Finite Group Databasebeta

← #10595004

Subgroups — #10595004

Order 576.

625 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#235C211yesmaximal, characteristic, complemented
2C2×#3795C211yesmaximal, characteristic, complemented
3#64133C211yesmaximal, characteristic, complemented
4C3×#930C211yesmaximal, characteristic, complemented
5#63970C212yesmaximal, complemented
6#63970C212yesmaximal, complemented
7#63908C211yesmaximal, characteristic, complemented
8D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
9#63887C212yesmaximal, complemented
10#63885C212yesmaximal, complemented
11#63857C211yesmaximal, characteristic, complemented
12#5833—31nomaximal
13#5608—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#634C2211yescharacteristic
16C2×C3×#67C2211yescharacteristic
17C2×C3×#67C2211yescharacteristic
18C2×C3×#67C2212yes—
19C3×#235—24no—
20C2×C3×D12C2211yescharacteristic
21C2×C3×D12C2211yescharacteristic
22C2×C3×C4×D3C2211yescharacteristic
23C2×C3×C4×D3C2211yescharacteristic
24C2×C3×Dic6C2211yescharacteristic
25C2×#625C2212yes—
26C2×#625C2212yes—
27C2×D3×Dic3C2212yes—
28C2×D3×Dic3C2212yes—
29#3795—24no—
30#3757C2212yes—
31C4⋊(C3⋊Dic3)C2211yescharacteristic
32#3750C2212yes—
33C3×(C2×D3)⋊C4C2212yes—
34C3×C4⋊Dic3C2211yescharacteristic
35Dic3⋊Dic3C2212yes—
36Dic3⋊Dic3C2212yes—
37(C2×D3)⋊Dic3C2212yes—
38(C2×D3)⋊Dic3C2212yes—
39C2×C3×#41—31no—
40C2×#237—31no—
41C2×#235—31no—
42C2×#235D311yescharacteristic, complemented
43C3×#121—31no—
44#946—32no—
45#939—32no—
46#939—32no—
47#939—32no—
48#938—32no—
49#938—32no—
50#938—31no—
51#930—31no—
52#930D311yescharacteristic, complemented
53C2×C4⋊Dic3—31no—
54#924—31no—
55#924—31no—
56#924—31no—
57#924—61no—
58#897—31no—
59D3×C4⋊C4—31no—
60#884—32no—
61#878—32no—
62C23×C32C2311yescharacteristic
63C22×C3×D3C2311yescharacteristic
64C22×C3×D3C2312yes—
65C22×C3×D3C2311yescharacteristic
66C2×C32×C4—22no—
67C2×C32×C4C2311yescharacteristic
68C2×C32×C4C2311yescharacteristic
69C2×C3⋊Dic3C2312yes—
70C2×C3⋊Dic3C2312yes—
71#634—22no—
72C3×#67—22no—
73C3×#67—22no—
74C3×#67—42no—
75C2×C3×Dic3C2311yescharacteristic
76C2×C3×Dic3C2312yes—
77C2×C3×Dic3C2311yescharacteristic
78C3×D12—22no—
79C3×D12—22no—
80C3×C4×D3—22no—
81C3×C4×D3D414yescomplemented
82C3×Dic6—22no—
83#625—24no—
84#625—42no—
85D3×Dic3—24no—
86D3×Dic3—42no—
87#537—91noSylow
88C3×#41—64no—
89C2×C3×Q8—31no—
90C2×C3×D4—31no—
91C2×C3×D4—31no—
92C2×C3×D4—32no—
93C2×C3×D4—31no—
94C2×C3×D4—31no—
95C22×C3×C4—21no—
96C22×C3×C4—31no—
97C22×C3×C4—31no—
98C22×C3×C4C2×D311yescharacteristic
99C22×C3×C4C2×D311yescharacteristic
100C2×#67—32no—
101C2×#67—32no—
102C2×#67—32no—
103C2×#67—32no—
104C2×#67C2×D311yescharacteristic
105C2×#67C2×D311yescharacteristic
106C2×#67C2×D312yes—
107C22×Dic3—32no—
108C22×Dic3—32no—
109#237—64no—
110#235—24no—
111#235—64no—
112C2×D12C2×D311yescharacteristic
