Finite Group Databasebeta

← #13159305

Subgroups — #13159305

Order 1728.

856 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 856 conjugacy classes of subgroups — too many to draw. Only its 146 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#11723928C211yesmaximal, characteristic
2#11723536C211yesmaximal, characteristic
3C3×#63635C211yesmaximal, characteristic
4C32×#808C211yesmaximal, characteristic
5#11722050C211yesmaximal, characteristic
6#11722042C211yesmaximal, characteristic
7#11722041C211yesmaximal, characteristic
8#10592553—31nomaximal
9#10590431—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×#6293—21no—
12C33×C4⋊C4C2211yescharacteristic
13#88857—21no—
14C4×C3⋊(C3⋊Dic3)—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21#64097—31no—
22#64096—31no—
23#64096—34no—
24#64096—64no—
25#63919—34no—
26C32×#106—31no—
27#63643—31no—
28#63641—31no—
29#63637—31no—
30#63635D311yescharacteristic
31C3×#808—34no—
32C3×#808D314yes—
33#63572—34no—
34#63567—34no—
35#63567—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)D411yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41#6308—21no—
42#6308Q812yescomplemented
43C2×C3×C3⋊Dic3—21no—
44C2×C32×Dic3—21no—
45#6293—42no—
46C3×C3⋊(C3⋊C8)—21no—
47C32×C3⋊C8—21no—
48#4854—94no—
49#4854—91no—
50C2×#634—31no—
51C2×#634—34no—
52C2×#634—64no—
53C2×#634—61no—
54C2×#627—64no—
55C2×C32×C8—31no—
56C32×C4⋊C4—24no—
57C32×C4⋊C4—31no—
58C32×C4⋊C4C2×D314yes—
59C32×C4⋊C4C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61#3750—61no—
62#3750—61no—
63#3750—64no—
64#3750—68no—
65C4×C3⋊Dic3—31no—
66C4×C3⋊Dic3—34no—
67C4×C3⋊Dic3—61no—
68C4×C3⋊Dic3—64no—
69C4×C3⋊Dic3—64no—
70C4×C3⋊Dic3—68no—
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74C2×C3×C3⋊C8—34no—
75C2×C3×C3⋊C8C2×D314yes—
76#3724—64no—
77C22×C33C2×D411yescharacteristic
78C33×C4—41no—
79C33×C4C2×Q811yescharacteristic
80C33×C4#4111yescharacteristic
81C3⋊(C3⋊Dic3)—22no—
82C3⋊(C3⋊Dic3)—21no—
83C3⋊(C3⋊Dic3)SD1612yes—
84C3×C3⋊Dic3—41no—
85C32×Dic3—41no—
86#888—94no—
87#888—91no—
88#887—94no—
89#887—91no—
90#887—184no—
91C3×#106—34no—
92C3×#106—91no—
93#816—94no—
94#816—91no—
95Dic3⋊C8—94no—
96Dic3⋊C8—91no—
97#810—94no—
98#810—91no—
99#808—34no—
100#808C3⋊D311yescharacteristic
101C2×C32×C4—28no—
102C2×C32×C4—24no—
103C2×C32×C4—21no—
104C2×C32×C4—24no—
105C2×C32×C4—61no—
106C2×C32×C4C22×D314yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—21no—
109C2×C3⋊Dic3—31no—
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—34no—
112C2×C3⋊Dic3—34no—
113C2×C3⋊Dic3—64no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3—68no—
116C2×C3⋊Dic3—64no—
117C2×C3⋊Dic3—61no—
118#634—32no—
119#634—38no—
120#634—68no—
121#634—68no—
122#634—64no—
123#634—61no—
124#634—122no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—64no—
127#627—128no—
128C32×C8—61no—
129C3⋊(C3⋊C8)—21no—
130C3×C3⋊C8—24no—
131C3×C3⋊C8—64no—
132#475—271noSylow
133C2×C33C22⋊Q811yescharacteristic
134C2×C33C2×SD1611yescharacteristic
135C2×C33#13711yescharacteristic
136C2×Dic6—94no—
137C2×Dic6—91no—
138C2×Dic6—184no—
139C2×Dic6—181no—
140C2×Dic6—184no—
141C2×C3×C8—34no—
142C2×C3×C8—91no—
143C3×C4⋊C4—24no—
144C3×C4⋊C4—34no—
145C3×C4⋊C4—91no—
146C3×C4⋊C4D3214yes—
147C3×C4⋊C4C2×C3⋊D311yescharacteristic
148C4⋊Dic3—34no—
149C4⋊Dic3C2×C3⋊D311yescharacteristic
150Dic3⋊C4—181no—
151Dic3⋊C4—184no—
152Dic3⋊C4—184no—
153Dic3⋊C4—181no—
154Dic3⋊C4—188no—
155C4×Dic3—94no—
156C4×Dic3—91no—
157C4×Dic3—184no—
158C4×Dic3—184no—
