Finite Group Databasebeta

← #10595158

Subgroups — #10595158

Order 576.

671 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×#237C211yesmaximal, characteristic, complemented
2C2×#3796C211yesmaximal, characteristic, complemented
3#64147C211yesmaximal, characteristic, complemented
4C3×#940C211yesmaximal, characteristic, complemented
5#63979C212yesmaximal, complemented
6#63978C212yesmaximal, complemented
7Dic3×#67C212yesmaximal, complemented
8#63970C212yesmaximal, complemented
9#63908C211yesmaximal, characteristic, complemented
10C4×#626C211yesmaximal, characteristic, complemented
11Dic3×Dic6C211yesmaximal, characteristic, complemented
12#5834—31nomaximal
13#5718—31nomaximal
14C2×C32×D4C2211yescharacteristic
15C2×#638C2212yes—
16C2×C4×C3⋊D3C2211yescharacteristic
17C2×C3×#67C2212yes—
18C2×C3×#67C2212yes—
19C2×C3×#67C2212yes—
20C22×C3×Dic3C2212yes—
21C3×#237—24no—
22C2×C3×D12C2211yescharacteristic
23C2×C3×C4×D3C2211yescharacteristic
24C2×C3×Dic6C2211yescharacteristic
25C2×#626C2212yes—
26C2×#626C2211yescharacteristic
27C2×#625C2212yes—
28C2×D3×Dic3C2212yes—
29#3796—24no—
30#3757C2212yes—
31C4⋊(C3⋊Dic3)C2211yescharacteristic
32C3×C4×Dic3C2211yescharacteristic
33Dic3⋊Dic3C2212yes—
34Dic3⋊Dic3C2211yescharacteristic
35#3722C2211yescharacteristic
36(C2×D3)⋊Dic3C2211yescharacteristic
37(C2×D3)⋊Dic3C2212yes—
38Dic32C2212yes—
39C2×C3×#41—31no—
40C2×#239—31no—
41C2×#237—31no—
42C2×#237D311yescharacteristic, complemented
43C3×#127—31no—
44Q8×Dic3—31no—
45#940—32no—
46#940—32no—
47#940D311yescharacteristic, complemented
48#939—32no—
49#939—32no—
50#938—31no—
51#938—31no—
52#938—32no—
53#938—31no—
54#938—61no—
55D4×Dic3—32no—
56#930—32no—
57#930—31no—
58C4×#67—32no—
59C4×#67—31no—
60C4×D12—31no—
61C4×Dic6—31no—
62C23×C32C2312yes—
63C22×C3⋊D3C2311yescharacteristic
64C22×C3×D3C2312yes—
65C22×C3×D3C2311yescharacteristic
66C32×D4—22no—
67C2×C32×C4C2311yescharacteristic
68#638—42no—
69C2×C3⋊Dic3C2312yes—
70C2×C3⋊Dic3C2311yescharacteristic
71C4×C3⋊D3—22no—
72C3×#67—24no—
73C3×#67—24no—
74C3×#67—42no—
75C2×C3×Dic3—24no—
76C2×C3×Dic3C2311yescharacteristic
77C2×C3×Dic3C2311yescharacteristic
78C2×C3×Dic3C2312yes—
79C2×C3×Dic3C2311yescharacteristic
80C3×D12—22no—
81C3×C4×D3—22no—
82C3×Dic6D414yescomplemented
83#626—24no—
84#626—22no—
85#625—42no—
86D3×Dic3—24no—
87#550—91noSylow
88C3×#41—64no—
89C2×C3×Q8—31no—
90C2×C3×D4—21no—
91C2×C3×D4—32no—
92C2×C3×D4—32no—
93C2×C3×D4—32no—
94C2×C3×D4—31no—
95C2×C3×D4C2×D311yescharacteristic
96C2×C3×D4C2×D311yescharacteristic
97C22×C3×C4—32no—
98C22×C3×C4—31no—
99C2×#67—32no—
100C2×#67—32no—
101C2×#67—32no—
102C2×#67—32no—
103C2×#67—31no—
104C2×#67—32no—
105C2×#67—62no—
106C2×#67C2×D312yes—
107C2×#67C2×D312yes—
108C2×#67C2×D312yes—
109C22×Dic3—32no—
110C22×Dic3C2×D312yes—
111#239—64no—
112#237—24no—
113#237—64no—
114C2×D12—31no—
115C2×D12—32no—