113C2×D12C2×D311yescharacteristic
114C2×C4×D3—32no—
115C2×C4×D3—32no—
116C2×C4×D3C2×D311yescharacteristic
117C2×C4×D3C2×D311yescharacteristic
118C2×Dic6—31no—
119C2×Dic6—31no—
120C2×Dic6—61no—
121C2×Dic6C2×D311yescharacteristic
122C3×C4⋊C4—31no—
123C3×C22⋊C4—32no—
124#217—32no—
125#217—32no—
126#217—32no—
127#217—32no—
128#217—62no—
129(C2×D3)⋊C4—32no—
130(C2×D3)⋊C4—32no—
131(C2×D3)⋊C4C2×D312yes—
132C4⋊Dic3—31no—
133C4⋊Dic3—31no—
134C4⋊Dic3—32no—
135C4⋊Dic3—32no—
136C4⋊Dic3—61no—
137C4⋊Dic3C2×D311yescharacteristic
138Dic3⋊C4—32no—
139Dic3⋊C4—32no—
140Dic3⋊C4—32no—
141Dic3⋊C4—32no—
142Dic3⋊C4—62no—
143C22×C32—21no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32C2411yescharacteristic
147C2×C3×D3—21no—
148C2×C3×D3—21no—
149C2×C3×D3—22no—
150C2×C3×D3—21no—
151C2×C3×D3—21no—
152C2×C3×D3—22no—
153C2×C3×D3—21no—
154C2×C3×D3—22no—
155C2×C3×D3C2×D412yes—
156C32×C4—21no—
157C32×C4C2×D412yes—
158C3⋊Dic3—22no—
159C3⋊Dic3—22no—
160C3×Dic3—21no—
161C3×Dic3—22no—
162C3×Dic3C2×D412yes—
163C2×#41—31no—
164C2×#41—91no—
165C22⋊Q8—92no—
166C22⋊Q8—91no—
167C22⋊Q8—91no—
168#121—31no—
169#121—92no—
170#121—91no—
171#121—92no—
172#121—92no—
173C2×C4⋊C4—91no—
174C23×C3—21no—
175C23×C3—31no—
176C23×C3—31no—
177C23×C3—32no—
178C23×C3C22×D311yescharacteristic
179C23×C3C22×D311yescharacteristic
180C22×D3C22×D311yescharacteristic
181C22×D3C22×D312yes—
182C22×D3C22×D311yescharacteristic
183C3×Q8—62no—
184C3×D4—62no—
185C3×D4—62no—
186C3×D4—62no—
187C3×D4—62no—
188C3×D4—122no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—31no—
194C2×C3×C4—34no—
195C2×C3×C4—31no—
196C2×C3×C4—32no—
197C2×C3×C4—42no—
198C2×C3×C4—62no—
199C2×C3×C4C22×D311yescharacteristic
200C2×C3×C4C22×D311yescharacteristic
201C2×C3×C4C22×D311yescharacteristic
202C2×C3×C4C22×D311yescharacteristic
203#67—22no—
204#67—22no—
205#67—42no—
206#67—64no—
207#67—64no—
208#67—122no—
209#67—122no—
210C2×Dic3—32no—
211C2×Dic3—32no—
212C2×Dic3—32no—
213C2×Dic3—32no—
214C2×Dic3—62no—
215C2×Dic3—62no—
216C2×Dic3—64no—
217C2×Dic3—122no—
218C2×Dic3C22×D312yes—
219C2×Dic3C22×D311yescharacteristic
220C2×Dic3C22×D311yescharacteristic
221D12—22no—
222D12—22no—
223C4×D3—22no—
224C4×D3—64no—
225C4×D3—122no—
226C4×D3#6714yescomplemented
227Dic6—22no—
228Dic6—62no—
229Dic6—62no—
230Dic6—122no—
231C2×C32—41no—
232C2×C32C22×D411yescharacteristic
233C2×C32H(5,𝔽2)11yescharacteristic
234C2×C322−1+411yescharacteristic
235C3×D3—22no—
236C3×D3—41no—
237C3×D3—42no—
238#41—64no—
239#41—184no—
240C2×Q8—31no—
241C2×Q8—91no—
242C2×D4—31no—
243C2×D4—31no—
244C2×D4—32no—
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—92no—
248C2×D4—92no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—92no—
252C22×C4—92no—
253C22×C4D3211yescharacteristic
254C4⋊C4—31no—
255C4⋊C4—92no—
256C4⋊C4—92no—
257C4⋊C4—91no—
258C4⋊C4—92no—
