Finite Group Databasebeta

← #13139939

Subgroups — #13139939

Order 1728.

438 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 85 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
1#11721824C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic
4#11720326C212yesmaximal, complemented
5#11720306C212yesmaximal, complemented
6#10591132—31nomaximal
7#10591103—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#6229C2211yescharacteristic
10C22×(C3⋊Dic3)⋊C3—21no—
11#88552—21no—
12C4⋊#1084C2211yescharacteristic
13#88540—21no—
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C2×C4⋊(C3⋊Dic3)D311yescharacteristic
18C2×C3×C4⋊Dic3—31no—
19C3×#924—31no—
20C3×#924—31no—
21C3×#924—61no—
22#63657D311yescharacteristic
23C3×#830—31no—
24#63576—32no—
25#63575—32no—
26#63566—32no—
27#63565—32no—
28C23×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—21no—
33C2×#1084—21no—
34#6229—22no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3—42no—
37(C3⋊(C3⋊C8))⋊C3—22no—
38#5044—91no—
39#4970—91no—
40#4722—91no—
41C22×C32×C4—21no—
42C22×C32×C4C2×D311yescharacteristic
43C22×C32×C4C2×D311yescharacteristic
44C22×C3⋊Dic3—21no—
45C22×C3×Dic3—61no—
46C2×C3×Dic6—31no—
47C2×C3×Dic6—31no—
48C2×C3×Dic6—61no—
49C4⋊(C3⋊Dic3)—21no—
50C4⋊(C3⋊Dic3)C2×D312yes—
51C2×C3⋊(C3⋊C8)C2×D312yes—
52C3×#217—62no—
53C3×#217—61no—
54C3×#217—61no—
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3—31no—
57C3×C4⋊Dic3—32no—
58C3×C4⋊Dic3—61no—
59C3×C4⋊Dic3—61no—
60C3×Dic3⋊C4—62no—
61C3×Dic3⋊C4—61no—
62C3×Dic3⋊C4—61no—
63C2×C3×C3⋊C8—32no—
64C22×H(3,𝔽3)—21no—
65C22×H(3,𝔽3)—21no—
66C22×H(3,𝔽3)Q1612yes—
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C4×H(3,𝔽3)—21no—
69C4×H(3,𝔽3)#4112yes—
70#1084—41no—
71#1084—41no—
72(C3⋊Dic3)⋊C3—42no—
73C3×C22⋊Q8—91no—
74C2×C3×C4⋊C4—91no—
75C2×C4⋊Dic3—31no—
76C2×C4⋊Dic3—91no—
77#924—31no—
78#924—31no—
79#924—61no—
80C3×C22⋊C8—91no—
81#835—92no—
82#830—31no—
83#830—91no—
84#818—92no—
85#816—92no—
86#810—92no—
87#807—92no—
88#807—92no—
89C23×C32—21no—
90C23×C32D1211yescharacteristic
91C23×C32#6711yescharacteristic
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—42no—
97C2×C32×C4D1211yescharacteristic
98C2×C32×C4#6711yescharacteristic
99C2×C32×C4C22×D311yescharacteristic
100C2×C32×C4C22×D311yescharacteristic
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3—42no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—122no—
111C3×Dic6—62no—
112C3×Dic6—62no—
113C3×Dic6—122no—
114C3⋊(C3⋊C8)—22no—
115C3×C3⋊C8—62no—
116#484—271noSylow
117C2×H(3,𝔽3)—22no—
118C2×H(3,𝔽3)#12311yescharacteristic
119C2×H(3,𝔽3)C2×Q1611yescharacteristic
120C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
121C2×C3×Q8—91no—
122C22×C3×C4—31no—
123C22×C3×C4—31no—
124C22×C3×C4—61no—
125C22×C3×C4—181no—
126C22×C3×C4D3211yescharacteristic
127C22×Dic3—61no—
128C22×Dic3—181no—
129C2×Dic6—31no—
130C2×Dic6—31no—
131C2×Dic6—61no—
132C2×C3×C8—92no—
133C3×C4⋊C4—92no—
134C3×C4⋊C4—91no—
135C3×C4⋊C4—181no—
136C3×C4⋊C4—181no—
137C3×C22⋊C4—181no—
138#217—61no—
139#217—61no—
140#217—62no—
141C4⋊Dic3—31no—
142C4⋊Dic3—31no—
143C4⋊Dic3—32no—
144C4⋊Dic3—61no—
145C4⋊Dic3—61no—
146C4⋊Dic3—92no—
147C4⋊Dic3—181no—
148Dic3⋊C4—61no—
149Dic3⋊C4—61no—
150Dic3⋊C4—62no—
151C2×C3⋊C8—32no—
152C2×C3⋊C8—92no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32—22no—
158C22×C32—21no—
159C22×C32—41no—
160C22×C32—41no—
161C22×C32Dic1212yes—
162C22×C32#21612yes—
163C22×C32C2×D1211yescharacteristic
164C22×C32C2×#6711yescharacteristic
165C32×C4—21no—
166C32×C4—21no—
