Finite Group Databasebeta

← #10594523

Subgroups — #10594523

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×#3799C211yesmaximal, characteristic, complemented
2C2×#3798C211yesmaximal, characteristic, complemented
3#64104C211yesmaximal, characteristic, complemented
4C3×#897C211yesmaximal, characteristic, complemented
5C3×D3×C4⋊C4C211yesmaximal, characteristic, complemented
6#63910C212yesmaximal, complemented
7#63909C211yesmaximal, characteristic, complemented
8#63909C212yesmaximal, complemented
9#63906C212yesmaximal, complemented
10#63895C212yesmaximal, complemented
11#63871C211yesmaximal, characteristic, complemented
12#5629—31nomaximal
13#5510—31nomaximal
14C2×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C4⋊(C3⋊D3)C2212yes—
16C2×C4×C3⋊D3C2211yescharacteristic
17C2×C3×C4×D3C2211yescharacteristic
18C2×C3×C4×D3C2212yes—
19C2×C3×C4×D3C2211yescharacteristic
20C2×C3×Dic6C2211yescharacteristic
21C2×#627C2211yescharacteristic
22C2×#626C2211yescharacteristic
23C2×#626C2211yescharacteristic
24C2×#626C2212yes—
25C2×#626C2212yes—
26C2×#625C2211yescharacteristic
27C2×#624C2212yes—
28#3799—24no—
29#3798—24no—
30C32×C4⋊C4C2211yescharacteristic
31#3752C2212yes—
32C3×(C2×D3)⋊C4C2212yes—
33C3×C4⋊Dic3C2211yescharacteristic
34C3×C4⋊Dic3C2212yes—
35C3×Dic3⋊C4C2212yes—
36Dic3⋊Dic3C2212yes—
37#3722C2212yes—
38#3722C2212yes—
39C2×#239—31no—
40C2×#235—31no—
41C2×#235—31no—
42C2×#235—31no—
43C3×C22⋊Q8—31no—
44C2×C3×C4⋊C4—31no—
45#946—31no—
46#930—32no—
47#930—32no—
48#930—31no—
49#897—31no—
50#897—32no—
51#897D311yescharacteristic, complemented
52#896—32no—
53#895—31no—
54#895—32no—
55#895—31no—
56#895—31no—
57#895—32no—
58#895—61no—
59D3×C4⋊C4D311yescharacteristic, complemented
60#884—32no—
61#884—32no—
62C22×C3⋊D3C2311yescharacteristic
63C22×C3⋊D3C2312yes—
64C22×C3×D3C2311yescharacteristic
65C22×C3×D3C2311yescharacteristic
66C2×C32×C4C2312yes—
67C2×C32×C4C2311yescharacteristic
68C2×C3⋊Dic3C2311yescharacteristic
69C4⋊(C3⋊D3)—22no—
70C4⋊(C3⋊D3)—42no—
71C4×C3⋊D3—22no—
72C2×C3×Dic3C2312yes—
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2312yes—
75C2×C3×Dic3C2311yescharacteristic
76C3×C4×D3—22no—
77C3×C4×D3—42no—
78C3×C4×D3D414yescomplemented
79C3×Dic6—22no—
80#627—22no—
81#626—22no—
82#626—24no—
83#626—22no—
84#626—42no—
85#625—22no—
86#624—24no—
87#537—91noSylow
88C2×C3×Q8—31no—
89C22×C3×C4—31no—
90C22×C3×C4—31no—
91C22×C3×C4—32no—
92C2×#67—31no—
93C2×#67—31no—
94C2×#67—31no—
95C2×#67—32no—
96C2×#67—31no—
97C2×#67—32no—
98#239—64no—
99#235—64no—
100#235—64no—
101#235—64no—
102C2×D12—32no—
103C2×D12—31no—
104C2×D12—31no—
105C2×D12—31no—
106C2×D12—31no—
107C2×D12—32no—
108C2×D12—32no—
109C2×D12—32no—
110C2×D12—61no—
111C2×D12—62no—
112C2×C4×D3—31no—
113C2×C4×D3—31no—
