Finite Group Databasebeta

← C3×#10268 #10628875

Subgroups — C3×#10268 #10628875

Order 768.

658 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#492C211yesmaximal, characteristic
2C2×C3×#475C211yesmaximal, characteristic, complemented
3C2×C3×#460C211yesmaximal, characteristic
4C3×#1984C211yesmaximal, characteristic, complemented
5C2×C3×#332C211yesmaximal, characteristic, complemented
6C2×C3×#331C211yesmaximal, characteristic
7C3×#1277C211yesmaximal, characteristic, complemented
8#10268C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×SD16—21no—
10C22×C3×SD16—22no—
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×#127C2211yescharacteristic
13C2×C3×C4×Q8C2211yescharacteristic
14C3×#492—22no—
15C3×#492—22no—
16C3×#475—22no—
17C3×#475—22no—
18C3×#460—22no—
19C3×#460—22no—
20C2×C3×#106—21no—
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103—21no—
25C2×C3×#102—21no—
26C2×C3×C4×C8C2211yescharacteristic
27C3×#336—22no—
28C3×#332—22no—
29C3×#331—22no—
30C2×#492C2×C311yescharacteristic
31C2×#475C2×C311yescharacteristic, complemented
32C2×#460C2×C311yescharacteristic
33#1984C2×C311yescharacteristic, complemented
34C2×#332C2×C311yescharacteristic, complemented
35C2×#331C2×C311yescharacteristic
36#1277C2×C311yescharacteristic, complemented
37C22×C3×Q8—21no—
38C22×C3×Q8D411yescharacteristic
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C2×C3×SD16—24no—
43C2×C3×SD16—28no—
44C2×C3×SD16—44no—
45C2×C3×SD16—42no—
46C2×C3×SD16—42no—
47C2×C3×SD16—44no—
48C22×C3×C8—21no—
49C22×C3×C8—21no—
50C22×C3×C8—21no—
51C22×C3×C8D412yes—
52C3×C4⋊Q8—21no—
53C3×C4⋊Q8—21no—
54C3×C4⋊Q8—21no—
55C3×C4⋊Q8D412yes—
56C3×#127—21no—
57C3×#127—21no—
58C3×#127—21no—
59C3×#127D412yes—
60C3×C4×Q8—21no—
61C3×C4×Q8—21no—
62C3×C4×Q8—21no—
63C3×C4×Q8—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C42—21no—
69C2×C3×C42C2311yescharacteristic
70C3×#106—24no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8D412yes—
76C3×C4⋊C8D412yes—
77C3×#103—24no—
78C3×#102—24no—
79C3×C4×C8—21no—
80C3×C4×C8—21no—
81C22×SD16—21no—
82C22×SD16—22no—
83C2×C4⋊Q8C22×C311yescharacteristic
84C2×#127C22×C311yescharacteristic
85C2×C4×Q8C22×C311yescharacteristic
86#492—22no—
87#492—22no—
88#475—22no—
89#475—22no—
90#460—22no—
91#460—22no—
92C2×#106—21no—
93C2×C4⋊C8C22×C311yescharacteristic
94C2×C4⋊C8C22×C311yescharacteristic
95C2×C4⋊C8C22×C311yescharacteristic
96C2×#103—21no—
97C2×#102—21no—
98C2×C4×C8C22×C311yescharacteristic
99#336—22no—
100#332—22no—
101#331—22no—
102C24×C3—41no—
103C2×C3×Q8—21no—
104C2×C3×Q8—21no—
105C2×C3×Q8—21no—
106C2×C3×Q8—22no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—41no—
110C2×C3×Q8—42no—
111C2×C3×Q8—42no—
112C2×C3×Q8SD1612yescomplemented
113C2×C3×D4—22no—
114C2×C3×D4—22no—
115C2×C3×D4—22no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—42no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—42no—
124C2×C3×D4—41no—
125C2×C3×D4—82no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—41no—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C3×SD16—416no—
133C3×SD16—48no—
134C2×C3×C8—22no—
135C2×C3×C8—22no—
136C2×C3×C8—22no—
137C2×C3×C8—22no—
138C2×C3×C8—41no—
139C2×C3×C8—41no—
140C2×C3×C8—41no—
141C2×C3×C8—41no—
142C2×C3×C8—41no—
143C2×C3×C8—42no—
144C2×C3×C8SD1614yescomplemented
145C3×C4⋊C4—21no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C42—41no—
154C3×C42—41no—
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C22×Q8—21no—
160C22×Q8C3×D411yescharacteristic
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C2×SD16—24no—
165C2×SD16—28no—
166C2×SD16—44no—
167C2×SD16—42no—
168C2×SD16—42no—
169C2×SD16—44no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8C3×D412yes—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8C3×D412yes—
178#127—21no—
179#127—21no—
180#127—21no—
181#127C3×D412yes—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4C3×D411yescharacteristic
190C2×C42—21no—
191C2×C42C23×C311yescharacteristic
192#106—24no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8C3×D412yes—
198C4⋊C8C3×D412yes—
199#103—24no—
200#102—24no—
201C4×C8—21no—
202C4×C8—21no—
203C23×C3—42no—
204C23×C3—82no—
205C23×C3—81no—
206C23×C3—81no—
207C23×C3—82no—
208C23×C3#12111yescharacteristic
209C3×Q8—22no—
210C3×Q8—22no—
211C3×Q8—41no—
