Finite Group Databasebeta

← #10594522

Subgroups — #10594522

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×#3798C211yesmaximal, characteristic, complemented
2C2×#3795C211yesmaximal, characteristic, complemented
3#64106C211yesmaximal, characteristic, complemented
4C3×#895C211yesmaximal, characteristic, complemented
5C3×D3×C4⋊C4C211yesmaximal, characteristic, complemented
6#63916C211yesmaximal, characteristic, complemented
7#63908C212yesmaximal, complemented
8#63906C212yesmaximal, complemented
9#63893C212yesmaximal, complemented
10#63886C212yesmaximal, complemented
11#63857C211yesmaximal, characteristic, complemented
12#5608—31nomaximal
13#5510—31nomaximal
14C2×C4×C3⋊D3C2211yescharacteristic
15C2×#634C2211yescharacteristic
16C2×C3×D12C2211yescharacteristic
17C2×C3×D12C2212yes—
18C2×C3×C4×D3C2211yescharacteristic
19C2×C3×C4×D3C2212yes—
20C2×C3×C4×D3C2211yescharacteristic
21C2×#627C2211yescharacteristic
22C2×#626C2211yescharacteristic
23C2×#626C2211yescharacteristic
24C2×#626C2212yes—
25C2×#625C2212yes—
26C2×#625C2211yescharacteristic
27C2×D3×Dic3C2212yes—
28#3798—24no—
29#3795—24no—
30C32×C4⋊C4C2211yescharacteristic
31#3752C2212yes—
32C4⋊(C3⋊Dic3)C2212yes—
33C3×(C2×D3)⋊C4C2212yes—
34C3×C4⋊Dic3C2211yescharacteristic
35C3×Dic3⋊C4C2212yes—
36Dic3⋊Dic3C2212yes—
37(C2×D3)⋊Dic3C2212yes—
38(C2×D3)⋊Dic3C2212yes—
39C2×#237—31no—
40C2×#235—31no—
41C2×#235—31no—
42C2×#235—31no—
43C3×#121—31no—
44C2×C3×C4⋊C4—31no—
45#939—32no—
46#938—32no—
47#938—31no—
48#930—32no—
49#930—32no—
50#924—31no—
51#897—31no—
52#897—31no—
53#897—31no—
54#897—32no—
55#897—61no—
56#895—32no—
57#895—31no—
58#895D311yescharacteristic, complemented
59D3×C4⋊C4D311yescharacteristic, complemented
60#884—32no—
61#878—32no—
62C22×C3⋊D3C2311yescharacteristic
63C22×C3×D3C2311yescharacteristic
64C22×C3×D3C2312yes—
65C22×C3×D3C2311yescharacteristic
66C2×C32×C4C2312yes—
67C2×C32×C4C2311yescharacteristic
68C2×C3⋊Dic3C2311yescharacteristic
69C2×C3⋊Dic3C2312yes—
70C4×C3⋊D3—22no—
71#634—22no—
72C2×C3×Dic3C2312yes—
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2311yescharacteristic
75C3×D12—22no—
76C3×D12—42no—
77C3×C4×D3—22no—
78C3×C4×D3—42no—
79C3×C4×D3D414yescomplemented
80#627—22no—
81#626—22no—
82#626—22no—
83#626—42no—
84#625—24no—
85#625—22no—
86D3×Dic3—24no—
87#537—91noSylow
88C2×C3×D4—31no—
89C2×C3×D4—32no—
90C22×C3×C4—31no—
91C22×C3×C4—31no—
92C22×C3×C4—32no—
93C2×#67—32no—
94C2×#67—31no—
95C2×#67—32no—
96C2×#67—31no—
97C2×#67—31no—
98C2×#67—31no—
99C2×#67—32no—
100C22×Dic3—32no—
101#237—64no—
102#235—64no—
103#235—64no—
104#235—64no—
105C2×D12—31no—
106C2×D12—31no—
107C2×D12—32no—
108C2×D12C2×D311yescharacteristic
109C2×D12C2×D312yes—
110C2×C4×D3—32no—
111C2×C4×D3—31no—
112C2×C4×D3—31no—
113C2×C4×D3—61no—
114C2×C4×D3C2×D311yescharacteristic
