Finite Group Databasebeta

← C3×#5646 #10595743

Subgroups — C3×#5646 #10595743

Order 576.

429 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C22⋊Q8C211yesmaximal, characteristic, complemented
2C3×#946C211yesmaximal, characteristic, complemented
3C3×#944C211yesmaximal, characteristic, complemented
4C3×#929C211yesmaximal, characteristic
5C3×#924C211yesmaximal, characteristic, complemented
6C3×#898C211yesmaximal, characteristic, complemented
7C3×#898C211yesmaximal, characteristic, complemented
8C3×#896C211yesmaximal, characteristic, complemented
9C3×#890C211yesmaximal, characteristic, complemented
10C3×#889C211yesmaximal, characteristic, complemented
11C3×#888C211yesmaximal, characteristic, complemented
12C3×#885C211yesmaximal, characteristic
13C3×#883C211yesmaximal, characteristic
14C3×#879C211yesmaximal, characteristic, complemented
15C3×#879C211yesmaximal, characteristic, complemented
16C3×#563—31nomaximal
17#5646C311yesmaximal, characteristic, direct product factor
18C2×C32×Q8C2211yescharacteristic
19C22×C32×C4C2211yescharacteristic
20C2×C3×#67C2211yescharacteristic
21C2×C3×C4×D3C2211yescharacteristic
22C2×C3×Dic6C2211yescharacteristic
23C2×C3×Dic6C2211yescharacteristic
24C32×C4⋊C4C2211yescharacteristic
25C32×C4⋊C4C2211yescharacteristic
26C32×C4⋊C4C2211yescharacteristic
27C32×C22⋊C4C2211yescharacteristic
28C32×C22⋊C4C2211yescharacteristic
29C3×#217C2211yescharacteristic
30C3×#217C2211yescharacteristic
31C3×#217C2211yescharacteristic
32C3×(C2×D3)⋊C4C2211yescharacteristic
33C3×(C2×D3)⋊C4C2211yescharacteristic
34C3×(C2×D3)⋊C4C2211yescharacteristic
35C3×(C2×D3)⋊C4C2211yescharacteristic
36C3×(C2×D3)⋊C4C2211yescharacteristic
37C3×C4⋊Dic3C2211yescharacteristic
38C3×C4⋊Dic3C2211yescharacteristic
39C3×C4⋊Dic3C2211yescharacteristic
40C3×C4⋊Dic3C2211yescharacteristic
41C3×Dic3⋊C4C2211yescharacteristic
42C3×Dic3⋊C4C2211yescharacteristic
43C3×Dic3⋊C4C2211yescharacteristic
44C3×Dic3⋊C4C2211yescharacteristic
45C3×Dic3⋊C4C2211yescharacteristic
46C3×Dic3⋊C4C2211yescharacteristic
47C3×Dic3⋊C4C2211yescharacteristic
48C3×Dic3⋊C4C2211yescharacteristic
49C3×Dic3⋊C4C2211yescharacteristic
50C3×C4×Dic3C2211yescharacteristic
51C3×C4×Dic3C2211yescharacteristic
52C3×C4×Dic3C2211yescharacteristic
53C3×C4⋊Q8—31no—
54C3×C4⋊Q8—31no—
55C3×#126—31no—
56C3×#126—31no—
57C3×#126—31no—
58C3×#126—31no—
59C3×#125—31no—
60C3×#125—31no—
61C3×#124—31no—
62C3×#123—31no—
63C3×#123—31no—
64C3×C22⋊Q8—21no—
65C3×C22⋊Q8—31no—
66C3×C22⋊Q8—31no—
67C3×C22⋊Q8—31no—
68C3×C22⋊Q8D311yescharacteristic, complemented
69C3×C22⋊Q8C2×C311yescharacteristic, complemented
70#946C2×C311yescharacteristic, complemented
71#944C2×C311yescharacteristic, complemented
72#929C2×C311yescharacteristic
73#924C2×C311yescharacteristic, complemented
74#898C2×C311yescharacteristic, complemented
75#898C2×C311yescharacteristic, complemented
76#896C2×C311yescharacteristic, complemented
