Finite Group Databasebeta

← #10594894

Subgroups — #10594894

Order 576.

635 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×#235C211yesmaximal, characteristic, complemented
2C2×#3799C211yesmaximal, characteristic, complemented
3#64139C211yesmaximal, characteristic, complemented
4C3×#924C211yesmaximal, characteristic, complemented
5#63971C212yesmaximal, complemented
6#63958C212yesmaximal, complemented
7#63909C211yesmaximal, characteristic, complemented
8D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
9#63888C212yesmaximal, complemented
10#63886C212yesmaximal, complemented
11#63856C211yesmaximal, characteristic, complemented
12#5833—31nomaximal
13#5629—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#638C2212yes—
16C2×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#67C2211yescharacteristic
18C2×C3×#67C2211yescharacteristic
19C3×#235—24no—
20C2×C3×D12C2211yescharacteristic
21C2×C3×C4×D3C2211yescharacteristic
22C2×C3×C4×D3C2211yescharacteristic
23C2×C3×Dic6C2211yescharacteristic
24C2×C3×Dic6C2211yescharacteristic
25C2×#626C2212yes—
26C2×#626C2212yes—
27C2×#624C2212yes—
28C2×D3×Dic3C2212yes—
29#3799—24no—
30#3752C2212yes—
31C4⋊(C3⋊Dic3)C2211yescharacteristic
32C3×#217C2212yes—
33C3×C4⋊Dic3C2211yescharacteristic
34C3×Dic3⋊C4C2212yes—
35#3724C2212yes—
36Dic3⋊Dic3C2212yes—
37#3722C2212yes—
38(C2×D3)⋊Dic3C2212yes—
39C2×C3×#41—31no—
40C2×#239—31no—
41C2×#235—31no—
42C2×#235D311yescharacteristic, complemented
43C3×C22⋊Q8—31no—
44#946—31no—
45#946—32no—
46#939—32no—
47#939—32no—
48#938—32no—
49#930—31no—
50#930—31no—
51#930—31no—
52#930—61no—
53C2×C4⋊Dic3—31no—
54#924—31no—
55#924D311yescharacteristic, complemented
56#896—32no—
57#895—32no—
58#895—31no—
59D3×C4⋊C4—31no—
60#884—32no—
61#884—32no—
62C23×C32C2311yescharacteristic
63C22×C3⋊D3C2312yes—
64C22×C3×D3C2311yescharacteristic
65C22×C3×D3C2311yescharacteristic
66C2×C32×C4—22no—
67C2×C32×C4C2311yescharacteristic
68C2×C32×C4C2311yescharacteristic
69#638—42no—
70C2×C3⋊Dic3C2312yes—
71C4⋊(C3⋊D3)—22no—
72C3×#67—22no—
73C3×#67—22no—
74C2×C3×Dic3C2312yes—
75C2×C3×Dic3C2311yescharacteristic
76C2×C3×Dic3C2312yes—
77C2×C3×Dic3C2311yescharacteristic
78C3×D12—22no—
79C3×C4×D3—22no—
80C3×C4×D3D414yescomplemented
81C3×Dic6—22no—
82C3×Dic6—22no—
83#626—24no—
84#626—42no—
85#624—24no—
86D3×Dic3—42no—
87#537—91noSylow
88C3×#41—64no—
89C2×C3×Q8—31no—
90C2×C3×Q8—31no—
91C2×C3×D4—31no—
92C2×C3×D4—31no—
93C2×C3×D4—31no—
94C22×C3×C4—21no—
95C22×C3×C4—31no—
96C22×C3×C4—31no—
97C22×C3×C4C2×D311yescharacteristic
98C22×C3×C4C2×D311yescharacteristic
99C2×#67—32no—
100C2×#67—32no—
101C2×#67—32no—
102C2×#67—32no—
103C2×#67—62no—
104C2×#67C2×D311yescharacteristic
105C2×#67C2×D311yescharacteristic
106C22×Dic3—32no—
107#239—64no—
108#235—24no—
