Finite Group Databasebeta

← C3×#10056 #10628663

Subgroups — C3×#10056 #10628663

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#494C211yesmaximal, characteristic, complemented
2C2×C3×#476C211yesmaximal, characteristic
3C3×#2563C211yesmaximal, characteristic
4C3×#1987C211yesmaximal, characteristic, complemented
5C2×C3×#333C211yesmaximal, characteristic, complemented
6C2×C3×D4⋊C8C211yesmaximal, characteristic
7C3×#1279C211yesmaximal, characteristic, complemented
8#10056C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×Q16—22no—
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4×D4C2211yescharacteristic
13C3×#494—22no—
14C3×#494—22no—
15C3×#476—22no—
16C3×#476—22no—
17C2×C3×#106—21no—
18C2×C3×C4⋊C8C2211yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×#102—21no—
21C2×C3×C4×C8C2211yescharacteristic
22C3×#393—22no—
23C3×#390—21no—
24C3×#382—21no—
25C3×#380—22no—
26C3×C4×C4⋊C4C2211yescharacteristic
27C3×#340—22no—
28C3×#333—22no—
29C3×D4⋊C8—22no—
30C2×#494C2×C311yescharacteristic, complemented
31C2×#476C2×C311yescharacteristic
32#2563C2×C311yescharacteristic
33#1987C2×C311yescharacteristic, complemented
34C2×#333C2×C311yescharacteristic, complemented
35C2×D4⋊C8C2×C311yescharacteristic
36#1279C2×C311yescharacteristic, complemented
37C22×C3×Q8—21no—
38C22×C3×Q8D412yes—
39C22×C3×D4D411yescharacteristic
40C23×C3×C4—21no—
41C2×C3×Q16—28no—
42C2×C3×Q16—48no—
43C22×C3×C8—21no—
44C22×C3×C8—21no—
45C22×C3×C8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8—21no—
51C3×C4⋊Q8—21no—
52C3×C4⋊Q8—21no—
53C3×C4⋊Q8D412yes—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4D412yes—
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C42—21no—
67C2×C3×C42C2311yescharacteristic
68C3×#106—24no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×#102—24no—
74C3×C4×C8—21no—
75C3×C4×C8D412yes—
76C3×#95—24no—
77C3×#95—21no—
78C22×Q16—22no—
79C2×C4⋊Q8C22×C311yescharacteristic
80C2×C4⋊Q8C22×C311yescharacteristic
81C2×C4×D4C22×C311yescharacteristic
82#494—22no—
83#494—22no—
84#476—22no—
85#476—22no—
86C2×#106—21no—
87C2×C4⋊C8C22×C311yescharacteristic
88C2×C4⋊C8C22×C311yescharacteristic
89C2×#102—21no—
90C2×C4×C8C22×C311yescharacteristic
91#393—22no—
92#390—21no—
93#382—21no—
94#380—22no—
95C4×C4⋊C4C22×C311yescharacteristic
96#340—22no—
97#333—22no—
98D4⋊C8—22no—
99C24×C3—21no—
100C2×C3×Q8—22no—
101C2×C3×Q8—22no—
102C2×C3×Q8—22no—
103C2×C3×Q8—44no—
104C2×C3×Q8—44no—
105C2×C3×Q8—41no—
106C2×C3×D4—21no—
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4Q1612yescomplemented
112C22×C3×C4—21no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4—42no—
119C22×C3×C4—42no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C3×Q16—416no—
124C2×C3×C8—24no—
125C2×C3×C8—22no—
126C2×C3×C8—22no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C2×C3×C8—42no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145C3×C22⋊C4—84no—
146C3×C42—41no—
147C3×C42—41no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152C22×Q8—21no—
153C22×Q8C3×D412yes—
154C22×D4C3×D411yescharacteristic
155C23×C4—21no—
156C2×Q16—28no—
157C2×Q16—48no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8C3×D412yes—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4C3×D411yescharacteristic
178C2×C4⋊C4C3×D412yes—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—24no—
181C2×C42—21no—
182C2×C42C23×C311yescharacteristic
183#106—24no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8C3×D412yes—
188#102—24no—
189C4×C8—21no—
190C4×C8C3×D412yes—
191#95—24no—
192#95—21no—
193C23×C3—22no—
194C23×C3—41no—
195C23×C3—42no—
196C23×C3—41no—
197C23×C3—42no—
198C23×C3#12011yescharacteristic
199C3×Q8—44no—
200C3×Q8—44no—
201C3×Q8—82no—
202C3×Q8—82no—
203C3×D4—22no—
204C3×D4—22no—
205C3×D4—41no—
206C3×D4—41no—
207C2×C3×C4—22no—