116C2×D12C2×D311yescharacteristic
117C2×C4×D3—32no—
118C2×C4×D3—31no—
119C2×C4×D3—31no—
120C2×C4×D3—61no—
121C2×C4×D3C2×D311yescharacteristic
122C2×Dic6C2×D311yescharacteristic
123C3×C42—31no—
124#217—32no—
125#217—32no—
126#217—31no—
127#217—32no—
128#217—62no—
129(C2×D3)⋊C4—31no—
130(C2×D3)⋊C4—32no—
131(C2×D3)⋊C4—31no—
132(C2×D3)⋊C4—31no—
133C4⋊Dic3—31no—
134C4⋊Dic3—31no—
135C4⋊Dic3—31no—
136C4⋊Dic3—32no—
137C4⋊Dic3—61no—
138Dic3⋊C4—32no—
139Dic3⋊C4—31no—
140C4×Dic3—32no—
141C4×Dic3—32no—
142C4×Dic3C2×D311yescharacteristic
143C22×C32—22no—
144C22×C32—22no—
145C22×C32—22no—
146C22×C32C2411yescharacteristic
147C2×C3⋊D3—21no—
148C2×C3⋊D3—21no—
149C2×C3⋊D3—21no—
150C2×C3×D3—22no—
151C2×C3×D3—21no—
152C2×C3×D3—21no—
153C2×C3×D3—22no—
154C2×C3×D3—21no—
155C2×C3×D3—22no—
156C32×C4C2×D412yes—
157C3⋊Dic3—21no—
158C3⋊Dic3—22no—
159C3×Dic3—21no—
160C3×Dic3—21no—
161C3×Dic3C2×D414yes—
162C3×Dic3#4112yescomplemented
163C2×#41—31no—
164C2×#41—91no—
165#127—31no—
166#127—92no—
167#121—91no—
168#121—92no—
169#121—92no—
170#121—91no—
171C4×Q8—91no—
172C4×D4—91no—
173C4×D4—92no—
174C23×C3—22no—
175C23×C3—32no—
176C23×C3—31no—
177C23×C3C22×D312yes—
178C23×C3C22×D312yes—
179C22×D3—31no—
180C22×D3—31no—
181C22×D3—61no—
182C22×D3C22×D312yes—
183C22×D3C22×D311yescharacteristic
184C3×Q8—34no—
185C3×D4—22no—
186C3×D4—22no—
187C3×D4—42no—
188C3×D4—64no—
189C3×D4—64no—
190C3×D4—62no—
191C3×D4—122no—
192C2×C3×C4—21no—
193C2×C3×C4—31no—
194C2×C3×C4—32no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—64no—
198C2×C3×C4—62no—
199C2×C3×C4C22×D311yescharacteristic
200C2×C3×C4C22×D311yescharacteristic
201#67—24no—
202#67—24no—
203#67—42no—
204#67—62no—
205#67—64no—
206#67—122no—
207#67—122no—
208#67—122no—
209#67—122no—
210#67—124no—
211C2×Dic3—24no—
212C2×Dic3—32no—
213C2×Dic3—32no—
214C2×Dic3—31no—
215C2×Dic3—31no—
216C2×Dic3—62no—
217C2×Dic3—61no—
218C2×Dic3—64no—
219C2×Dic3C22×D311yescharacteristic
220C2×Dic3C22×D311yescharacteristic
221C2×Dic3C22×D311yescharacteristic
222C2×Dic3C22×D312yes—
223D12—22no—
224D12—62no—
225D12—64no—
226C4×D3—22no—
227C4×D3—64no—
228C4×D3—62no—
229C4×D3—62no—
230C4×D3—122no—
231Dic6#6714yescomplemented
232C2×C32—42no—
233C2×C32C22×D411yescharacteristic
234C2×C32C2×#4111yescharacteristic
235C2×C32H(5,𝔽2)11yescharacteristic
236C3⋊D3—41no—
237C3×D3—42no—
238C3×D3—41no—
239#41—64no—
240#41—184no—
241C2×Q8—31no—
242C2×D4—32no—
243C2×D4—32no—
244C2×D4—32no—
245C2×D4—31no—
246C2×D4—91no—
247C2×D4—92no—
248C2×D4—92no—
249C2×D4—92no—
250C2×D4D3211yescharacteristic
251C22×C4—32no—
252C22×C4—31no—
253C22×C4—92no—
254C22×C4—91no—
255C4⋊C4—91no—
256C4⋊C4—92no—
257C4⋊C4—91no—
258C22⋊C4—91no—
259C22⋊C4—91no—
260C22⋊C4—92no—
261C22⋊C4—92no—
262C42—31no—
263C42—92no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—22no—
267C22×C3—22no—
268C22×C3—22no—
