Finite Group Databasebeta

← #10594830

Subgroups — #10594830

Order 576.

627 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#235C211yesmaximal, characteristic, complemented
2C2×#3797C211yesmaximal, characteristic, complemented
3C4×#638C211yesmaximal, characteristic, complemented
4C3×#924C211yesmaximal, characteristic, complemented
5#63959C212yesmaximal, complemented
6#63958C212yesmaximal, complemented
7#63916C211yesmaximal, characteristic, complemented
8#63904C212yesmaximal, complemented
9#63863C211yesmaximal, characteristic, complemented
10#63851C212yesmaximal, complemented
11#63843C211yesmaximal, characteristic, complemented
12#5831—31nomaximal
13#5633—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#638C2211yescharacteristic
16C2×C4×C3⋊D3C2211yescharacteristic
17C2×C3×#67C2212yes—
18C3×#235—24no—
19C2×C3×D12C2211yescharacteristic
20C2×C3×C4×D3C2212yes—
21C2×C3×Dic6C2211yescharacteristic
22C2×C3×Dic6C2211yescharacteristic
23C2×#627C2212yes—
24C2×#626C2212yes—
25C2×#624C2212yes—
26#3797—24no—
27#3757C2211yescharacteristic
28#3752C2211yescharacteristic
29#3750C2211yescharacteristic
30C4×C3⋊Dic3C2211yescharacteristic
31C3×#217C2212yes—
32C3×C4⋊Dic3C2211yescharacteristic
33C3×Dic3⋊C4C2212yes—
34Dic3⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37(C2×D3)⋊Dic3C2212yes—
38Dic32C2212yes—
39C2×C3×#41—31no—
40C2×D3×Q8—31no—
41C2×D3×Q8—31no—
42C2×#235D311yescharacteristic, complemented
43C3×C22⋊Q8—31no—
44#947—31no—
45Q8×Dic3—31no—
46#938—32no—
47#938—31no—
48#936—31no—
49#936—32no—
50C4×#67—31no—
51C4×#67—31no—
52C4×#67—32no—
53C4×#67—61no—
54#924—32no—
55#924D311yescharacteristic, complemented
56#896—32no—
57#895—31no—
58#893—32no—
59#887—31no—
60#885—32no—
61#884—32no—
62C23×C32C2311yescharacteristic
63C22×C3⋊D3C2311yescharacteristic
64C22×C3×D3C2312yes—
65C2×C32×C4—22no—
66C2×C32×C4C2311yescharacteristic
67C2×C32×C4C2311yescharacteristic
68#638—22no—
69C2×C3⋊Dic3C2311yescharacteristic
70C2×C3⋊Dic3C2311yescharacteristic
71C2×C3⋊Dic3C2311yescharacteristic
72C4×C3⋊D3—22no—
73C3×#67—24no—
74C2×C3×Dic3C2312yes—
75C2×C3×Dic3C2312yes—
76C2×C3×Dic3C2312yes—
77C3×D12—22no—
78C3×C4×D3—24no—
79C3×Dic6—22no—
80C3×Dic6D414yescomplemented
81#627—24no—
82#626—24no—
83#624—24no—
84#548—91noSylow
85C3×#41—64no—
86C2×C3×Q8—31no—
87C2×C3×Q8—31no—
88C2×C3×D4—32no—
89C2×C3×D4—31no—
90C22×C3×C4—21no—
91C22×C3×C4—32no—
92C22×C3×C4C2×D311yescharacteristic
93C22×C3×C4C2×D311yescharacteristic
94C2×#67—31no—
95C2×#67—31no—
96C2×#67—32no—
97C2×#67—61no—
98C2×#67C2×D312yes—
99D3×Q8—64no—
100D3×Q8—64no—
101#235—24no—
102C2×D12—32no—
103C2×D12C2×D311yescharacteristic
104C2×C4×D3—31no—
105C2×C4×D3—31no—
106C2×C4×D3—32no—
107C2×C4×D3—32no—