77#890C2×C311yescharacteristic, complemented
78#889C2×C311yescharacteristic, complemented
79#888C2×C311yescharacteristic, complemented
80#885C2×C311yescharacteristic
81#883C2×C311yescharacteristic
82#879C2×C311yescharacteristic, complemented
83#879C2×C311yescharacteristic, complemented
84C23×C32C2311yescharacteristic
85C22×C3×D3C2311yescharacteristic
86C32×Q8—41no—
87C2×C32×C4—41no—
88C2×C32×C4C2311yescharacteristic
89C2×C32×C4C2311yescharacteristic
90C2×C32×C4C2311yescharacteristic
91C2×C32×C4C2311yescharacteristic
92C2×C32×C4C2311yescharacteristic
93C2×C32×C4C2311yescharacteristic
94C3×#67—41no—
95C2×C3×Dic3C2311yescharacteristic
96C2×C3×Dic3C2311yescharacteristic
97C2×C3×Dic3C2311yescharacteristic
98C2×C3×Dic3C2311yescharacteristic
99C2×C3×Dic3C2311yescharacteristic
100C2×C3×Dic3C2311yescharacteristic
101C2×C3×Dic3C2311yescharacteristic
102C3×C4×D3—41no—
103C3×Dic6—41no—
104C3×Dic6—41no—
105#563—31noSylow
106C2×C3×Q8—21no—
107C2×C3×Q8—31no—
108C2×C3×Q8—31no—
109C2×C3×Q8C2×D311yescharacteristic
110C2×C3×Q8C22×C311yescharacteristic
111C2×C3×D4—31no—
112C22×C3×C4—21no—
113C22×C3×C4—31no—
114C22×C3×C4C2×D311yescharacteristic
115C22×C3×C4C22×C311yescharacteristic
116C2×#67C22×C311yescharacteristic
117C2×C4×D3C22×C311yescharacteristic
118C2×Dic6C22×C311yescharacteristic
119C2×Dic6C22×C311yescharacteristic
120C3×C4⋊C4—21no—
121C3×C4⋊C4—21no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—31no—
124C3×C4⋊C4—31no—
125C3×C4⋊C4—31no—
126C3×C4⋊C4—31no—
127C3×C4⋊C4—31no—
128C3×C4⋊C4—31no—
129C3×C4⋊C4—31no—
130C3×C4⋊C4—31no—
131C3×C4⋊C4—31no—
132C3×C4⋊C4—31no—
133C3×C4⋊C4—31no—
134C3×C4⋊C4—31no—
135C3×C4⋊C4—31no—
136C3×C4⋊C4C2×D311yescharacteristic
137C3×C4⋊C4C2×D311yescharacteristic
138C3×C4⋊C4C2×D311yescharacteristic
139C3×C4⋊C4C22×C311yescharacteristic
140C3×C4⋊C4C22×C311yescharacteristic
141C3×C4⋊C4C22×C311yescharacteristic
142C3×C22⋊C4—21no—
143C3×C22⋊C4—21no—
144C3×C22⋊C4—31no—
145C3×C22⋊C4—31no—
146C3×C22⋊C4—31no—
147C3×C22⋊C4—31no—
148C3×C22⋊C4—31no—
149C3×C22⋊C4—31no—
150C3×C22⋊C4—31no—
151C3×C22⋊C4—31no—
152C3×C22⋊C4C2×D311yescharacteristic
153C3×C22⋊C4C2×D311yescharacteristic
154C3×C22⋊C4C22×C311yescharacteristic
155C3×C22⋊C4C22×C311yescharacteristic
156C3×C42—31no—
157C3×C42—31no—
158C3×C42—31no—
159#217C22×C311yescharacteristic
160#217C22×C311yescharacteristic
161#217C22×C311yescharacteristic
162(C2×D3)⋊C4C22×C311yescharacteristic
163(C2×D3)⋊C4C22×C311yescharacteristic
164(C2×D3)⋊C4C22×C311yescharacteristic
165(C2×D3)⋊C4C22×C311yescharacteristic
166(C2×D3)⋊C4C22×C311yescharacteristic
167C4⋊Dic3C22×C311yescharacteristic
168C4⋊Dic3C22×C311yescharacteristic
169C4⋊Dic3C22×C311yescharacteristic
170C4⋊Dic3C22×C311yescharacteristic
171Dic3⋊C4C22×C311yescharacteristic
172Dic3⋊C4C22×C311yescharacteristic
173Dic3⋊C4C22×C311yescharacteristic
174Dic3⋊C4C22×C311yescharacteristic
175Dic3⋊C4C22×C311yescharacteristic
176Dic3⋊C4C22×C311yescharacteristic
