C311yescharacteristic
204C2×C4⋊C4C23×C312yes—
205C2×C4⋊C4C23×C312yes—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42C23×C312yes—
210C2×C42C23×C311yescharacteristic
211#107—24no—
212#106—28no—
213C4⋊C8—22no—
214#103—24no—
215#103C3×Q818yescomplemented
216C4×C8—22no—
217#95—21no—
218#95—24no—
219C23×C3C22×D411yescharacteristic
220C3×Q8—22no—
221C3×Q8—24no—
222C3×Q8—44no—
223C3×Q8—48no—
224C3×Q8—84no—
225C3×Q8C4⋊Q814yes—
226C2×C3×C4—22no—
227C2×C3×C4—24no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—24no—
233C2×C3×C4—24no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—24no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—44no—
244C2×C3×C4#12012yes—
245C2×C3×C4#12012yes—
246C2×C3×C4#13512yes—
247C2×C3×C4#13712yes—
248C2×C3×C4C22×D411yescharacteristic
249C2×C3×C4C22×D411yescharacteristic
250C2×C3×C4C22×Q812yes—
251C2×C3×C42−1+412yes—
252C3×C8—24no—
253C3×C8—42no—
254C2×Q8—22no—
255C2×Q8—24no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—21no—
259C2×Q8—22no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—82no—
264C2×Q8C2×C3×D412yes—
265C2×Q8C2×C3×Q814yes—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4C2×C3×D412yes—
272C22×C4C2×C3×D412yes—
273C22×C4C2×C3×D411yescharacteristic
274C22×C4C2×C3×D411yescharacteristic
275C22×C4C24×C311yescharacteristic
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8C2×C3×Q814yes—
281C4⋊C4—28no—
282C4⋊C4—28no—
283C4⋊C4—28no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—21no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—42no—
294C4⋊C4—84no—
295C4⋊C4C2×C3×D414yes—
296C4⋊C4C2×C3×D412yes—
297C4⋊C4C2×C3×Q814yes—
298C42—28no—
299C42—24no—
300C42—22no—
301C42—22no—
302C42—42no—
303C42C2×C3×D412yes—
304C42C2×C3×D412yes—
305C22×C3C2×#12011yescharacteristic
306C22×C3D4×Q812yes—
307C22×C3C2×#13511yescharacteristic
308C22×C3C2×#13711yescharacteristic
309C22×C3#57712yes—
310C3×C4—24no—
311C3×C4—22no—
312C3×C4—44no—
313C3×C4—42no—
314C3×C4—44no—
315C3×C4—42no—
316C3×C4—42no—
317C3×C4—82no—
318C3×C4#45214yes—
319C3×C4C2×C4⋊Q811yescharacteristic
320C3×C4#53611yescharacteristic
321C3×C4D4×Q812yes—
322C23C22×C3×D411yescharacteristic
323Q8—22no—
324Q8—24no—
325Q8—44no—
326Q8—48no—
327Q8—84no—
328Q8C3×C4⋊Q814yes—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4C3×#12012yes—
348C2×C4C3×#12012yes—
349C2×C4C3×#13512yes—
350C2×C4C3×#13712yes—
351C2×C4C22×C3×D411yescharacteristic
352C2×C4C22×C3×D411yescharacteristic
353C2×C4C22×C3×Q812yes—
354C2×C4C3×2−1+412yes—
355C8—24no—
356C8—42no—
357C2×C3#243411yescharacteristic
358C2×C3C2×#45211yescharacteristic
359C2×C3#301011yescharacteristic
360C2×C3#338112yes—
361C2×C3#338712yes—
362C22C2×C3×#12011yescharacteristic
363C22C3×D4×Q812yes—
364C22C2×C3×#13511yescharacteristic
365C22C2×C3×#13711yescharacteristic
366C22C3×#57712yes—
367C4—24no—
368C4—22no—
369C4—42no—
370C4—44no—
371C4—42no—
372C4—42no—
373C4—44no—
374C4—82no—
375C4C3×#45214yes—
376C4C2×C3×C4⋊Q811yescharacteristic
377C4C3×#53611yescharacteristic
378C4C3×D4×Q812yes—
379C3#2583611yescharacteristic, direct product factor, Sylow
380C2C3×#243411yescharacteristic
381C2C2×C3×#45211yescharacteristic
382C2C3×#301011yescharacteristic
383C2C3×#338112yes—
384C2C3×#338712yes—
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. #25836C3direct (N × H)
90. C3×#103Q8semidirect
215. #103C3×Q8semidirect
379. C3#25836direct (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×#4982
C2×C3×#4772
C3×#25672
C3×#20592
C3×#19882
C3×#19842
C3×#18652
C3×C4×#1032
#258363