108C2×C4×D3—61no—
109C2×C4×D3C2×D312yes—
110C2×Dic6—32no—
111C2×Dic6—32no—
112C2×Dic6C2×D311yescharacteristic
113C2×Dic6C2×D311yescharacteristic
114C3×C4⋊C4—31no—
115C3×C4⋊C4—32no—
116C3×C22⋊C4—32no—
117#217—31no—
118#217—31no—
119#217—32no—
120#217—32no—
121#217—32no—
122#217—61no—
123#217C2×D312yes—
124(C2×D3)⋊C4—32no—
125(C2×D3)⋊C4—32no—
126(C2×D3)⋊C4—32no—
127(C2×D3)⋊C4—31no—
128(C2×D3)⋊C4—31no—
129(C2×D3)⋊C4—61no—
130C4⋊Dic3—32no—
131C4⋊Dic3C2×D311yescharacteristic
132Dic3⋊C4—31no—
133Dic3⋊C4—31no—
134Dic3⋊C4—32no—
135Dic3⋊C4—61no—
136Dic3⋊C4C2×D312yes—
137C4×Dic3—31no—
138C4×Dic3—31no—
139C4×Dic3—32no—
140C4×Dic3—32no—
141C4×Dic3—61no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32—21no—
145C22×C32C2411yescharacteristic
146C2×C3⋊D3—21no—
147C2×C3⋊D3—21no—
148C2×C3⋊D3#4112yescomplemented
149C2×C3×D3—22no—
150C2×C3×D3—22no—
151C2×C3×D3—22no—
152C32×C4—21no—
153C32×C4C2×D412yes—
154C3⋊Dic3—21no—
155C3⋊Dic3—21no—
156C3⋊Dic3—21no—
157C3×Dic3—22no—
158C3×Dic3—22no—
159C3×Dic3C2×D414yes—
160C2×#41—31no—
161C22×Q8—91no—
162#124—92no—
163#124—91no—
164C22⋊Q8—31no—
165C22⋊Q8—92no—
166#121—92no—
167#121—91no—
168C4×Q8—91no—
169C4×D4—92no—
170C4×D4—91no—
171C23×C3—21no—
172C23×C3—32no—
173C23×C3C22×D311yescharacteristic
174C23×C3C22×D311yescharacteristic
175C22×D3—31no—
176C22×D3—31no—
177C22×D3—61no—
178C22×D3C22×D312yes—
179C3×Q8—34no—
180C3×Q8—62no—
181C3×D4—64no—
182C3×D4—62no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—32no—
188C2×C3×C4—32no—
189C2×C3×C4—32no—
190C2×C3×C4—42no—
191C2×C3×C4—64no—
192C2×C3×C4C22×D311yescharacteristic
193C2×C3×C4C22×D311yescharacteristic
194C2×C3×C4C22×D311yescharacteristic
195C2×C3×C4C22×D311yescharacteristic
196#67—24no—
197#67—64no—
198#67—62no—
199#67—62no—
200#67—64no—
201C2×Dic3—31no—
202C2×Dic3—31no—
203C2×Dic3—31no—
204C2×Dic3—31no—
205C2×Dic3—31no—
206C2×Dic3—31no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3C22×D312yes—
211C2×Dic3C22×D312yes—
212C2×Dic3C22×D312yes—
213D12—22no—
214D12—64no—
215C4×D3—24no—
216C4×D3—64no—
217C4×D3—62no—
218C4×D3—62no—
219C4×D3—64no—
220C4×D3—122no—
221Dic6—22no—
222Dic6—64no—
223Dic6—64no—
224Dic6#6714yescomplemented
225C2×C32—41no—
226C2×C32C22×D411yescharacteristic
227C2×C32C2×#4111yescharacteristic
228C2×C322−1+411yescharacteristic
229C3⋊D3—22no—
230C3×D3—42no—
231#41—64no—
232C2×Q8—31no—
233C2×Q8—31no—
234C2×Q8—92no—
235C2×Q8—184no—
236C2×D4—32no—
237C2×D4—31no—
238C2×D4—92no—
239C2×D4—91no—
240C22×C4—32no—
241C22×C4—92no—
242C22×C4—91no—
243C22×C4D3211yescharacteristic
244C4⋊C4—32no—
245C4⋊C4—31no—
246C4⋊C4—92no—
247C4⋊C4—91no—
248C22⋊C4—32no—
249C22⋊C4—92no—
250C22⋊C4—92no—
251C22⋊C4—92no—
252C22⋊C4—91no—