177Dic3⋊C4C22×C311yescharacteristic
178Dic3⋊C4C22×C311yescharacteristic
179Dic3⋊C4C22×C311yescharacteristic
180C4×Dic3C22×C311yescharacteristic
181C4×Dic3C22×C311yescharacteristic
182C4×Dic3C22×C311yescharacteristic
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32C2411yescharacteristic
187C2×C3×D3—21no—
188C2×C3×D3—21no—
189C2×C3×D3—21no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4—21no—
196C3×Dic3—21no—
197C3×Dic3—21no—
198C3×Dic3—21no—
199C3×Dic3—21no—
200C3×Dic3—21no—
201C3×Dic3—21no—
202C3×Dic3—21no—
203C4⋊Q8—31no—
204C4⋊Q8—31no—
205#126—31no—
206#126—31no—
207#126—31no—
208#126—31no—
209#125—31no—
210#125—31no—
211#124—31no—
212#123—31no—
213#123—31no—
214C22⋊Q8—31no—
215C22⋊Q8—31no—
216C22⋊Q8—31no—
217C22⋊Q8C3×D311yescharacteristic, complemented
218C23×C3—21no—
219C23×C3—31no—
220C23×C3C22×D311yescharacteristic
221C23×C3C23×C311yescharacteristic
222C22×D3C23×C311yescharacteristic
223C3×Q8—41no—
224C3×Q8—41no—
225C3×Q8—42no—
226C3×Q8—121no—
227C3×Q8—121no—
228C3×D4—121no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—31no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—121no—
246C2×C3×C4C22×D311yescharacteristic
247C2×C3×C4C22×D311yescharacteristic
248C2×C3×C4C22×D311yescharacteristic
249C2×C3×C4C22×D311yescharacteristic
250C2×C3×C4C22×D311yescharacteristic
251C2×C3×C4C22×D311yescharacteristic
252C2×C3×C4C23×C311yescharacteristic
253C2×C3×C4C23×C311yescharacteristic
254C2×C3×C4C23×C311yescharacteristic
255C2×C3×C4C23×C311yescharacteristic
256C2×C3×C4C23×C311yescharacteristic
257C2×C3×C4C23×C311yescharacteristic
258#67—41no—
259C2×Dic3C23×C311yescharacteristic
260C2×Dic3C23×C311yescharacteristic
261C2×Dic3C23×C311yescharacteristic
262C2×Dic3C23×C311yescharacteristic
263C2×Dic3C23×C311yescharacteristic
264C2×Dic3C23×C311yescharacteristic
265C2×Dic3C23×C311yescharacteristic
266C4×D3—41no—
267Dic6—41no—
268Dic6—41no—
269C2×C32—41no—
270C2×C32H(5,𝔽2)11yescharacteristic
271C2×C322−1+411yescharacteristic
272C2×C322−1+411yescharacteristic
273C3×D3—41no—
274C2×Q8—31no—
275C2×Q8—31no—
276C2×Q8C2×C3×D311yescharacteristic
277C2×D4—31no—
278C22×C4—31no—
279C22×C4C2×C3×D311yescharacteristic
280C4⋊C4—31no—
281C4⋊C4—31no—
282C4⋊C4—31no—
283C4⋊C4—31no—
284C4⋊C4—31no—
285C4⋊C4—31no—
286C4⋊C4—31no—
287C4⋊C4—31no—
288C4⋊C4—31no—
289C4⋊C4—31no—
290C4⋊C4—31no—
291C4⋊C4—31no—
292C4⋊C4—31no—
293C4⋊C4C2×C3×D311yescharacteristic
294C4⋊C4C2×C3×D311yescharacteristic
295C4⋊C4C2×C3×D311yescharacteristic
296C22⋊C4—31no—
297C22⋊C4—31no—
298C22⋊C4—31no—
299C22⋊C4—31no—
300C22⋊C4—31no—
301C22⋊C4—31no—
302C22⋊C4—31no—
303C22⋊C4—31no—
304C22⋊C4C2×C3×D311yescharacteristic
305C22⋊C4C2×C3×D311yescharacteristic
306C42—31no—
307C42—31no—
308C42—31no—
309C22×C3—22no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—61no—
320C22×C3—61no—
321C22×C3—61no—
322C22×C3C23×D311yescharacteristic
323C22×C3C24×C311yescharacteristic
324C2×D3—21no—