109#235—64no—
110C2×D12—32no—
111C2×D12—31no—
112C2×D12—31no—
113C2×D12—32no—
114C2×D12—61no—
115C2×D12C2×D311yescharacteristic
116C2×C4×D3—32no—
117C2×C4×D3—32no—
118C2×C4×D3—32no—
119C2×C4×D3C2×D311yescharacteristic
120C2×C4×D3C2×D311yescharacteristic
121C2×Dic6C2×D311yescharacteristic
122C2×Dic6C2×D311yescharacteristic
123C3×C4⋊C4—31no—
124C3×C4⋊C4—32no—
125C3×C22⋊C4—32no—
126#217—32no—
127#217C2×D312yes—
128(C2×D3)⋊C4—32no—
129(C2×D3)⋊C4—32no—
130(C2×D3)⋊C4—32no—
131(C2×D3)⋊C4—32no—
132(C2×D3)⋊C4—32no—
133(C2×D3)⋊C4—62no—
134C4⋊Dic3—31no—
135C4⋊Dic3—31no—
136C4⋊Dic3—32no—
137C4⋊Dic3—61no—
138C4⋊Dic3C2×D311yescharacteristic
139Dic3⋊C4—32no—
140Dic3⋊C4—32no—
141Dic3⋊C4—32no—
142Dic3⋊C4C2×D312yes—
143C22×C32—21no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32C2411yescharacteristic
147C2×C3⋊D3—22no—
148C2×C3⋊D3—22no—
149C2×C3⋊D3—22no—
150C2×C3×D3—21no—
151C2×C3×D3—21no—
152C2×C3×D3—21no—
153C2×C3×D3—21no—
154C2×C3×D3—21no—
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—31no—
166C22⋊Q8—92no—
167C22⋊Q8—91no—
168#121—92no—
169#121—92no—
170#121—91no—
171#121—92no—
172#121—91no—
173C2×C4⋊C4—91no—
174C23×C3—21no—
175C23×C3—31no—
176C23×C3—31no—
177C23×C3C22×D311yescharacteristic
178C23×C3C22×D311yescharacteristic
179C22×D3—32no—
180C22×D3—32no—
181C22×D3—62no—
182C22×D3C22×D311yescharacteristic
183C22×D3C22×D311yescharacteristic
184C3×Q8—62no—
185C3×Q8—62no—
186C3×D4—62no—
187C3×D4—62no—
188C3×D4—62no—
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—32no—
198C2×C3×C4—42no—
199C2×C3×C4—62no—
200C2×C3×C4C22×D311yescharacteristic
201C2×C3×C4C22×D311yescharacteristic
202C2×C3×C4C22×D311yescharacteristic
203C2×C3×C4C22×D311yescharacteristic
204#67—22no—
205#67—22no—
206#67—64no—
207#67—122no—
208#67—122no—
209#67—122no—
210#67—124no—
211C2×Dic3—32no—
212C2×Dic3—32no—
213C2×Dic3—62no—
214C2×Dic3—122no—
215C2×Dic3C22×D311yescharacteristic
216C2×Dic3C22×D312yes—
217C2×Dic3C22×D312yes—
218C2×Dic3C22×D311yescharacteristic
219D12—22no—
220D12—64no—
221D12—62no—
222D12—62no—
223D12—122no—
224D12—122no—
225C4×D3—22no—
226C4×D3—64no—
227C4×D3—64no—
228C4×D3—122no—
229C4×D3#6714yescomplemented
230Dic6—22no—
231Dic6—22no—
232C2×C32—41no—
233C2×C32C22×D411yescharacteristic
234C2×C32H(5,𝔽2)11yescharacteristic
235C2×C322−1+411yescharacteristic
236C3⋊D3—42no—
237C3×D3—22no—
238C3×D3—41no—
239#41—64no—
240#41—184no—
241C2×Q8—31no—
242C2×Q8—31no—
243C2×D4—31no—
244C2×D4—31no—
245C2×D4—31no—
246C2×D4—92no—
247C2×D4—92no—
248C2×D4—92no—
249C2×D4—91no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—92no—
253C22×C4—92no—
254C22×C4D3211yescharacteristic
255C4⋊C4—31no—
256C4⋊C4—32no—
257C4⋊C4—92no—
258C4⋊C4—92no—
259C4⋊C4—91no—
260C22⋊C4—32no—
261C22⋊C4—92no—
262C22⋊C4—92no—