114C2×C4×D3—32no—
115C2×C4×D3—32no—
116C2×C4×D3—61no—
117C2×C4×D3C2×D311yescharacteristic
118C2×C4×D3C2×D312yes—
119C2×C4×D3C2×D311yescharacteristic
120C2×Dic6—31no—
121C2×Dic6—31no—
122C2×Dic6C2×D311yescharacteristic
123C3×C4⋊C4—21no—
124C3×C4⋊C4—32no—
125C3×C4⋊C4—32no—
126C3×C4⋊C4—31no—
127C3×C4⋊C4C2×D311yescharacteristic
128C3×C4⋊C4C2×D311yescharacteristic
129C3×C22⋊C4—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—32no—
135(C2×D3)⋊C4—32no—
136(C2×D3)⋊C4—62no—
137(C2×D3)⋊C4C2×D312yes—
138C4⋊Dic3—32no—
139C4⋊Dic3C2×D311yescharacteristic
140C4⋊Dic3C2×D312yes—
141Dic3⋊C4—32no—
142Dic3⋊C4C2×D312yes—
143C22×C32C2411yescharacteristic
144C2×C3⋊D3—21no—
145C2×C3⋊D3—22no—
146C2×C3⋊D3—21no—
147C2×C3⋊D3—22no—
148C2×C3⋊D3—21no—
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—22no—
157C32×C4C2×D412yes—
158C3⋊Dic3—21no—
159C3×Dic3—22no—
160C3×Dic3—21no—
161C3×Dic3—22no—
162C3×Dic3C2×D412yes—
163C2×#41—91no—
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—31no—
174C23×C3—31no—
175C23×C3—31no—
176C22×D3—32no—
177C22×D3—31no—
178C22×D3—31no—
179C22×D3—32no—
180C22×D3—61no—
181C22×D3—62no—
182C22×D3C22×D311yescharacteristic
183C22×D3C22×D311yescharacteristic
184C3×Q8—62no—
185C2×C3×C4—21no—
186C2×C3×C4—22no—
187C2×C3×C4—31no—
188C2×C3×C4—34no—
189C2×C3×C4—32no—
190C2×C3×C4—32no—
191C2×C3×C4—31no—
192C2×C3×C4—62no—
193C2×C3×C4—122no—
194C2×C3×C4C22×D312yes—
195C2×C3×C4C22×D312yes—
196C2×C3×C4C22×D311yescharacteristic
197C2×C3×C4C22×D311yescharacteristic
198#67—64no—
199#67—62no—
200#67—62no—
201#67—62no—
202#67—62no—
203#67—122no—
204C2×Dic3—31no—
205C2×Dic3—31no—
206C2×Dic3—61no—
207C2×Dic3C22×D312yes—
208C2×Dic3C22×D312yes—
209C2×Dic3C22×D311yescharacteristic
210C2×Dic3C22×D311yescharacteristic
211D12—62no—
212D12—62no—
213D12—64no—
214D12—62no—
215D12—62no—
216D12—122no—
217D12—122no—
218D12—122no—
219D12—124no—
220D12—122no—
221C4×D3—22no—
222C4×D3—42no—
223C4×D3—64no—
224C4×D3—62no—
225C4×D3—62no—
226C4×D3—64no—
227C4×D3—122no—
228C4×D3D1214yescomplemented
229Dic6—22no—
230Dic6—62no—
231Dic6—62no—
232C2×C32C22×D411yescharacteristic
233C2×C32H(5,𝔽2)11yescharacteristic
234C2×C322−1+411yescharacteristic
235C3⋊D3—41no—
236C3⋊D3—42no—
237C3×D3—22no—
238C3×D3—41no—
239#41—184no—
240#41—184no—
241C2×Q8—31no—
242C2×Q8—91no—
243C2×D4—91no—
244C2×D4—92no—
245C2×D4—91no—
246C2×D4—92no—
247C2×D4—91no—
248C2×D4—91no—
249C2×D4—92no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—32no—
253C22×C4—92no—
254C22×C4—91no—
255C4⋊C4—32no—
256C4⋊C4—31no—
257C4⋊C4—32no—
258C4⋊C4—92no—
259C4⋊C4D3211yescharacteristic
260C22⋊C4—32no—
261C22⋊C4—92no—
262C22⋊C4—92no—
263C22⋊C4—92no—
264C22×C3—21no—
265C22×C3—32no—
266C22×C3—61no—
267C22×C3—61no—
268C22×C3—61no—
269C22×C3—61no—
270C22×C3—61no—