269C22×C3—22no—
270C22×C3—21no—
271C22×C3—42no—
272C22×C3—42no—
273C22×C3—42no—
274C22×C3—61no—
275C22×C3—61no—
276C22×C3—61no—
277C22×C3—62no—
278C22×C3—62no—
279C22×C3—62no—
280C22×C3C23×D311yescharacteristic
281C22×C3C23×D311yescharacteristic
282C2×D3—22no—
283C2×D3—22no—
284C2×D3—22no—
285C2×D3—21no—
286C2×D3—21no—
287C2×D3—21no—
288C2×D3—62no—
289C2×D3—61no—
290C2×D3—61no—
291C2×D3—61no—
292C2×D3—61no—
293C2×D3—61no—
294C2×D3—61no—
295C2×D3—121no—
296C2×D3—121no—
297C3×C4—22no—
298C3×C4—34no—
299C3×C4—32no—
300C3×C4—61no—
301C3×C4—61no—
302C3×C4D3×D412yes—
303C3×C4C2×#6712yes—
304Dic3—21no—
305Dic3—21no—
306Dic3—61no—
307Dic3—62no—
308Dic3—62no—
309Dic3—61no—
310Dic3—121no—
311Dic3—122no—
312Dic3#23712yescomplemented
313Dic3C2×#6714yes—
314C32#55011yescharacteristic, complemented, Sylow
315C23—31no—
316C23—32no—
317C23—91no—
318C23C2×D3212yes—
319Q8—34no—
320D4—22no—
321D4—64no—
322D4—64no—
323D4—62no—
324D4—122no—
325D4—184no—
326D4—182no—
327D4—362no—
328D4—362no—
329C2×C4—31no—
330C2×C4—32no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—64no—
334C2×C4—62no—
335C2×C4—92no—
336C2×C4—91no—
337C2×C4—184no—
338C2×C4—182no—
339C2×C4C2×D3211yescharacteristic
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—42no—
344C2×C3—42no—
345C2×C3—82no—
346C2×C3—121no—
347C2×C3—122no—
348C2×C3C2×D3×D411yescharacteristic
349C2×C3C2×#23711yescharacteristic
350C2×C3#100411yescharacteristic
351C2×C3D3×#4111yescharacteristic
352C2×C3#100911yescharacteristic
353C2×C3C22×#6711yescharacteristic
354D3—42no—
355D3—41no—
356D3—122no—
357D3—121no—
358D3—121no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—62no—
363C22—61no—
364C22—62no—
365C22—61no—
366C22—61no—
367C22—62no—
368C22—181no—
369C22—181no—
370C22—181no—
371C22C22×D3211yescharacteristic
372C4—32no—
373C4—34no—
374C4—61no—
375C4—61no—
376C4—182no—
377C4—181no—
378C4D3×#6712yes—
379C3—21no—
380C3#571811yescharacteristic, complemented
381C3#583411yescharacteristic, complemented
382C2—42no—
383C2—122no—
384C2—121no—
385C2—361no—
386C2D3×#23711yescharacteristic
387C2#6431411yescharacteristic
388C2C2×D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#550
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#237C2semidirect
2. C2×#3796C2semidirect
3. #64147C2semidirect
4. C3×#940C2semidirect
5. #63979C2semidirect
6. #63978C2semidirect
7. Dic3×#67C2semidirect
8. #63970C2semidirect
9. #63908C2semidirect
10. C4×#626C2semidirect
11. Dic3×Dic6C2semidirect
42. C2×#237D3semidirect
47. #940D3semidirect
82. C3×Dic6D4semidirect
162. C3×Dic3#41semidirect
231. Dic6#67semidirect
312. Dic3#237semidirect
314. C32#550semidirect
380. C3#5718semidirect
381. C3#5834semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#2372
C2×#37962
#641472
C3×#9402
#639792
#639782
Dic3×#672
#639702
#639082
C4×#6262
Dic3×Dic62
#58343
#57183