325C2×D3—21no—
326C2×D3—21no—
327C3×C4—22no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—41no—
341C3×C4—41no—
342C3×C4—41no—
343C3×C4—41no—
344C3×C4—41no—
345C3×C4—61no—
346C3×C4—61no—
347C3×C4—61no—
348C3×C4—61no—
349C3×C4—61no—
350C3×C4—61no—
351C3×C4—61no—
352Dic3—21no—
353Dic3—21no—
354Dic3—21no—
355Dic3—21no—
356Dic3—21no—
357Dic3—21no—
358Dic3—21no—
359C32#56311yescharacteristic, complemented, Sylow
360C23—31no—
361C23C22×C3×D311yescharacteristic
362Q8—41no—
363Q8—121no—
364Q8—121no—
365D4—121no—
366C2×C4—31no—
367C2×C4—31no—
368C2×C4—31no—
369C2×C4—31no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—31no—
373C2×C4—41no—
374C2×C4—121no—
375C2×C4C22×C3×D311yescharacteristic
376C2×C4C22×C3×D311yescharacteristic
377C2×C4C22×C3×D311yescharacteristic
378C2×C4C22×C3×D311yescharacteristic
379C2×C4C22×C3×D311yescharacteristic
380C2×C4C22×C3×D311yescharacteristic
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—42no—
385C2×C3—41no—
386C2×C3—41no—
387C2×C3—121no—
388C2×C3#100411yescharacteristic
389C2×C3#100711yescharacteristic
390C2×C3#101011yescharacteristic
391C2×C3C3×H(5,𝔽2)11yescharacteristic
392C2×C3C3×2−1+411yescharacteristic
393C2×C3C3×2−1+411yescharacteristic
394D3—41no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—61no—
399C22—61no—
400C22—61no—
401C22C23×C3×D311yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—61no—
409C4—61no—
410C4—61no—
411C4—61no—
412C4—61no—
413C4—61no—
414C4—61no—
415C3—21no—
416C3#564611yescharacteristic, direct product factor
417C3C3×#56311yescharacteristic, complemented
418C2—41no—
419C2—121no—
420C2C3×#100411yescharacteristic
421C2C3×#100711yescharacteristic
422C2C3×#101011yescharacteristic
subgroupG/H
centerC22×C3C23×D3
commutator, cosocleC22×C3C24×C3
FrattiniC22C23×C3×D3
FittingC32×C22⋊Q8C2
socleC22×C32C24
prime radicalC2×C3×#67C22
hypercenterC3×C22⋊Q8D3
hypocenterC3C3×#563
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C32×C22⋊Q8C2semidirect
2. C3×#946C2semidirect
3. C3×#944C2semidirect
5. C3×#924C2semidirect
6. C3×#898C2semidirect
7. C3×#898C2semidirect
8. C3×#896C2semidirect
9. C3×#890C2semidirect
10. C3×#889C2semidirect
11. C3×#888C2semidirect
14. C3×#879C2semidirect
15. C3×#879C2semidirect
17. #5646C3direct (N × H)
68. C3×C22⋊Q8D3semidirect
69. C3×C22⋊Q8C2×C3semidirect
70. #946C2×C3semidirect
71. #944C2×C3semidirect
73. #924C2×C3semidirect
74. #898C2×C3semidirect
75. #898C2×C3semidirect
76. #896C2×C3semidirect
77. #890C2×C3semidirect
78. #889C2×C3semidirect
79. #888C2×C3semidirect
82. #879C2×C3semidirect
83. #879C2×C3semidirect
217. C22⋊Q8C3×D3semidirect
359. C32#563semidirect
416. C3#5646direct (N × H)
417. C3C3×#563semidirect

Maximal subgroups

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

subgroupindex
C32×C22⋊Q82
C3×#9462
C3×#9442
C3×#9292
C3×#9242
C3×#8982
C3×#8962
C3×#8902
C3×#8892
C3×#8882
C3×#8852
C3×#8832
C3×#8792
C3×#5633
#56463