Finite Group Databasebeta

← #13167042

Subgroups — #13167042

Order 1728.

636 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C2×C3×#3797C211yesmaximal, characteristic
2C2×#88988C211yesmaximal, characteristic
3#11723689C211yesmaximal, characteristic
4#11722775C214yesmaximal, complemented
5#10593375—31nomaximal
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C32×Dic6—21no—
8C2×#6294—21no—
9C2×C3×#627C2212yes—
10C2×C3×#624—21no—
11C3×#3797—22no—
12C3×#3797C414yescomplemented
13C2×C3⋊#622C2212yes—
14#88988—22no—
15C3×C4⋊(C3⋊Dic3)—21no—
16#88815C2212yes—
17C2×C3×D3×Q8—61no—
18C2×#3797D311yescharacteristic
19C2×#3787—31no—
20#63900—31no—
21D3×C4⋊Dic3—61no—
22#63740—34no—
23C22×C3×C3⋊D3D411yescharacteristic
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C3×C4×C3⋊D3C2×C412yes—
28C3×C4×C3⋊D3D412yes—
29C2×C32×Dic3—22no—
30C32×Dic6—24no—
31#6294—42no—
32C3×#627—22no—
33C3×#627C2×C414yescomplemented
34C3×#624—24no—
35C3⋊#622—22no—
36C2×#835—91nomaximal
37C2×C32×Q8—61no—
38C2×C4×C3⋊D3C2×D311yescharacteristic
39C3×D3×Q8—64no—
40C3×D3×Q8—64no—
41C2×C3×C4×D3—62no—
42C2×C3×C4×D3—61no—
43C2×C3×C4×D3—61no—
44C2×C3×Dic6—21no—
45C2×C3×Dic6—21no—
46C2×C3×Dic6—41no—
47C2×C3×Dic6—62no—
48C2×#627C2×D312yes—
49C2×#624—21no—
50C2×#624—61no—
51C2×D3×Dic3—62no—
52#3797—22no—
53#3797Dic314yescomplemented
54C2×#622—32no—
55#3787—62no—
56C4⋊(C3⋊Dic3)—21no—
57C3×C4⋊Dic3—61no—
58C3×C4⋊Dic3—61no—
59C3×C4⋊Dic3—121no—
60#3724—32no—
61Dic3⋊Dic3—62no—
62C22×C33C2×D411yescharacteristic
63C2×C3×C3⋊D3C22⋊C412yes—
64C2×C3×C3⋊D3SD1612yes—
65C2×C3×C3⋊D3Q1612yes—
66C33×C4C22⋊C412yes—
67C3×C3⋊Dic3—42no—
68C3×C3⋊Dic3C22×C411yescharacteristic
69C3×C3⋊Dic3C2×D411yescharacteristic
70C32×Dic3—24no—
71C22×C3×Q8—91no—
72C2×D3×Q8—61no—
73C2×C4⋊Dic3—91no—
74C22×C3⋊C8—91no—
75D3×C4⋊C4—181no—
76#835—94no—
77C22×C3⋊D3#6711yescharacteristic
78C22×C3×D3—61no—
79C22×C3×D3—61no—
80C32×Q8—64no—
81C2×C32×C4—21no—
82C2×C32×C4—21no—
83C2×C32×C4—41no—
84C2×C32×C4—41no—
85C2×C32×C4—62no—
86C2×C32×C4#6711yescharacteristic
87C2×C3⋊Dic3—22no—
88C2×C3⋊Dic3C22×D311yescharacteristic
89C4×C3⋊D3C2×Dic312yes—
90C4×C3⋊D3#6712yes—
91C2×C3×Dic3—22no—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—42no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—122no—
99C3×C4×D3—62no—
100C3×C4×D3—62no—
101C3×C4×D3—62no—
102C3×C4×D3—62no—
103C3×C4×D3—68no—
104C3×Dic6—24no—
105C3×Dic6—24no—
106C3×Dic6—44no—
107C3×Dic6—64no—
108C3×Dic6—64no—
109#627—22no—
110#627C2×Dic314yescomplemented
111#624—24no—
112#624—122no—
113D3×Dic3—124no—
114#622—62no—
115C2×#103—271noSylow
116C2×C33C2×C22⋊C411yescharacteristic
117C2×C33C2×SD1611yescharacteristic
118C2×C33C2×Q1611yescharacteristic
119C3×C3⋊D3#10314yescomplemented
120C2×C3×Q8—61no—
121C2×C3×Q8—61no—
122C2×C3×Q8—92no—
123C2×C3×Q8—94no—
124C2×C3×Q8—121no—
125C2×C3×Q8—182no—
126C22×C3×C4—91no—
127C22×C3×C4—181no—
128C22×Dic3—181no—
129D3×Q8—64no—
130D3×Q8—64no—
131C2×C4×D3—61no—
132C2×C4×D3—61no—
133C2×C4×D3—62no—
134C2×C4×D3—182no—
135C2×Dic6—21no—
136C2×Dic6—62no—
137C3×C4⋊C4—181no—
138C4⋊Dic3—61no—
139C4⋊Dic3—61no—
140C4⋊Dic3—92no—
141C4⋊Dic3—121no—
142Dic3⋊C4—182no—
143C2×C3⋊C8—92no—
144C2×C3⋊C8—182no—
145C22×C32—21no—
146C22×C32—21no—
147C22×C32—41no—
148C22×C32—41no—
149C22×C32C2×#6711yescharacteristic
150C2×C3⋊D3#21512yes—
151C2×C3⋊D3#21612yes—
152C2×C3⋊D3#21712yes—
153C2×C3×D3—62no—
154C2×C3×D3—62no—
155C2×C3×D3—62no—
156C2×C3×D3—62no—