159C4×Dic3—181no—
160C4×Dic3—188no—
161C2×C3⋊C8—34no—
162C2×C3⋊C8C2×C3⋊D311yescharacteristic
163C22×C32—24no—
164C22×C32D3×D414yes—
165C22×C32D3×D411yescharacteristic
166C32×C4—24no—
167C32×C4—24no—
168C32×C4—48no—
169C32×C4—44no—
170C32×C4—41no—
171C32×C4—121no—
172C32×C4#23514yes—
173C32×C4#23511yescharacteristic
174C32×C4D3×Q814yes—
175C32×C4D3×Q811yescharacteristic
176C3⋊Dic3—32no—
177C3⋊Dic3—38no—
178C3⋊Dic3—41no—
179C3⋊Dic3—68no—
180C3⋊Dic3—616no—
181C3⋊Dic3—68no—
182C3⋊Dic3—62no—
183C3⋊Dic3—61no—
184C3⋊Dic3—64no—
185C3⋊Dic3—124no—
186C3×Dic3—44no—
187C3×Dic3—124no—
188C4⋊Q8—271no—
189C4×Q8—271no—
190#106—31no—
191#106—91no—
192C4⋊C8—271no—
193#103—271no—
194#103—271no—
195C33#47511yescharacteristic, complemented, Sylow
196C2×C3×C4—28no—
197C2×C3×C4—24no—
198C2×C3×C4—21no—
199C2×C3×C4—24no—
200C2×C3×C4—64no—
201C2×C3×C4—181no—
202C2×C3×C4C2×D3214yes—
203C2×C3×C4C22×C3⋊D311yescharacteristic
204C2×Dic3—21no—
205C2×Dic3—64no—
206C2×Dic3—94no—
207C2×Dic3—91no—
208C2×Dic3—94no—
209C2×Dic3—91no—
210C2×Dic3—184no—
211C2×Dic3—184no—
212C2×Dic3—188no—
213C2×Dic3—184no—
214C2×Dic3—181no—
215Dic6—98no—
216Dic6—92no—
217Dic6—188no—
218Dic6—188no—
219Dic6—184no—
220Dic6—181no—
221Dic6—362no—
222Dic6—368no—
223C3×C8—64no—
224C3×C8—181no—
225C3⋊C8—21no—
226C3⋊C8—64no—
227C2×C32—24no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32#89614yes—
231C2×C32#89611yescharacteristic
232C2×C32D3×SD1614yes—
233C2×C32D3×SD1611yescharacteristic
234C2×C32#91514yes—
235C2×C32#91511yescharacteristic
236C2×Q8—271no—
237C2×Q8—541no—
238C2×C8—31no—
239C2×C8—91no—
240C4⋊C4—31no—
241C4⋊C4—91no—
242C4⋊C4—541no—
243C4⋊C4—541no—
244C4⋊C4D3×C3⋊D311yescharacteristic
245C42—271no—
246C42—541no—
247C22×C3—24no—
248C22×C3#381114yes—
249C22×C3D4×C3⋊D311yescharacteristic
250C3×C4—24no—
251C3×C4—24no—
252C3×C4—48no—
253C3×C4—41no—
254C3×C4—44no—
255C3×C4—124no—
256C3×C4—361no—
257C3×C4#379714yes—
258C3×C4#379814yes—
259C3×C4#382811yescharacteristic
260C3×C4Q8×C3⋊D311yescharacteristic
261Dic3—41no—
262Dic3—92no—
263Dic3—98no—
264Dic3—124no—
265Dic3—188no—
266Dic3—1816no—
267Dic3—182no—
268Dic3—188no—
269Dic3—181no—
270Dic3—184no—
271Dic3—364no—
272C32—24no—
273C32#485414yescomplemented
274C32#485411yescharacteristic, complemented
275Q8—272no—
276Q8—541no—
277Q8—1082no—
278C2×C4—21no—
279C2×C4—61no—
280C2×C4—181no—
281C2×C4—271no—
282C2×C4—271no—
283C2×C4—541no—
284C2×C4C2×D3×C3⋊D311yescharacteristic
285C8—61no—
286C8—181no—
287C2×C3—24no—
288C2×C3—24no—
289C2×C3—24no—
290C2×C3#6386514yes—
291C2×C3#6393114yes—
292C2×C3#6394014yes—
293C2×C3#6410511yescharacteristic
294C2×C3SD16×C3⋊D311yescharacteristic
295C2×C3#6412411yescharacteristic
296C22#8903911yescharacteristic
297C4—41no—
298C4—121no—
299C4—272no—
300C4—361no—
301C4—541no—
302C4—542no—
303C4#8901911yescharacteristic
304C4#8902011yescharacteristic
305C3—24no—
306C3#1059043114yescomplemented
307C3#1059255311yescharacteristic, complemented
308C2#1172344211yescharacteristic
309C2#1172355411yescharacteristic
310C2#1172357111yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#475
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
42. #6308Q8semidirect
195. C33#475semidirect
273. C32#4854semidirect
274. C32#4854semidirect
306. C3#10590431semidirect
307. C3#10592553semidirect

Maximal subgroups

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

subgroupindex
#117239282
#117235362
C3×#636352
C32×#8082
#117220502
#117220422
#117220412
#105925533
#105904313