115C2×C4×D3C2×D312yes—
116C2×C4×D3C2×D311yescharacteristic
117C2×Dic6—31no—
118C2×Dic6—31no—
119C2×Dic6—31no—
120C2×Dic6—31no—
121C2×Dic6—61no—
122C3×C4⋊C4—21no—
123C3×C4⋊C4—32no—
124C3×C4⋊C4—31no—
125C3×C4⋊C4C2×D311yescharacteristic
126C3×C4⋊C4C2×D311yescharacteristic
127C3×C22⋊C4—32no—
128#217—32no—
129#217—32no—
130(C2×D3)⋊C4—32no—
131(C2×D3)⋊C4—32no—
132(C2×D3)⋊C4—32no—
133(C2×D3)⋊C4—32no—
134(C2×D3)⋊C4—62no—
135(C2×D3)⋊C4C2×D312yes—
136C4⋊Dic3—32no—
137C4⋊Dic3—32no—
138C4⋊Dic3—32no—
139C4⋊Dic3—62no—
140C4⋊Dic3C2×D311yescharacteristic
141Dic3⋊C4—32no—
142Dic3⋊C4C2×D312yes—
143C22×C32C2411yescharacteristic
144C2×C3⋊D3—21no—
145C2×C3⋊D3—21no—
146C2×C3⋊D3—21no—
147C2×C3×D3—21no—
148C2×C3×D3—21no—
149C2×C3×D3—22no—
150C2×C3×D3—21no—
151C2×C3×D3—22no—
152C2×C3×D3—21no—
153C2×C3×D3—21no—
154C2×C3×D3—22no—
155C2×C3×D3C2×D412yes—
156C32×C4—22no—
157C32×C4C2×D412yes—
158C3⋊Dic3—22no—
159C3⋊Dic3—21no—
160C3×Dic3—21no—
161C3×Dic3—22no—
162C3×Dic3C2×D412yes—
163C2×#41—91no—
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—31no—
174C23×C3—31no—
175C23×C3—31no—
176C23×C3—32no—
177C22×D3—31no—
178C22×D3—31no—
179C22×D3—61no—
180C22×D3C22×D311yescharacteristic
181C22×D3C22×D312yes—
182C22×D3C22×D311yescharacteristic
183C3×D4—62no—
184C3×D4—122no—
185C2×C3×C4—21no—
186C2×C3×C4—22no—
187C2×C3×C4—31no—
188C2×C3×C4—34no—
189C2×C3×C4—31no—
190C2×C3×C4—32no—
191C2×C3×C4—62no—
192C2×C3×C4—122no—
193C2×C3×C4C22×D312yes—
194C2×C3×C4C22×D312yes—
195C2×C3×C4C22×D311yescharacteristic
196C2×C3×C4C22×D311yescharacteristic
197#67—62no—
198#67—62no—
199#67—64no—
200#67—62no—
201#67—64no—
202#67—62no—
203#67—122no—
204C2×Dic3—32no—
205C2×Dic3—31no—
206C2×Dic3—32no—
207C2×Dic3—31no—
208C2×Dic3—61no—
209C2×Dic3—62no—
210C2×Dic3—64no—
211C2×Dic3C22×D312yes—
212C2×Dic3C22×D311yescharacteristic
213C2×Dic3C22×D311yescharacteristic
214D12—22no—
215D12—42no—
216D12—62no—
217D12—62no—
218D12—122no—
219C4×D3—22no—
220C4×D3—42no—
221C4×D3—64no—
222C4×D3—62no—
223C4×D3—62no—
224C4×D3—122no—
225C4×D3D1214yescomplemented
226Dic6—62no—
227Dic6—62no—
228Dic6—62no—
229Dic6—62no—
230Dic6—122no—
231C2×C32C22×D411yescharacteristic
232C2×C32H(5,𝔽2)11yescharacteristic
233C2×C322−1+411yescharacteristic
234C3⋊D3—41no—
235C3×D3—22no—
236C3×D3—42no—
237C3×D3—41no—
238#41—184no—
239#41—184no—
240C2×Q8—91no—
241C2×Q8—91no—
242C2×D4—31no—
243C2×D4—32no—
244C2×D4—91no—
245C2×D4—91no—
246C2×D4—92no—
247C2×D4—92no—
248C2×D4—91no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—32no—
252C22×C4—92no—
253C22×C4—91no—
254C4⋊C4—32no—
255C4⋊C4—31no—
256C4⋊C4—92no—
257C4⋊C4—92no—
258C4⋊C4D3211yescharacteristic
259C22⋊C4—32no—
260C22⋊C4—92no—
261C22⋊C4—92no—
262C22⋊C4—92no—
263C22×C3—21no—
264C22×C3—32no—
265C22×C3—61no—
266C22×C3—61no—
267C22×C3—62no—