263C22⋊C4—92no—
264C22×C3—22no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—32no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—61no—
276C22×C3—61no—
277C22×C3—61no—
278C22×C3—61no—
279C22×C3—61no—
280C22×C3C23×D311yescharacteristic
281C22×C3C23×D311yescharacteristic
282C2×D3—21no—
283C2×D3—21no—
284C2×D3—21no—
285C2×D3—21no—
286C2×D3—21no—
287C2×D3—62no—
288C2×D3—62no—
289C2×D3—62no—
290C2×D3—62no—
291C2×D3—62no—
292C2×D3—62no—
293C2×D3—122no—
294C2×D3—122no—
295C2×D3—122no—
296C2×D3C2×#6712yes—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—22no—
300C3×C4—32no—
301C3×C4—41no—
302C3×C4—61no—
303C3×C4—62no—
304C3×C4—62no—
305C3×C4D3×D412yes—
306C3×C4C2×#6712yes—
307Dic3—22no—
308Dic3—22no—
309Dic3—21no—
310Dic3—62no—
311Dic3—62no—
312Dic3—122no—
313Dic3C2×#6712yes—
314C32#53711yescharacteristic, complemented, Sylow
315C23—31no—
316C23—31no—
317C23—92no—
318C23C2×D3211yescharacteristic
319Q8—62no—
320Q8—62no—
321D4—62no—
322D4—62no—
323D4—62no—
324D4—184no—
325D4—182no—
326D4—362no—
327D4—362no—
328C2×C4—22no—
329C2×C4—31no—
330C2×C4—34no—
331C2×C4—32no—
332C2×C4—32no—
333C2×C4—31no—
334C2×C4—62no—
335C2×C4—92no—
336C2×C4—184no—
337C2×C4—362no—
338C2×C4C2×D3211yescharacteristic
339C2×C4C2×D3211yescharacteristic
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—42no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—62no—
347C2×C3—121no—
348C2×C3C2×D3×D411yescharacteristic
349C2×C3#100711yescharacteristic
350C2×C3#100911yescharacteristic
351C2×C3#100911yescharacteristic
352C2×C3#101011yescharacteristic
353C2×C3C22×#6711yescharacteristic
354D3—22no—
355D3—41no—
356D3—122no—
357D3—122no—
358D3—242no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—32no—
363C22—61no—
364C22—61no—
365C22—61no—
366C22—61no—
367C22—61no—
368C22—182no—
369C22—182no—
370C22—182no—
371C22C22×D3211yescharacteristic
372C4—21no—
373C4—32no—
374C4—62no—
375C4—61no—
376C4—62no—
377C4—182no—
378C4D3×#6712yes—
379C3—21no—
380C3#562911yescharacteristic, complemented
381C3#583311yescharacteristic, complemented
382C2—41no—
383C2—62no—
384C2—121no—
385C2—362no—
386C2#6429711yescharacteristic
387C2#6430811yescharacteristic
388C2C2×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×#3799C2semidirect
3. #64139C2semidirect
4. C3×#924C2semidirect
5. #63971C2semidirect
6. #63958C2semidirect
7. #63909C2semidirect
8. D3×C4⋊Dic3C2semidirect
9. #63888C2semidirect
10. #63886C2semidirect
11. #63856C2semidirect
42. C2×#235D3semidirect
55. #924D3semidirect
80. C3×C4×D3D4semidirect
229. C4×D3#67semidirect
314. C32#537semidirect
380. C3#5629semidirect
381. 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×#37992
#641392
C3×#9242
#639712
#639582
#639092
D3×C4⋊Dic32
#638882
#638862
#638562
#58333
#56293