212C3×Q8—41no—
213C3×Q8—44no—
214C3×Q8—81no—
215C3×Q8—81no—
216C3×D4—42no—
217C3×D4—44no—
218C3×D4—42no—
219C3×D4—82no—
220C3×D4—82no—
221C3×D4—81no—
222C3×D4—81no—
223C3×D4—81no—
224C3×D4—81no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—82no—
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12711yescharacteristic
239C2×C3×C4C2×SD1611yescharacteristic
240C2×C3×C4C2×SD1612yes—
241C2×C3×C4#13511yescharacteristic
242C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
243C2×C3×C4AGL(1,ℤ/8)12yes—
244C2×C3×C4#13711yescharacteristic
245C3×C8—24no—
246C3×C8—42no—
247C3×C8—42no—
248C3×C8—81no—
249C24—41no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—22no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—41no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8C3×SD1612yescomplemented
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—41no—
272C2×D4—82no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4C2×C3×D411yescharacteristic
277C22×C4C2×C3×D411yescharacteristic
278C22×C4C3×#4111yescharacteristic
279SD16—416no—
280SD16—48no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8C3×SD1614yescomplemented
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4—22no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C42—41no—
301C42—41no—
302C42C2×C3×D411yescharacteristic
303C42C2×C3×D411yescharacteristic
304C42C2×C3×D411yescharacteristic
305C42C2×C3×D411yescharacteristic
306C22×C3—82no—
307C22×C3—82no—
308C22×C3—82no—
309C22×C3—162no—
310C22×C3—162no—
311C22×C3#39211yescharacteristic
312C22×C3#46012yes—
313C22×C3#46511yescharacteristic
314C22×C3#46811yescharacteristic
315C22×C3#47112yes—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—42no—
319C3×C4—82no—
320C3×C4—81no—
321C3×C4#44811yescharacteristic
322C3×C4#44911yescharacteristic
323C3×C4#45011yescharacteristic
324C3×C4#45012yes—
325C3×C4#45211yescharacteristic
326C3×C4#49211yescharacteristic
327C3×C4#49611yescharacteristic
328C23—42no—
329C23—82no—
330C23—81no—
331C23—81no—
332C23—82no—
333C23C3×#12111yescharacteristic
334Q8—22no—
335Q8—22no—
336Q8—41no—
337Q8—41no—
338Q8—44no—
339Q8—81no—
340Q8—81no—
341D4—42no—
342D4—44no—
343D4—42no—
344D4—82no—
345D4—82no—
346D4—81no—
347D4—81no—
348D4—81no—
349D4—81no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4C3×#12011yescharacteristic
359C2×C4C3×#12011yescharacteristic
360C2×C4C3×#12011yescharacteristic
361C2×C4C3×#12111yescharacteristic
362C2×C4C3×#12111yescharacteristic
363C2×C4C3×#12711yescharacteristic
364C2×C4C2×C3×SD1611yescharacteristic
365C2×C4C2×C3×SD1612yes—
366C2×C4C3×#13511yescharacteristic
367C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
368C2×C4C3×AGL(1,ℤ/8)12yes—
369C2×C4C3×#13711yescharacteristic
370C8—24no—
371C8—42no—
372C8—42no—
373C8—81no—
374C2×C3—162no—
375C2×C3#166612yes—
376C2×C3#168612yes—
377C2×C3#200011yescharacteristic
378C2×C3#201411yescharacteristic
379C2×C3#201911yescharacteristic
380C22—82no—
381C22—82no—
382C22—82no—
383C22—162no—
384C22—162no—
385C22C3×#39211yescharacteristic
386C22C3×#46012yes—
387C22C3×#46511yescharacteristic
388C22C3×#46811yescharacteristic
389C22C3×#47112yes—
390C4—21no—
391C4—21no—
392C4—42no—
393C4—82no—
394C4—81no—
395C4C3×#44811yescharacteristic
396C4C3×#44911yescharacteristic
397C4C3×#45011yescharacteristic
398C4C3×#45012yes—
399C4C3×#45211yescharacteristic
400C4C3×#49211yescharacteristic
401C4C3×#49611yescharacteristic
402C3#1026811yescharacteristic, direct product factor, Sylow
403C2—162no—
404C2C3×#166612yes—
405C2C3×#168612yes—
406C2C3×#200011yescharacteristic
407C2C3×#201411yescharacteristic
408C2C3×#201911yescharacteristic
subgroupG/H
center, socleC23×C3#121
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC2×C3×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#475C2semidirect
4. C3×#1984C2semidirect
5. C2×C3×#332C2semidirect
7. C3×#1277C2semidirect
8. #10268C3direct (N × H)
31. C2×#475C2×C3semidirect
33. #1984C2×C3semidirect
34. C2×#332C2×C3semidirect
36. #1277C2×C3semidirect
112. C2×C3×Q8SD16semidirect
144. C2×C3×C8SD16semidirect
259. C2×Q8C3×SD16semidirect
291. C2×C8C3×SD16semidirect
402. C3#10268direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4922
C2×C3×#4752
C2×C3×#4602
C3×#19842
C2×C3×#3322
C2×C3×#3312
C3×#12772
#102683