271C22×C3C23×D311yescharacteristic
272C22×C3C23×D311yescharacteristic
273C2×D3—21no—
274C2×D3—21no—
275C2×D3—21no—
276C2×D3—21no—
277C2×D3—21no—
278C2×D3—62no—
279C2×D3—62no—
280C2×D3—61no—
281C2×D3—62no—
282C2×D3—61no—
283C2×D3—61no—
284C2×D3—62no—
285C2×D3—61no—
286C2×D3—62no—
287C2×D3—61no—
288C2×D3—62no—
289C2×D3—61no—
290C2×D3—62no—
291C2×D3—121no—
292C2×D3—121no—
293C2×D3—122no—
294C2×D3—122no—
295C2×D3—122no—
296C2×D3C2×D1212yes—
297C3×C4—22no—
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—32no—
301C3×C4—42no—
302C3×C4—61no—
303C3×C4—62no—
304C3×C4—62no—
305C3×C4C2×D1212yes—
306C3×C4D3×D412yes—
307Dic3—22no—
308Dic3—21no—
309Dic3—22no—
310Dic3—61no—
311Dic3—61no—
312Dic3—121no—
313Dic3C2×D1212yes—
314C32#53711yescharacteristic, complemented, Sylow
315C23—31no—
316C23—31no—
317C23—91no—
318C23—92no—
319Q8—62no—
320Q8—182no—
321D4—182no—
322D4—182no—
323D4—184no—
324D4—182no—
325D4—182no—
326D4—362no—
327D4—362no—
328C2×C4—31no—
329C2×C4—34no—
330C2×C4—32no—
331C2×C4—32no—
332C2×C4—31no—
333C2×C4—62no—
334C2×C4—91no—
335C2×C4—122no—
336C2×C4—184no—
337C2×C4—182no—
338C2×C4C2×D3212yes—
339C2×C4C2×D3211yescharacteristic
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—62no—
344C2×C3—121no—
345C2×C3C22×D1211yescharacteristic
346C2×C3C2×D3×D411yescharacteristic
347C2×C3#100411yescharacteristic
348C2×C3#100711yescharacteristic
349C2×C3#100711yescharacteristic
350C2×C3#100911yescharacteristic
351D3—22no—
352D3—41no—
353D3—122no—
354D3—121no—
355D3—122no—
356D3—121no—
357D3—122no—
358D3—242no—
359C22—32no—
360C22—61no—
361C22—61no—
362C22—61no—
363C22—61no—
364C22—61no—
365C22—181no—
366C22—182no—
367C22—181no—
368C22—182no—
369C22—181no—
370C22—182no—
371C22C22×D3211yescharacteristic
372C4—22no—
373C4—32no—
374C4—61no—
375C4—62no—
376C4—62no—
377C4—181no—
378C4D3×D1212yes—
379C3—21no—
380C3#551011yescharacteristic, complemented
381C3#562911yescharacteristic, complemented
382C2—62no—
383C2—121no—
384C2—361no—
385C2—362no—
386C2C2×D3×D1211yescharacteristic
387C2#6431411yescharacteristic
388C2#6431611yescharacteristic
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×#3799C2semidirect
2. C2×#3798C2semidirect
3. #64104C2semidirect
4. C3×#897C2semidirect
5. C3×D3×C4⋊C4C2semidirect
6. #63910C2semidirect
7. #63909C2semidirect
8. #63909C2semidirect
9. #63906C2semidirect
10. #63895C2semidirect
11. #63871C2semidirect
51. #897D3semidirect
59. D3×C4⋊C4D3semidirect
78. C3×C4×D3D4semidirect
228. C4×D3D12semidirect
314. C32#537semidirect
380. C3#5510semidirect
381. C3#5629semidirect

Maximal subgroups

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

subgroupindex
C2×#37992
C2×#37982
#641042
C3×#8972
C3×D3×C4⋊C42
#639102
#639092
#639062
#638952
#638712
#56293
#55103