Finite Group Databasebeta

← C3×#10248 #10628855

Subgroups — C3×#10248 #10628855

Order 768.

604 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3209C211yesmaximal, characteristic, complemented
2C3×#3077C211yesmaximal, characteristic
3C3×#2542C211yesmaximal, characteristic
4C3×#1548C211yesmaximal, characteristic, complemented
5C3×#1503C211yesmaximal, characteristic, complemented
6C3×#1478C211yesmaximal, characteristic
7C3×#1303C211yesmaximal, characteristic, complemented
8#10248C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C22⋊Q8—22no—
12C2×C3×C4×D4C2211yescharacteristic
13C3×#494—21no—
14C3×#487—21no—
15C3×#478—22no—
16C3×#476—21no—
17C3×#474—21no—
18C3×#462—22no—
19C3×#451—22no—
20C3×#431C2211yescharacteristic
21C3×#431C2211yescharacteristic
22C3×#428—21no—
23C3×#423C2211yescharacteristic
24C3×#418—21no—
25C3×#399—22no—
26C3×#390—21no—
27C3×#384—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C3×#333—22no—
30C3×#328—22no—
31C3×D4⋊C8—22no—
32#3209C2×C311yescharacteristic, complemented
33#3077C2×C311yescharacteristic
34#2542C2×C311yescharacteristic
35#1548C2×C311yescharacteristic, complemented
36#1503C2×C311yescharacteristic, complemented
37#1478C2×C311yescharacteristic
38#1303C2×C311yescharacteristic, complemented
39C22×C3×Q8D412yes—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C2×C3×Q16—24no—
43C2×C3×M16—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8D411yescharacteristic
49C3×C4⋊Q8D411yescharacteristic
50C3×#125—21no—
51C3×C22⋊Q8—21no—
52C3×C22⋊Q8—21no—
53C3×C22⋊Q8—48no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C3×#117—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4D411yescharacteristic
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—21no—
68C2×C3×C42—21no—
69C2×C3×C42C2311yescharacteristic
70C3×#107—22no—
71C3×#106—22no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8D412yes—
75C3×C4⋊C8D412yes—
76C3×#103—24no—
77C3×#102—22no—
78C3×#102—22no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8—21no—
81C3×C4×C8—21no—
82C3×C4×C8—21no—
83C3×#95D412yes—
84#533C22×C311yescharacteristic
85C2×C4⋊Q8C22×C311yescharacteristic
86C2×C22⋊Q8—22no—
87C2×C4×D4C22×C311yescharacteristic
88#494—21no—
89#487—21no—
90#478—22no—
91#476—21no—
92#474—21no—
93#462—22no—
94#451—22no—
95#431C22×C311yescharacteristic
96#431C22×C311yescharacteristic
97#428—21no—
98#423C22×C311yescharacteristic
99#418—21no—
100#399—22no—
101#390—21no—
102#384—21no—
103C4×C4⋊C4C22×C311yescharacteristic
104#333—22no—
105#328—22no—
106D4⋊C8—22no—
107C24×C3—21no—
108C2×C3×Q8—22no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—22no—
112C2×C3×Q8—44no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4Q1612yescomplemented
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—21no—
122C22×C3×C4—44no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C3×Q16—84no—
129C3×M16—41no—
130C3×M16—41no—
131C2×C3×C8—22no—
132C2×C3×C8—22no—
133C2×C3×C8—41no—
134C3×C4⋊C4—24no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—48no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C22⋊C4—48no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C42—41no—
153C3×C42—41no—
154C3×C42—41no—
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42#4111yescharacteristic
159C22×Q8C3×D412yes—
160C22×D4C3×D411yescharacteristic
161C23×C4—21no—
162C2×Q16—24no—
163C2×M16—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8C3×D411yescharacteristic
169C4⋊Q8C3×D411yescharacteristic
170#125—21no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8—48no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180#117—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C3×D411yescharacteristic
186C2×C22⋊C4—24no—
187C2×C22⋊C4—21no—
188C2×C42—21no—
189C2×C42C23×C311yescharacteristic
190#107—22no—
191#106—22no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8C3×D412yes—
195C4⋊C8C3×D412yes—
196#103—24no—
197#102—22no—
198#102—22no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201C4×C8—21no—
202C4×C8—21no—
203#95C3×D412yes—
