Finite Group Databasebeta

← #10594897

Subgroups — #10594897

Order 576.

579 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 579 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Dic6C211yesmaximal, characteristic, complemented
2C2×#3804C211yesmaximal, characteristic
3#64138C211yesmaximal, characteristic, complemented
4C3×#929C211yesmaximal, characteristic, complemented
5#63966C211yesmaximal, characteristic, complemented
6#63961C211yesmaximal, characteristic, complemented
7#63955C211yesmaximal, characteristic, complemented
8#63907C212yesmaximal, complemented
9#63888C212yesmaximal, complemented
10#63887C212yesmaximal, complemented
11#63839C212yesmaximal, complemented
12#5816—31nomaximal
13#5648—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#638C2211yescharacteristic
16C2×C3×#67C2211yescharacteristic
17C22×C3×Dic3C2211yescharacteristic
18C22×C3×Dic3C2211yescharacteristic
19C2×C3×Dic6—24no—
20C2×C3×Dic6C2212yes—
21C2×C3×Dic6C2212yes—
22C2×#627C2211yescharacteristic
23C2×#626C2211yescharacteristic
24C2×#624C2211yescharacteristic
25#3804—24no—
26C2×D3×Dic3C2211yescharacteristic
27#3757C2211yescharacteristic
28#3752C2212yes—
29#3750C2212yes—
30C3×#217C2211yescharacteristic
31C3×(C2×D3)⋊C4C2212yes—
32C3×Dic3⋊C4C2212yes—
33#3724C2212yes—
34Dic3⋊Dic3C2212yes—
35Dic3⋊Dic3C2212yes—
36#3722C2212yes—
37(C2×D3)⋊Dic3C2212yes—
38Dic32C2212yes—
39C22×C3×Q8—31no—
40C2×#237—31no—
41C2×#235—31no—
42C22×Dic6D311yescharacteristic, complemented
43C3×#123—31no—
44#947—32no—
45#946—32no—
46#946—32no—
47#944—32no—
48#935—31no—
49#929—31no—
50#929—31no—
51#929—31no—
52#929—31no—
53#929—61no—
54#929D311yescharacteristic, complemented
55#926—31no—
56#896—32no—
57#888—32no—
58#886—31no—
59#885—32no—
60#878—32no—
61#877—31no—
62C23×C32C2311yescharacteristic
63C22×C3⋊D3C2311yescharacteristic
64C22×C3×D3C2311yescharacteristic
65C2×C32×C4—22no—
66C2×C32×C4C2312yes—
67#638—22no—
68C2×C3⋊Dic3C2312yes—
69C2×C3⋊Dic3C2311yescharacteristic
70C3×#67—22no—
71C2×C3×Dic3—22no—
72C2×C3×Dic3D414yes—
73C2×C3×Dic3C2312yes—
74C2×C3×Dic3C2312yes—
75C2×C3×Dic3C2311yescharacteristic
76C2×C3×Dic3C2311yescharacteristic
77C2×C3×Dic3C2311yescharacteristic
78C3×Dic6—24no—
79C3×Dic6—24no—
80#627—22no—
81#626—22no—
82#624—22no—
83D3×Dic3—22no—
84#536—91noSylow
85C2×C3×Q8—32no—
86C2×C3×Q8—32no—
87C2×C3×Q8—64no—
88C2×C3×D4—31no—
89C22×C3×C4—21no—
90C22×C3×C4—31no—
91C22×C3×C4—31no—
92C22×C3×C4C2×D311yescharacteristic
93C22×C3×C4C2×D311yescharacteristic
94C2×#67—31no—
95C2×#67—31no—
96C2×#67—31no—
97C2×#67—61no—
