Finite Group Databasebeta

← #13167043

Subgroups — #13167043

Order 1728.

426 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 55 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#11723689C211yesmaximal, characteristic
2C3×#63865C211yesmaximal, characteristic
3#11723103C211yesmaximal, characteristic
4#11722778C212yesmaximal, complemented
5#11722775C212yesmaximal, complemented
6#10593386—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×#6294—21no—
9C2×C3×#627C2211yescharacteristic
10C2×C3⋊#622C2212yes—
11C3×C4⋊(C3⋊Dic3)—21no—
12C32×Dic3⋊C4—21no—
13#88815C2212yes—
14C3×#3724C2211yescharacteristic
15C3×#3722—21no—
16C3×#896—61no—
17#63900—31no—
18D3×C4⋊Dic3—61no—
19#63865D311yescharacteristic
20#63778—31no—
21#63742—32no—
22#63740—32no—
23C22×C3×C3⋊D3D411yescharacteristic
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C3×C4×C3⋊D3—22no—
28C2×C32×Dic3—21no—
29C2×C32×Dic3—21no—
30#6294—42no—
31C3×#627—22no—
32C3⋊#622—22no—
33#5043—91nomaximal
34C2×C4×C3⋊D3C2×D311yescharacteristic
35C2×C3×C4×D3—61no—
36C2×C3×C4×D3—61no—
37C2×C3×Dic6—61no—
38C2×#627C2×D311yescharacteristic
39C2×#624—61no—
40C2×D3×Dic3—62no—
41C2×#622—32no—
42C32×C4⋊C4—61no—
43C4⋊(C3⋊Dic3)—21no—
44C3×(C2×D3)⋊C4—61no—
45C3×(C2×D3)⋊C4—61no—
46C3×C4⋊Dic3—61no—
47C3×C4⋊Dic3—61no—
48C3×C4⋊Dic3—121no—
49C3×Dic3⋊C4—21no—
50C3×Dic3⋊C4—21no—
51C3×Dic3⋊C4—41no—
52C3×Dic3⋊C4—61no—
53#3724—32no—
54#3724C2×D311yescharacteristic
55Dic3⋊Dic3—62no—
56#3722—21no—
57C22×C33C2×D411yescharacteristic
58C2×C3×C3⋊D3—21no—
59C2×C3×C3⋊D3—21no—
60C2×C3×C3⋊D3SD1612yes—
61C33×C4—21no—
62C3×C3⋊Dic3—42no—
63C3×C3⋊Dic3#4112yes—
64C32×Dic3—41no—
65C32×Dic3—41no—
66C3×C22⋊Q8—91no—
67C2×C4⋊Dic3—91no—
68#896—61no—
69D3×C4⋊C4—181no—
70#835—92no—
71#830—91no—
72#808—92no—
73C22×C3⋊D3#6711yescharacteristic
74C22×C3×D3—61no—
75C22×C3×D3—61no—
76C2×C32×C4—21no—
77C2×C32×C4—21no—
78C2×C32×C4—41no—
79C2×C32×C4—41no—
80C2×C32×C4—61no—
81C2×C32×C4—61no—
82C2×C32×C4#6711yescharacteristic
83C2×C3⋊Dic3—22no—
84C2×C3⋊Dic3C22×D311yescharacteristic
85C4×C3⋊D3—22no—
86C2×C3×Dic3—21no—
87C2×C3×Dic3—21no—
88C2×C3×Dic3—21no—
89C2×C3×Dic3—21no—
90C2×C3×Dic3—41no—
91C2×C3×Dic3—41no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—122no—
97C3×C4×D3—64no—
98C3×C4×D3—122no—
99C3×Dic6—122no—
100#627—22no—
101#624—122no—
102D3×Dic3—124no—
103#622—62no—
104#483—271noSylow
105C2×C33#12311yescharacteristic
106C2×C33C2×SD1611yescharacteristic
107C2×C33#13711yescharacteristic
108C3×C3⋊D3—22no—
109C2×C3×Q8—91no—
110C22×C3×C4—91no—
111C22×Dic3—181no—
112C2×C4×D3—61no—
113C2×C4×D3—61no—
114C2×C4×D3—182no—
115C2×Dic6—61no—
116C3×C4⋊C4—61no—
117C3×C4⋊C4—61no—
118C3×C4⋊C4—91no—
119C3×C4⋊C4—121no—
120C3×C4⋊C4—181no—
121C3×C22⋊C4—181no—
122(C2×D3)⋊C4—61no—
123(C2×D3)⋊C4—61no—
124C4⋊Dic3—61no—
125C4⋊Dic3—61no—
126C4⋊Dic3—92no—
127C4⋊Dic3—121no—
128Dic3⋊C4—21no—
129Dic3⋊C4—61no—
130Dic3⋊C4—182no—
131C2×C3⋊C8—92no—
132C22×C32—21no—
133C22×C32—21no—
134C22×C32—41no—
135C22×C32—41no—
136C22×C32C2×#6711yescharacteristic
137C2×C3⋊D3—21no—
138C2×C3⋊D3—21no—
139C2×C3⋊D3#21512yes—
140C2×C3×D3—62no—
141C2×C3×D3—62no—
142C2×C3×D3—62no—
143C2×C3×D3—62no—
144C2×C3×D3—121no—
145C2×C3×D3—121no—