253C22⋊C4—91no—
254C42—92no—
255C42—91no—
256C22×C3—22no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—62no—
267C22×C3—62no—
268C22×C3—62no—
269C22×C3C23×D311yescharacteristic
270C22×C3C23×D311yescharacteristic
271C2×D3—22no—
272C2×D3—22no—
273C2×D3—22no—
274C2×D3—32no—
275C2×D3—32no—
276C2×D3—62no—
277C2×D3—61no—
278C2×D3—61no—
279C2×D3—61no—
280C2×D3—61no—
281C2×D3—121no—
282C2×D3—121no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—22no—
286C3×C4—34no—
287C3×C4—41no—
288C3×C4—62no—
289C3×C4—62no—
290C3×C4D3×D412yes—
291C3×C4C2×#6712yes—
292Dic3—22no—
293Dic3—22no—
294Dic3—62no—
295Dic3—61no—
296Dic3—61no—
297Dic3—61no—
298Dic3—61no—
299Dic3—61no—
300Dic3—61no—
301Dic3—121no—
302Dic3—121no—
303Dic3C2×#6714yes—
304C32#54811yescharacteristic, complemented, Sylow
305C23—32no—
306C23—91no—
307C23C2×D3211yescharacteristic
308Q8—34no—
309Q8—62no—
310Q8—184no—
311D4—64no—
312D4—62no—
313D4—184no—
314D4—182no—
315C2×C4—22no—
316C2×C4—32no—
317C2×C4—32no—
318C2×C4—32no—
319C2×C4—64no—
320C2×C4—91no—
321C2×C4—91no—
322C2×C4—91no—
323C2×C4—184no—
324C2×C4—182no—
325C2×C4C2×D3211yescharacteristic
326C2×C4C2×D3211yescharacteristic
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—42no—
331C2×C3—41no—
332C2×C3—41no—
333C2×C3—122no—
334C2×C3C2×D3×D411yescharacteristic
335C2×C3#100711yescharacteristic
336C2×C3D3×#4111yescharacteristic
337C2×C3D3×#4111yescharacteristic
338C2×C3#101011yescharacteristic
339C2×C3C22×#6711yescharacteristic
340D3—42no—
341D3—62no—
342D3—62no—
343D3—122no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—62no—
348C22—62no—
349C22—62no—
350C22—92no—
351C22—181no—
352C22—181no—
353C22C22×D3211yescharacteristic
354C4—21no—
355C4—34no—
356C4—62no—
357C4—62no—
358C4—181no—
359C4—181no—
360C4—181no—
361C4D3×#6712yes—
362C3—21no—
363C3#563311yescharacteristic, complemented
364C3#583111yescharacteristic, complemented
365C2—41no—
366C2—122no—
367C2—182no—
368C2#6429711yescharacteristic
369C2#6430611yescharacteristic
370C2C2×D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#548
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#235C2semidirect
2. C2×#3797C2semidirect
3. C4×#638C2semidirect
4. C3×#924C2semidirect
5. #63959C2semidirect
6. #63958C2semidirect
7. #63916C2semidirect
8. #63904C2semidirect
9. #63863C2semidirect
10. #63851C2semidirect
11. #63843C2semidirect
42. C2×#235D3semidirect
55. #924D3semidirect
80. C3×Dic6D4semidirect
148. C2×C3⋊D3#41semidirect
224. Dic6#67semidirect
304. C32#548semidirect
363. C3#5633semidirect
364. C3#5831semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#2352
C2×#37972
C4×#6382
C3×#9242
#639592
#639582
#639162
#639042
#638632
#638512
#638432
#58313
#56333