208C2×C3×C4—41no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—44no—
216C2×C3×C4—82no—
217C2×C3×C4—82no—
218C2×C3×C4—81no—
219C2×C3×C4—81no—
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12711yescharacteristic
226C2×C3×C4C2×Q1611yescharacteristic
227C2×C3×C4#13511yescharacteristic
228C2×C3×C4#13512yes—
229C2×C3×C4#13711yescharacteristic
230C2×C3×C4#13711yescharacteristic
231C2×C3×C4#13712yes—
232C3×C8—24no—
233C3×C8—42no—
234C3×C8—81no—
235C24—21no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—44no—
240C2×Q8—44no—
241C2×Q8—41no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4C3×Q1612yescomplemented
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4C2×C3×D411yescharacteristic
257C22×C4C2×C3×D411yescharacteristic
258C22×C4C2×C3×D411yescharacteristic
259Q16—416no—
260C2×C8—24no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—42no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—84no—
282C42—41no—
283C42—41no—
284C42C2×C3×D411yescharacteristic
285C42C2×C3×D411yescharacteristic
286C42C2×C3×D411yescharacteristic
287C42C3×#4111yescharacteristic
288C22×C3—44no—
289C22×C3—42no—
290C22×C3—82no—
291C22×C3—81no—
292C22×C3—81no—
293C22×C3#39211yescharacteristic
294C22×C3#45111yescharacteristic
295C22×C3#45212yes—
296C22×C3#45211yescharacteristic
297C22×C3#45312yes—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—82no—
301C3×C4—82no—
302C3×C4—82no—
303C3×C4—81no—
304C3×C4#45111yescharacteristic
305C3×C4#45211yescharacteristic
306C3×C4#46412yes—
307C3×C4#46711yescharacteristic
308C3×C4#47011yescharacteristic
309C3×C4#49511yescharacteristic
310C3×C4#49711yescharacteristic
311C23—22no—
312C23—41no—
313C23—42no—
314C23—41no—
315C23—42no—
316C23C3×#12011yescharacteristic
317Q8—44no—
318Q8—44no—
319Q8—82no—
320Q8—82no—
321D4—22no—
322D4—22no—
323D4—41no—
324D4—41no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4C3×#12011yescharacteristic
339C2×C4C3×#12011yescharacteristic
340C2×C4C3×#12111yescharacteristic
341C2×C4C3×#12111yescharacteristic
342C2×C4C3×#12111yescharacteristic
343C2×C4C3×#12711yescharacteristic
344C2×C4C2×C3×Q1611yescharacteristic
345C2×C4C3×#13511yescharacteristic
346C2×C4C3×#13512yes—
347C2×C4C3×#13711yescharacteristic
348C2×C4C3×#13711yescharacteristic
349C2×C4C3×#13712yes—
350C8—24no—
351C8—42no—
352C8—81no—
353C2×C3—82no—
354C2×C3#163212yes—
355C2×C3#163412yes—
356C2×C3#200111yescharacteristic
357C2×C3#200211yescharacteristic
358C2×C3#201511yescharacteristic
359C22—42no—
360C22—44no—
361C22—81no—
362C22—81no—
363C22—82no—
364C22C3×#39211yescharacteristic
365C22C3×#45111yescharacteristic
366C22C3×#45211yescharacteristic
367C22C3×#45212yes—
368C22C3×#45312yes—
369C4—21no—
370C4—21no—
371C4—82no—
372C4—82no—
373C4—81no—
374C4—82no—
375C4C3×#45111yescharacteristic
376C4C3×#45211yescharacteristic
377C4C3×#46412yes—
378C4C3×#46711yescharacteristic
379C4C3×#47011yescharacteristic
380C4C3×#49511yescharacteristic
381C4C3×#49711yescharacteristic
382C3#1005611yescharacteristic, direct product factor, Sylow
383C2—82no—
384C2C3×#163212yes—
385C2C3×#163412yes—
386C2C3×#200111yescharacteristic
387C2C3×#200211yescharacteristic
388C2C3×#201511yescharacteristic
subgroupG/H
center, socleC23×C3#120
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC22×C3×D4D4
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
1. C2×C3×#494C2semidirect
4. C3×#1987C2semidirect
5. C2×C3×#333C2semidirect
7. C3×#1279C2semidirect
8. #10056C3direct (N × H)
30. C2×#494C2×C3semidirect
33. #1987C2×C3semidirect
34. C2×#333C2×C3semidirect
36. #1279C2×C3semidirect
111. C2×C3×D4Q16semidirect
247. C2×D4C3×Q16semidirect
382. C3#10056direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4942
C2×C3×#4762
C3×#25632
C3×#19872
C2×C3×#3332
C2×C3×D4⋊C82
C3×#12792
#100563