146C32×C4—22no—
147C32×C4—21no—
148C32×C4—41no—
149C32×C4—42no—
150C32×C4—81no—
151C32×C4—121no—
152C32×C4—121no—
153C3⋊Dic3—42no—
154C3⋊Dic3#23712yes—
155C3×Dic3—41no—
156C3×Dic3—41no—
157C3×Dic3—42no—
158C3×Dic3—41no—
159C3×Dic3—41no—
160C3×Dic3—62no—
161C3×Dic3—62no—
162C3×Dic3—81no—
163C3×Dic3—122no—
164C3×Dic3—122no—
165C3×Dic3—242no—
166C22⋊Q8—91no—
167C2×C4⋊C4—271no—
168#106—272no—
169#103—272no—
170C22⋊C8—271no—
171C33#48311yescharacteristic, complemented, Sylow
172C23×C3—91no—
173C22×D3—61no—
174C22×D3—61no—
175C3×Q8—182no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—61no—
181C2×C3×C4—61no—
182C2×C3×C4—61no—
183C2×C3×C4—61no—
184C2×C3×C4—91no—
185C2×C3×C4—121no—
186C2×C3×C4—121no—
187C2×C3×C4—182no—
188C2×C3×C4—182no—
189C2×C3×C4GO+(4,𝔽2)11yescharacteristic
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—61no—
193C2×Dic3—61no—
194C2×Dic3—62no—
195C2×Dic3—62no—
196C2×Dic3—122no—
197C2×Dic3—362no—
198C4×D3—64no—
199C4×D3—122no—
200C4×D3—364no—
201Dic6—122no—
202C3⋊C8—182no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32#93511yescharacteristic
216C2×C32C2×#21511yescharacteristic
217C2×C32#95111yescharacteristic
218C3⋊D3—22no—
219C3×D3—64no—
220C3×D3—122no—
221C2×Q8—91no—
222C22×C4—91no—
223C22×C4—541no—
224C2×C8—272no—
225C4⋊C4—61no—
226C4⋊C4—91no—
227C4⋊C4—181no—
228C4⋊C4—272no—
229C22⋊C4—181no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—92no—
235C22×C3—181no—
236C22×C3—181no—
237C22×C3C2×GO+(4,𝔽2)11yescharacteristic
238C2×D3—62no—
239C2×D3—62no—
240C2×D3—62no—
241C2×D3—62no—
242C2×D3—121no—
243C2×D3—121no—
244C3×C4—22no—
245C3×C4—21no—
246C3×C4—41no—
247C3×C4—42no—
248C3×C4—81no—
249C3×C4—92no—
250C3×C4—121no—
251C3×C4—121no—
252C3×C4—122no—
253C3×C4—121no—
254C3×C4—121no—
255C3×C4—241no—
256C3×C4—362no—
257Dic3—41no—
258Dic3—41no—
259Dic3—62no—
260Dic3—62no—
261Dic3—122no—
262Dic3—122no—
263Dic3—242no—
264C32—21no—
265C32—21no—
266C32—41no—
267C32—41no—
268C32#504311yescharacteristic, complemented
269C23—91no—
270Q8—182no—
271C2×C4—61no—
272C2×C4—61no—
273C2×C4—91no—
274C2×C4—182no—
275C2×C4—182no—
276C2×C4—1082no—
277C2×C4D32⋊D311yescharacteristic
278C8—542no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—41no—
286C2×C3—41no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—182no—
292C2×C3#6422611yescharacteristic
293C2×C3#6422711yescharacteristic
294C2×C3#6423311yescharacteristic
295D3—64no—
296D3—122no—
297C22—92no—
298C22—181no—
299C22—181no—
300C22C2×D32⋊D311yescharacteristic
301C4—21no—
302C4—92no—
303C4—121no—
304C4—121no—
305C4—362no—
306C3—21no—
307C3—21no—
308C3—41no—
309C3—41no—
310C3#1059338611yescharacteristic, complemented
311C2—182no—
312C2#1172430811yescharacteristic
313C2#1172430911yescharacteristic
314C2#1172431911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C4D32⋊D3
hypocenterC33#483
LCSG ▷ C2×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. #11722778C2semidirect
5. #11722775C2semidirect
171. C33#483semidirect
268. C32#5043semidirect
310. C3#10593386semidirect

Maximal subgroups

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

subgroupindex
#117236892
C3×#638652
#117231032
#117227782
#117227752
#105933863
#50439