98C2×#67C2×D311yescharacteristic
99C22×Dic3—31no—
100C22×Dic3C2×D311yescharacteristic
101C22×Dic3C2×D311yescharacteristic
102#237—64no—
103#235—64no—
104C2×D12—31no—
105C2×C4×D3—31no—
106C2×C4×D3—31no—
107C2×C4×D3—31no—
108C2×Dic6—24no—
109C2×Dic6—31no—
110C2×Dic6—31no—
111C2×Dic6C2×D312yes—
112C2×Dic6C2×D312yes—
113C3×C4⋊C4—32no—
114C3×C22⋊C4—31no—
115C3×C22⋊C4—32no—
116#217—31no—
117#217—31no—
118#217—32no—
119#217—61no—
120#217C2×D311yescharacteristic
121(C2×D3)⋊C4—32no—
122(C2×D3)⋊C4—32no—
123(C2×D3)⋊C4—32no—
124(C2×D3)⋊C4—32no—
125(C2×D3)⋊C4—32no—
126(C2×D3)⋊C4—62no—
127(C2×D3)⋊C4C2×D312yes—
128C4⋊Dic3—32no—
129C4⋊Dic3—32no—
130Dic3⋊C4—32no—
131Dic3⋊C4—32no—
132Dic3⋊C4—32no—
133Dic3⋊C4—32no—
134Dic3⋊C4—32no—
135Dic3⋊C4—32no—
136Dic3⋊C4—62no—
137Dic3⋊C4C2×D312yes—
138C4×Dic3—32no—
139C4×Dic3—32no—
140C22×C32—22no—
141C22×C32C2×D412yes—
142C22×C32C2411yescharacteristic
143C2×C3⋊D3—21no—
144C2×C3⋊D3—22no—
145C2×C3×D3—21no—
146C2×C3×D3—22no—
147C32×C4—22no—
148C3⋊Dic3—21no—
149C3⋊Dic3—22no—
150C3×Dic3—21no—
151C3×Dic3—22no—
152C3×Dic3—22no—
153C3×Dic3C2×D412yes—
154C3×Dic3C2×D412yes—
155C2×#41—91no—
156C22×Q8—31no—
157C4⋊Q8—92no—
158#124—92no—
159#123—31no—
160#123—91no—
161#123—91no—
162#123—91no—
163C22⋊Q8—92no—
164C22⋊Q8—92no—
165#117—91no—
166C23×C3—21no—
167C23×C3—31no—
168C23×C3C22×D311yescharacteristic
169C23×C3C22×D311yescharacteristic
170C22×D3—31no—
171C22×D3—31no—
172C22×D3—61no—
173C22×D3C22×D311yescharacteristic
174C3×Q8—64no—
175C3×Q8—64no—
176C3×D4—62no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—34no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—32no—
184C2×C3×C4—32no—
185C2×C3×C4—31no—
186C2×C3×C4—42no—
187C2×C3×C4—62no—
188C2×C3×C4C22×D312yes—
189C2×C3×C4C22×D312yes—
190#67—22no—
191#67—62no—
192#67—62no—
193#67—62no—
194#67—122no—
195C2×Dic3—22no—
196C2×Dic3—32no—
197C2×Dic3—32no—
198C2×Dic3—31no—
199C2×Dic3—31no—
200C2×Dic3—61no—
201C2×Dic3—62no—
202C2×Dic3—62no—
203C2×Dic3#6714yes—
204C2×Dic3C22×D312yes—
205C2×Dic3C22×D311yescharacteristic
206C2×Dic3C22×D312yes—
207C2×Dic3C22×D311yescharacteristic
208C2×Dic3C22×D311yescharacteristic
209D12—62no—
210C4×D3—62no—
211C4×D3—62no—
212C4×D3—62no—
213Dic6—24no—
214Dic6—24no—
215Dic6—62no—
216Dic6—62no—
217C2×C32—22no—
218C2×C32C22×D411yescharacteristic
219C2×C322−1+412yes—
220C3⋊D3—41no—
221C3×D3—41no—
222#41—184no—
223C2×Q8—32no—
224C2×Q8—32no—
225C2×Q8—64no—
226C2×Q8—91no—
227C2×D4—31no—
228C2×D4—91no—
229C2×D4—91no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—91no—
233C22×C4—91no—
234C22×C4D3211yescharacteristic