204C23×C3—22no—
205C23×C3—42no—
206C23×C3—42no—
207C23×C3—41no—
208C23×C3—41no—
209C23×C3#12011yescharacteristic
210C3×Q8—42no—
211C3×Q8—44no—
212C3×Q8—81no—
213C3×Q8—82no—
214C3×D4—22no—
215C3×D4—22no—
216C3×D4—41no—
217C3×D4—41no—
218C2×C3×C4—22no—
219C2×C3×C4—24no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—44no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—42no—
229C2×C3×C4—82no—
230C2×C3×C4—82no—
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12011yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12711yescharacteristic
237C2×C3×C4C2×Q1611yescharacteristic
238C2×C3×C4#13511yescharacteristic
239C2×C3×C4#13711yescharacteristic
240C2×C3×C4#13711yescharacteristic
241C3×C8—42no—
242C3×C8—42no—
243C3×C8—81no—
244C24—21no—
245C2×Q8—22no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—22no—
249C2×Q8—44no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4C3×Q1612yescomplemented
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—44no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4C2×C3×D411yescharacteristic
263C22×C4C2×C3×D411yescharacteristic
264C22×C4C2×C3×D411yescharacteristic
265Q16—84no—
266M16—41no—
267M16—41no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—41no—
271C4⋊C4—24no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—48no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C22⋊C4—48no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C42—41no—
290C42—41no—
291C42—41no—
292C42C2×C3×D411yescharacteristic
293C42C2×C3×D411yescharacteristic
294C42C2×C3×D411yescharacteristic
295C42C3×#4111yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—44no—
299C22×C3—42no—
300C22×C3—82no—
301C22×C3—81no—
302C22×C3—81no—
303C22×C3#39211yescharacteristic
304C22×C3#45111yescharacteristic
305C22×C3#45211yescharacteristic
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—44no—
310C3×C4—81no—
311C3×C4—82no—
312C3×C4—81no—
313C3×C4—81no—
314C3×C4#45111yescharacteristic
315C3×C4#45211yescharacteristic
316C3×C4#45312yes—
317C3×C4#46711yescharacteristic
318C3×C4#47011yescharacteristic
319C23—22no—
320C23—42no—
321C23—42no—
322C23—41no—
323C23—41no—
324C23C3×#12011yescharacteristic
325Q8—42no—
326Q8—44no—
327Q8—82no—
328Q8—81no—
329D4—22no—
330D4—22no—
331D4—41no—
332D4—41no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4C3×#12011yescharacteristic
347C2×C4C3×#12011yescharacteristic
348C2×C4C3×#12111yescharacteristic
349C2×C4C3×#12111yescharacteristic
350C2×C4C3×#12111yescharacteristic
351C2×C4C3×#12711yescharacteristic
352C2×C4C2×C3×Q1611yescharacteristic
353C2×C4C3×#13511yescharacteristic
354C2×C4C3×#13711yescharacteristic
355C2×C4C3×#13711yescharacteristic
356C8—42no—
357C8—42no—
358C8—81no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—82no—
362C2×C3#200111yescharacteristic
363C2×C3#200211yescharacteristic
364C2×C3#200911yescharacteristic
365C22—21no—
366C22—21no—
367C22—44no—
368C22—42no—
369C22—82no—
370C22—81no—
371C22—81no—
372C22C3×#39211yescharacteristic
373C22C3×#45111yescharacteristic
374C22C3×#45211yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—44no—
379C4—81no—
380C4—82no—
381C4—81no—
382C4—81no—
383C4C3×#45111yescharacteristic
384C4C3×#45211yescharacteristic
385C4C3×#45312yes—
386C4C3×#46711yescharacteristic
387C4C3×#47011yescharacteristic
388C3#1024811yescharacteristic, direct product factor, Sylow
389C2—21no—
390C2—21no—
391C2—82no—
392C2C3×#200111yescharacteristic
393C2C3×#200211yescharacteristic
394C2C3×#200911yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC22×C3×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3209C2semidirect
4. C3×#1548C2semidirect
5. C3×#1503C2semidirect
7. C3×#1303C2semidirect
8. #10248C3direct (N × H)
32. #3209C2×C3semidirect
35. #1548C2×C3semidirect
36. #1503C2×C3semidirect
38. #1303C2×C3semidirect
118. C2×C3×D4Q16semidirect
255. C2×D4C3×Q16semidirect
388. C3#10248direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32092
C3×#30772
C3×#25422
C3×#15482
C3×#15032
C3×#14782
C3×#13032
#102483