157C2×C3×D3—62no—
158C2×C3×D3—62no—
159C32×C4—22no—
160C32×C4—22no—
161C32×C4—42no—
162C32×C4—42no—
163C32×C4—64no—
164C32×C4#21712yes—
165C3⋊Dic3—42no—
166C3⋊Dic3C22×Dic311yescharacteristic
167C3⋊Dic3C2×#6711yescharacteristic
168C3×Dic3—24no—
169C3×Dic3—24no—
170C3×Dic3—44no—
171C3×Dic3—61no—
172C3×Dic3—61no—
173C3×Dic3—61no—
174C3×Dic3—61no—
175C3×Dic3—122no—
176C3×Dic3—122no—
177C3×Dic3—242no—
178C22×Q8—91no—
179C22×C8—271no—
180C2×C4⋊C4—271no—
181#103—274no—
182C33C2×#10311yescharacteristic, complemented, Sylow
183C23×C3—91no—
184C22×D3—61no—
185C22×D3—61no—
186C3×Q8—64no—
187C3×Q8—64no—
188C3×Q8—94no—
189C3×Q8—124no—
190C3×Q8—182no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—62no—
196C2×C3×C4—62no—
197C2×C3×C4—91no—
198C2×C3×C4—92no—
199C2×C3×C4—92no—
200C2×C3×C4—122no—
201C2×C3×C4—184no—
202C2×C3×C4—182no—
203C2×C3×C4GO+(4,𝔽2)11yescharacteristic
204C2×Dic3—22no—
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—62no—
208C2×Dic3—62no—
209C2×Dic3—122no—
210C2×Dic3—362no—
211C4×D3—62no—
212C4×D3—62no—
213C4×D3—62no—
214C4×D3—62no—
215C4×D3—68no—
216C4×D3—364no—
217Dic6—24no—
218Dic6—64no—
219Dic6—64no—
220C3⋊C8—182no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32C2×#21511yescharacteristic
234C2×C32C2×#21611yescharacteristic
235C2×C32C2×#21711yescharacteristic
236C3⋊D3#83514yescomplemented
237C3×D3—64no—
238C3×D3—64no—
239C2×Q8—61no—
240C2×Q8—92no—
241C2×Q8—94no—
242C2×Q8—182no—
243C22×C4—91no—
244C22×C4—181no—
245C22×C4—541no—
246C2×C8—272no—
247C2×C8—542no—
248C4⋊C4—181no—
249C4⋊C4—272no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—92no—
255C22×C3—92no—
256C22×C3—92no—
257C22×C3C2×GO+(4,𝔽2)11yescharacteristic
258C2×D3—62no—
259C2×D3—62no—
260C2×D3—62no—
261C2×D3—62no—
262C2×D3—62no—
263C2×D3—62no—
264C3×C4—22no—
265C3×C4—22no—
266C3×C4—42no—
267C3×C4—42no—
268C3×C4—64no—
269C3×C4—64no—
270C3×C4—91no—
271C3×C4—91no—
272C3×C4—124no—
273C3×C4—362no—
274C3×C4D32⋊C412yes—
275Dic3—24no—
276Dic3—61no—
277Dic3—61no—
278Dic3—61no—
279Dic3—61no—
280Dic3—122no—
281Dic3—122no—
282Dic3—242no—
283C32—21no—
284C32—21no—
285C32—41no—
286C32—41no—
287C32C2×#83511yescharacteristic, complemented
288C23—91no—
289Q8—64no—
290Q8—94no—
291Q8—182no—
292C2×C4—62no—
293C2×C4—91no—
294C2×C4—92no—
295C2×C4—92no—
296C2×C4—184no—
297C2×C4—182no—
298C2×C4—1082no—
299C2×C4D32⋊D311yescharacteristic
300C8—542no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—41no—
313C2×C3—94no—
314C2×C3#6422711yescharacteristic
315C2×C3#6422811yescharacteristic
316C2×C3C2×D32⋊C411yescharacteristic
317D3—64no—
318D3—64no—
319C22—92no—
320C22—92no—
321C22—92no—
322C22C2×D32⋊D311yescharacteristic
323C4—64no—
324C4—91no—
325C4—91no—
326C4—362no—
327C4D32⋊Dic312yes—
328C3—21no—
329C3—21no—
330C3—41no—
331C3—41no—
332C3#1059337511yescharacteristic, complemented
333C2—94no—
334C2#1172430811yescharacteristic
335C2#1172431111yescharacteristic
336C2C2×D32⋊Dic311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D32⋊D3
hypocenterC33C2×#103
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722775C2semidirect
12. C3×#3797C4semidirect
33. C3×#627C2×C4semidirect
53. #3797Dic3semidirect
110. #627C2×Dic3semidirect
119. C3×C3⋊D3#103semidirect
182. C33C2×#103semidirect
236. C3⋊D3#835semidirect
287. C32C2×#835semidirect
332. C3#10593375semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37972
C2×#889882
#117236892
#117227752
#105933753
C2×#8359