235C4⋊C4—32no—
236C4⋊C4—92no—
237C4⋊C4—92no—
238C4⋊C4—92no—
239C4⋊C4—92no—
240C22⋊C4—31no—
241C22⋊C4—32no—
242C22⋊C4—92no—
243C22⋊C4—92no—
244C22⋊C4—91no—
245C22⋊C4—92no—
246C42—92no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3—21no—
250C22×C3—22no—
251C22×C3—42no—
252C22×C3—62no—
253C22×C3—61no—
254C22×C3D3×D412yes—
255C22×C3C2×#6712yes—
256C22×C3C23×D311yescharacteristic
257C22×C3C23×D311yescharacteristic
258C2×D3—21no—
259C2×D3—22no—
260C2×D3—61no—
261C2×D3—62no—
262C2×D3—62no—
263C2×D3—61no—
264C2×D3—121no—
265C2×D3—122no—
266C3×C4—22no—
267C3×C4—22no—
268C3×C4—32no—
269C3×C4—32no—
270C3×C4—42no—
271C3×C4—62no—
272C3×C4—62no—
273C3×C4—61no—
274Dic3—21no—
275Dic3—22no—
276Dic3—22no—
277Dic3—61no—
278Dic3—62no—
279Dic3—62no—
280Dic3—61no—
281Dic3—121no—
282Dic3—122no—
283Dic3C2×#6712yes—
284Dic3C2×#6712yes—
285C32#53611yescharacteristic, complemented, Sylow
286C23—31no—
287C23—91no—
288C23C2×D3211yescharacteristic
289Q8—64no—
290Q8—64no—
291Q8—182no—
292D4—62no—
293D4—182no—
294D4—182no—
295C2×C4—22no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—34no—
299C2×C4—32no—
300C2×C4—32no—
301C2×C4—31no—
302C2×C4—62no—
303C2×C4—92no—
304C2×C4—91no—
305C2×C4—182no—
306C2×C4—182no—
307C2×C4C2×D3212yes—
308C2×C3—22no—
309C2×C3—22no—
310C2×C3—22no—
311C2×C3—21no—
312C2×C3—42no—
313C2×C3—121no—
314C2×C3C2×D3×D411yescharacteristic
315C2×C3#100712yes—
316C2×C3#101012yes—
317C2×C3C22×#6711yescharacteristic
318D3—41no—
319D3—121no—
320D3—121no—
321D3—241no—
322C22—22no—
323C22—62no—
324C22—61no—
325C22—182no—
326C22—181no—
327C22D3×#6712yes—
328C22C22×D3211yescharacteristic
329C4—22no—
330C4—32no—
331C4—32no—
332C4—62no—
333C4—62no—
334C4—61no—
335C4—182no—
336C4—181no—
337C3—21no—
338C3#564811yescharacteristic, complemented
339C3#581611yescharacteristic, complemented
340C2—22no—
341C2—121no—
342C2—361no—
343C2#6429712yes—
344C2C2×D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radicalC2×#3804C2
hypercenterC22×C4D32
hypocenterC32#536
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×Dic6C2semidirect
3. #64138C2semidirect
4. C3×#929C2semidirect
5. #63966C2semidirect
6. #63961C2semidirect
7. #63955C2semidirect
8. #63907C2semidirect
9. #63888C2semidirect
10. #63887C2semidirect
11. #63839C2semidirect
42. C22×Dic6D3semidirect
54. #929D3semidirect
285. C32#536semidirect
338. C3#5648semidirect
339. C3#5816semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic62
C2×#38042
#641382
C3×#9292
#639662
#639612
#639552
#639072
#638882
#638872
#638392
#58163
#56483