259C22⋊C4—32no—
260C22⋊C4—92no—
261C22⋊C4—92no—
262C22⋊C4—92no—
263C22×C3—22no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—32no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—61no—
275C22×C3—61no—
276C22×C3—61no—
277C22×C3—61no—
278C22×C3—62no—
279C22×C3—62no—
280C22×C3—61no—
281C22×C3—62no—
282C22×C3C23×D311yescharacteristic
283C22×C3C23×D311yescharacteristic
284C2×D3—21no—
285C2×D3—21no—
286C2×D3—22no—
287C2×D3—22no—
288C2×D3—21no—
289C2×D3—21no—
290C2×D3—22no—
291C2×D3—21no—
292C2×D3C2×#6712yes—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—22no—
296C3×C4—32no—
297C3×C4—41no—
298C3×C4—61no—
299C3×C4—62no—
300C3×C4D3×D412yes—
301C3×C4C2×#6712yes—
302Dic3—22no—
303Dic3—21no—
304Dic3—62no—
305Dic3—62no—
306Dic3—62no—
307Dic3—62no—
308Dic3—122no—
309Dic3—122no—
310Dic3C2×#6712yes—
311C32#53711yescharacteristic, complemented, Sylow
312C23—31no—
313C23—32no—
314C23—31no—
315C23C2×D3211yescharacteristic
316Q8—62no—
317Q8—182no—
318D4—62no—
319D4—62no—
320D4—62no—
321D4—62no—
322D4—122no—
323D4—184no—
324D4—362no—
325C2×C4—22no—
326C2×C4—31no—
327C2×C4—34no—
328C2×C4—32no—
329C2×C4—31no—
330C2×C4—62no—
331C2×C4—92no—
332C2×C4—92no—
333C2×C4—184no—
334C2×C4—362no—
335C2×C4C2×D3211yescharacteristic
336C2×C4C2×D3211yescharacteristic
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—42no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—62no—
344C2×C3—122no—
345C2×C3—121no—
346C2×C3C2×D3×D411yescharacteristic
347C2×C3#100411yescharacteristic
348C2×C3#100911yescharacteristic
349C2×C3#101011yescharacteristic
350C2×C3#101011yescharacteristic
351C2×C3C22×#6711yescharacteristic
352D3—22no—
353D3—41no—
354D3—42no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—32no—
359C22—61no—
360C22—61no—
361C22—62no—
362C22—61no—
363C22—62no—
364C22—61no—
365C22—61no—
366C22—62no—
367C22C22×D3211yescharacteristic
368C4—21no—
369C4—32no—
370C4—62no—
371C4—61no—
372C4—182no—
373C4—182no—
374C4D3×#6712yes—
375C3—21no—
376C3#560811yescharacteristic, complemented
377C3#583311yescharacteristic, complemented
378C2—41no—
379C2—62no—
380C2—122no—
381C2—121no—
382C2#6429811yescharacteristic
383C2#6430711yescharacteristic
384C2C2×D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#537
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×#3795C2semidirect
3. #64133C2semidirect
4. C3×#930C2semidirect
5. #63970C2semidirect
6. #63970C2semidirect
7. #63908C2semidirect
8. D3×C4⋊Dic3C2semidirect
9. #63887C2semidirect
10. #63885C2semidirect
11. #63857C2semidirect
42. C2×#235D3semidirect
52. #930D3semidirect
81. C3×C4×D3D4semidirect
226. C4×D3#67semidirect
311. C32#537semidirect
376. C3#5608semidirect
377. C3#5833semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#2352
C2×#37952
#641332
C3×#9302
#639702
#639082
D3×C4⋊Dic32
#638872
#638852
#638572
#58333
#56083