Finite Group Databasebeta

← #13140902

Subgroups — #13140902

Order 1728.

357 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 79 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
1Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2#11721093C211yesmaximal, characteristic
3#11720583C211yesmaximal, characteristic
4#63662⋊C3C211yesmaximal, characteristic
5#11720315C211yesmaximal, characteristic
6#11720311C211yesmaximal, characteristic
7#11720307C211yesmaximal, characteristic
8#10591615—31nomaximal
9#10591547—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C4⋊#1084C2211yescharacteristic
12C2×#6180C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)—21no—
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4×(C3⋊Dic3)⋊C3—21no—
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88449—21no—
20#88448—21no—
21Q8×C3⋊Dic3D311yescharacteristic
22C3×Q8×Dic3—31no—
23#63872—31no—
24#63869—31no—
25#63662D311yescharacteristic
26C3×#835—31no—
27C3×#835—31no—
28C3×#835—31no—
29C3×#835—61no—
30#63575—31no—
31#63575—31no—
32#63574—31no—
33#63573—31no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36Q8×H(3,𝔽3)—21no—
37Q8×H(3,𝔽3)Q812yes—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C2×#1084—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45#6180—21no—
46(C3⋊(C3⋊C8))⋊C3—21no—
47#5022—91no—
48#4799—91no—
49#4796—91no—
50C2×C32×Q8—21no—
51C2×C32×Q8C2×D311yescharacteristic
52C2×C32×Q8C2×D311yescharacteristic
53C4⋊(C3⋊Dic3)—21no—
54C4⋊(C3⋊Dic3)C2×D311yescharacteristic
55C4⋊(C3⋊Dic3)C2×D311yescharacteristic
56C4×C3⋊Dic3—21no—
57C4×C3⋊Dic3C2×D311yescharacteristic
58C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3—31no—
61C3×C4⋊Dic3—31no—
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—61no—
64C3×C4⋊Dic3—61no—
65C3×C4×Dic3—31no—
66C3×C4×Dic3—61no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—61no—
71#3724—61no—
72Dic3⋊Dic3—61no—
73Dic3⋊Dic3—61no—
74Dic3⋊Dic3—61no—
75C22×H(3,𝔽3)C2×D411yescharacteristic
76C4×H(3,𝔽3)—22no—
77C4×H(3,𝔽3)C2×Q811yescharacteristic
78C4×H(3,𝔽3)#4111yescharacteristic
79#1084—41no—
80(C3⋊Dic3)⋊C3—21no—
81(C3⋊Dic3)⋊C3—21no—
82(C3⋊Dic3)⋊C3—41no—
83(C3⋊Dic3)⋊C3—41no—
84C3×C4×Q8—91no—
85Q8×Dic3—31no—
86Q8×Dic3—91no—
87#890—91no—
88#890—91no—
89#870—91no—
90C3×#103—91no—
91C3×#103—91no—
92#835—31no—
93#835—31no—
94#835—31no—
95#835—61no—
96#835—91no—
97#818—91no—
98#817—91no—
99Dic3⋊C8—91no—
100Dic3⋊C8—91no—
101#808—91no—
102#808—91no—
103#807—91no—
104#807—91no—
105C4⋊(C3⋊C8)—91no—
106C32×Q8—21no—
107C32×Q8—21no—
108C32×Q8—22no—
109C32×Q8—41no—
110C32×Q8Dic612yes—
111C32×Q8Dic612yes—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—41no—
116C2×C32×C4C22×D311yescharacteristic
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3#6711yescharacteristic
121C2×C3⋊Dic3#6711yescharacteristic
122C2×C3×Dic3—31no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—121no—
129C3⋊(C3⋊C8)—21no—
130C3×C3⋊C8—61no—
131C3×C3⋊C8—61no—
132C3×C3⋊C8—61no—
133C3×C3⋊C8—121no—
134#479—271noSylow
135C2×H(3,𝔽3)C22⋊Q811yescharacteristic
136C2×H(3,𝔽3)#13511yescharacteristic
137C2×H(3,𝔽3)#13711yescharacteristic
138C2×C3×Q8—31no—
139C2×C3×Q8—31no—
140C2×C3×Q8—61no—
141C2×C3×Q8D3211yescharacteristic
142C2×C3×C8—91no—
143C2×C3×C8—91no—
144C3×C4⋊C4—91no—
145C3×C4⋊C4—91no—
146C3×C4⋊C4—91no—
147C3×C4⋊C4—181no—
148C3×C42—91no—
149C3×C42—181no—
150C4⋊Dic3—31no—
151C4⋊Dic3—31no—
152C4⋊Dic3—31no—
153C4⋊Dic3—31no—
154C4⋊Dic3—61no—
155C4⋊Dic3—61no—
156C4⋊Dic3—91no—
157C4⋊Dic3—91no—
158C4⋊Dic3—181no—
159C4⋊Dic3—181no—
160C4⋊Dic3—181no—
161Dic3⋊C4—181no—
162Dic3⋊C4—181no—
163Dic3⋊C4—181no—
164Dic3⋊C4—181no—
165C4×Dic3—31no—
166C4×Dic3—61no—
167C4×Dic3—91no—
168C4×Dic3—181no—
169C2×C3⋊C8—31no—