167C32×C4—22no—
168C32×C4—41no—
169C32×C4#23512yes—
170C32×C4#23712yes—
171C3⋊Dic3—42no—
172C3×Dic3—122no—
173C3×Dic3—122no—
174C3×Dic3—121no—
175C3×Dic3—122no—
176C3×Dic3—121no—
177C3×Dic3—121no—
178C3×Dic3—121no—
179C22⋊Q8—91no—
180C2×C4⋊C4—91no—
181#107—272no—
182#103—272no—
183C22⋊C8—91no—
184H(3,𝔽3)#48411yescharacteristic, complemented, Sylow
185C23×C3—31no—
186C23×C3—31no—
187C23×C3—61no—
188C23×C3#62611yescharacteristic
189C3×Q8—182no—
190C2×C3×C4—22no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—62no—
196C2×C3×C4—62no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—122no—
200C2×C3×C4—182no—
201C2×C3×C4—181no—
202C2×C3×C4—181no—
203C2×C3×C4—362no—
204C2×C3×C4#62611yescharacteristic
205C2×C3×C4C2×D3211yescharacteristic
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—62no—
211C2×Dic3—62no—
212C2×Dic3—62no—
213C2×Dic3—122no—
214C2×Dic3—182no—
215C2×Dic3—362no—
216Dic6—62no—
217Dic6—62no—
218Dic6—122no—
219C3×C8—182no—
220C3⋊C8—62no—
221C3⋊C8—182no—
222C2×C32—22no—
223C2×C32—22no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—42no—
228C2×C32#88611yescharacteristic
229C2×C32C2×Dic1211yescharacteristic
230C2×C32#90811yescharacteristic
231C2×C32#92911yescharacteristic
232C2×C32#93211yescharacteristic
233C2×C32C2×#21611yescharacteristic
234C2×Q8—91no—
235C22×C4—181no—
236C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
237C2×C8—92no—
238C4⋊C4—92no—
239C4⋊C4—91no—
240C4⋊C4—181no—
241C4⋊C4—181no—
242C22⋊C4—181no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—31no—
246C22×C3—31no—
247C22×C3—32no—
248C22×C3—32no—
249C22×C3—61no—
250C22×C3—61no—
251C22×C3—61no—
252C22×C3—61no—
253C22×C3—61no—
254C22×C3—62no—
255C22×C3—121no—
256C22×C3—121no—
257C22×C3#371912yes—
258C22×C3C2×#62611yescharacteristic
259C3×C4—21no—
260C3×C4—32no—
261C3×C4—32no—
262C3×C4—61no—
263C3×C4—61no—
264C3×C4—62no—
265C3×C4—121no—
266C3×C4—362no—
267C3×C4—361no—
268C3×C4—361no—
269C3×C4#380412yes—
270Dic3—121no—
271Dic3—121no—
272Dic3—121no—
273Dic3—121no—
274Dic3—122no—
275Dic3—122no—
276Dic3—122no—
277Dic3—362no—
278C32—21no—
279C32#472211yescharacteristic, complemented
280C32#497011yescharacteristic, complemented
281C23#620011yescharacteristic
282Q8—182no—
283C2×C4—22no—
284C2×C4—181no—
285C2×C4—181no—
286C2×C4—182no—
287C2×C4—362no—
288C2×C4#620011yescharacteristic
289C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
290C8—182no—
291C2×C3—22no—
292C2×C3—31no—
293C2×C3—31no—
294C2×C3—31no—
295C2×C3—31no—
296C2×C3—31no—
297C2×C3—31no—
298C2×C3—62no—
299C2×C3—62no—
300C2×C3—61no—
301C2×C3—61no—
302C2×C3—61no—
303C2×C3—122no—
304C2×C3#6383011yescharacteristic
305C2×C3C2×#371911yescharacteristic
306C2×C3#6396611yescharacteristic
307C22—21no—
308C22—21no—
309C22#8843912yes—
310C22C2×#620011yescharacteristic
311C4—21no—
312C4—361no—
313C4—362no—
314C4—361no—
315C4#8867112yes—
316C3—31no—
317C3—31no—
318C3—61no—
319C3#1059113511yescharacteristic
320C2—22no—
321C2#1172107111yescharacteristic
322C2C2×#8843911yescharacteristic
323C2#1172130111yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC23×H(3,𝔽3)D4
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#484
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11720326C2semidirect
5. #11720306C2semidirect
184. H(3,𝔽3)#484semidirect
279. C32#4722semidirect
280. C32#4970semidirect

Maximal subgroups

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

subgroupindex
#117218242
C2×C4⋊((C3⋊Dic3)⋊C3)2
#63657⋊C32
#117203262
#117203062
#105911323
#105911033