268C22×C3—61no—
269C22×C3—62no—
270C22×C3—61no—
271C22×C3—61no—
272C22×C3—62no—
273C22×C3C23×D311yescharacteristic
274C22×C3C23×D311yescharacteristic
275C2×D3—21no—
276C2×D3—21no—
277C2×D3—21no—
278C2×D3—22no—
279C2×D3—21no—
280C2×D3—22no—
281C2×D3—21no—
282C2×D3—22no—
283C2×D3—62no—
284C2×D3—61no—
285C2×D3—61no—
286C2×D3—61no—
287C2×D3—61no—
288C2×D3—61no—
289C2×D3—61no—
290C2×D3—121no—
291C2×D3—121no—
292C2×D3C2×D1212yes—
293C3×C4—22no—
294C3×C4—22no—
295C3×C4—22no—
296C3×C4—32no—
297C3×C4—42no—
298C3×C4—61no—
299C3×C4—62no—
300C3×C4C2×D1212yes—
301C3×C4D3×D412yes—
302Dic3—21no—
303Dic3—22no—
304Dic3—61no—
305Dic3—62no—
306Dic3—61no—
307Dic3—62no—
308Dic3—122no—
309Dic3—121no—
310Dic3C2×D1212yes—
311C32#53711yescharacteristic, complemented, Sylow
312C23—31no—
313C23—32no—
314C23—31no—
315C23—91no—
316Q8—182no—
317Q8—182no—
318D4—62no—
319D4—122no—
320D4—182no—
321D4—182no—
322D4—182no—
323D4—184no—
324D4—362no—
325C2×C4—31no—
326C2×C4—34no—
327C2×C4—31no—
328C2×C4—32no—
329C2×C4—62no—
330C2×C4—91no—
331C2×C4—92no—
332C2×C4—122no—
333C2×C4—184no—
334C2×C4—182no—
335C2×C4C2×D3212yes—
336C2×C4C2×D3211yescharacteristic
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—62no—
341C2×C3—122no—
342C2×C3—121no—
343C2×C3C22×D1211yescharacteristic
344C2×C3C2×D3×D411yescharacteristic
345C2×C3#100411yescharacteristic
346C2×C3#100411yescharacteristic
347C2×C3#100711yescharacteristic
348C2×C3#101011yescharacteristic
349D3—22no—
350D3—41no—
351D3—42no—
352D3—122no—
353D3—121no—
354D3—121no—
355C22—32no—
356C22—61no—
357C22—61no—
358C22—61no—
359C22—62no—
360C22—61no—
361C22—62no—
362C22—62no—
363C22—61no—
364C22—181no—
365C22—181no—
366C22—181no—
367C22C22×D3211yescharacteristic
368C4—22no—
369C4—32no—
370C4—61no—
371C4—62no—
372C4—182no—
373C4—181no—
374C4D3×D1212yes—
375C3—21no—
376C3#551011yescharacteristic, complemented
377C3#560811yescharacteristic, complemented
378C2—62no—
379C2—121no—
380C2—122no—
381C2—361no—
382C2C2×D3×D1211yescharacteristic
383C2#6431311yescharacteristic
384C2#6431511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#537
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#3798C2semidirect
2. C2×#3795C2semidirect
3. #64106C2semidirect
4. C3×#895C2semidirect
5. C3×D3×C4⋊C4C2semidirect
6. #63916C2semidirect
7. #63908C2semidirect
8. #63906C2semidirect
9. #63893C2semidirect
10. #63886C2semidirect
11. #63857C2semidirect
58. #895D3semidirect
59. D3×C4⋊C4D3semidirect
79. C3×C4×D3D4semidirect
225. C4×D3D12semidirect
311. C32#537semidirect
376. C3#5510semidirect
377. C3#5608semidirect

Maximal subgroups

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

subgroupindex
C2×#37982
C2×#37952
#641062
C3×#8952
C3×D3×C4⋊C42
#639162
#639082
#639062
#638932
#638862
#638572
#56083
#55103