170C2×C3⋊C8—31no—
171C2×C3⋊C8—31no—
172C2×C3⋊C8—61no—
173C2×C3⋊C8—91no—
174C22×C32—21no—
175C22×C32D3×D411yescharacteristic
176C22×C32C2×#6711yescharacteristic
177C32×C4—22no—
178C32×C4—22no—
179C32×C4—21no—
180C32×C4—21no—
181C32×C4—42no—
182C32×C4C2×Dic611yescharacteristic
183C32×C4C2×Dic611yescharacteristic
184C32×C4#23511yescharacteristic
185C32×C4#23711yescharacteristic
186C3⋊Dic3—21no—
187C3⋊Dic3—21no—
188C3⋊Dic3—41no—
189C3⋊Dic3—41no—
190C3×Dic3—61no—
191C3×Dic3—61no—
192C3×Dic3—121no—
193C3×Dic3—121no—
194C3×Dic3—121no—
195C3×Dic3—121no—
196C3×Dic3—241no—
197#125—271no—
198C4×Q8—91no—
199#107—271no—
200#106—271no—
201C4⋊C8—271no—
202#103—91no—
203#103—91no—
204H(3,𝔽3)#47911yescharacteristic, complemented, Sylow
205C3×Q8—21no—
206C3×Q8—32no—
207C3×Q8—32no—
208C3×Q8—61no—
209C3×Q8—61no—
210C3×Q8—62no—
211C3×Q8—121no—
212C3×Q8#62712yes—
213C2×C3×C4—21no—
214C2×C3×C4—31no—
215C2×C3×C4—31no—
216C2×C3×C4—61no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—91no—
220C2×C3×C4—91no—
221C2×C3×C4—121no—
222C2×C3×C4—181no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×C3×C4C2×D3211yescharacteristic
226C2×Dic3—31no—
227C2×Dic3—31no—
228C2×Dic3—61no—
229C2×Dic3—61no—
230C2×Dic3—61no—
231C2×Dic3—61no—
232C2×Dic3—91no—
233C2×Dic3—91no—
234C2×Dic3—121no—
235C2×Dic3—181no—
236C2×Dic3—181no—
237C3×C8—181no—
238C3×C8—181no—
239C3⋊C8—61no—
240C3⋊C8—61no—
241C3⋊C8—61no—
242C3⋊C8—121no—
243C3⋊C8—181no—
244C2×C32—21no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32#87811yescharacteristic
248C2×C32#91611yescharacteristic
249C2×C32#91811yescharacteristic
250C2×C32#92411yescharacteristic
251C2×C32#94211yescharacteristic
252C2×C32#95011yescharacteristic
253C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
254C2×C8—91no—
255C2×C8—91no—
256C4⋊C4—91no—
257C4⋊C4—91no—
258C4⋊C4—91no—
259C4⋊C4—181no—
260C4⋊C4—541no—
261C4⋊C4—541no—
262C42—91no—
263C42—181no—
264C22×C3—31no—
265C22×C3—31no—
266C22×C3—61no—
267C22×C3D3×#6711yescharacteristic
268C3×C4—22no—
269C3×C4—31no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—62no—
274C3×C4—62no—
275C3×C4—61no—
276C3×C4—61no—
277C3×C4—122no—
278C3×C4—181no—
279C3×C4—181no—
280C3×C4—361no—
281C3×C4—361no—
282C3×C4—361no—
283C3×C4#380411yescharacteristic
284C3×C4C2×#62711yescharacteristic
285Dic3—61no—
286Dic3—61no—
287Dic3—121no—
288Dic3—121no—
289Dic3—121no—
290Dic3—121no—
291Dic3—181no—
292Dic3—181no—
293Dic3—241no—
294Dic3—361no—
295Dic3—361no—
296C32—21no—
297C32#479911yescharacteristic, complemented
298C32#502211yescharacteristic, complemented
299Q8—21no—
300Q8#619612yes—
301C2×C4—21no—
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—181no—
305C2×C4—181no—
306C2×C4—181no—
307C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
308C8—181no—
309C8—181no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—31no—
315C2×C3—31no—
316C2×C3—61no—
317C2×C3—61no—
318C2×C3—61no—
319C2×C3#6394711yescharacteristic
320C2×C3#6394811yescharacteristic
321C2×C3#6396411yescharacteristic
322C22#8867611yescharacteristic
323C4—22no—
324C4—181no—
325C4—181no—
326C4—361no—
327C4—361no—
328C4—361no—
329C4C2×#619611yescharacteristic
330C4#8867111yescharacteristic
331C3—31no—
332C3—31no—
333C3—61no—
334C3#1059154511yescharacteristic
335C2#1172123411yescharacteristic
336C2#1172123911yescharacteristic
337C2#1172129911yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#479
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
204. H(3,𝔽3)#479semidirect
297. C32#4799semidirect
298. C32#5022semidirect

Maximal subgroups

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

subgroupindex
Q8×(C3⋊Dic3)⋊C32
#117210932
#117205832
#63662⋊C32
#117203152
#117203112
#117203072
#105916153
#105915473