Finite Group Databasebeta

← #13167304

Subgroups — #13167304

Order 1728.

462 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 59 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×#88815C211yesmaximal, characteristic
2C3×#63982C211yesmaximal, characteristic
3#11723112C211yesmaximal, characteristic
4#11722779C212yesmaximal, complemented
5#11722775C212yesmaximal, complemented
6#10593500—31nomaximal
7C22×C3×C3⋊Dic3—21no—
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×C3×#627C2211yescharacteristic
10C2×C3⋊#622C2212yes—
11C32×#217—21no—
12#88815—21no—
13#88815C2212yes—
14C3×#3724C2211yescharacteristic
15C3×Dic3⋊Dic3—21no—
16C3×#878—61no—
17#63982D311yescharacteristic
18C2×#3724—31no—
19C2×Dic3⋊Dic3—61no—
20C22⋊#622—31no—
21#63743—32no—
22#63740—32no—
23C23×C33D411yescharacteristic
24C2×C3×C3⋊Dic3—22no—
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3—42no—
27C2×C3×C3⋊Dic3D411yescharacteristic
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C2×C32×Dic3—21no—
30C2×C32×Dic3—21no—
31C3×#627—22no—
32C3⋊#622—22no—
33#5044—91nomaximal
34C22×C3⋊Dic3—21no—
35C22×C3⋊Dic3C2×D311yescharacteristic
36C22×C3×Dic3—61no—
37C22×C3×Dic3—61no—
38C22×C3×Dic3—61no—
39C22×C3×Dic3—61no—
40C22×C3×Dic3—121no—
41C2×C3×Dic6—61no—
42C2×#627C2×D311yescharacteristic
43C2×#622—32no—
44C32×C22⋊C4—61no—
45C3×#217—21no—
46C3×#217—21no—
47C3×#217—41no—
48C3×C4⋊Dic3—61no—
49C3×Dic3⋊C4—61no—
50C3×Dic3⋊C4—61no—
51#3724—32no—
52#3724—61no—
53#3724C2×D311yescharacteristic
54Dic3⋊Dic3—21no—
55Dic3⋊Dic3—62no—
56Dic3⋊Dic3—62no—
57C22×C33—21no—
58C22×C33—21no—
59C22×C33Q1612yes—
60C22×C33C2×D411yescharacteristic
61C3×C3⋊Dic3—21no—
62C3×C3⋊Dic3—42no—
63C3×C3⋊Dic3#4112yes—
64C32×Dic3—41no—
65C32×Dic3—41no—
66C3×C22⋊Q8—91no—
67C2×C4⋊Dic3—91no—
68C2×Dic3⋊C4—181no—
69#878—61no—
70#835—92no—
71#830—91no—
72#807—92no—
73C23×C32—21no—
74C23×C32—21no—
75C23×C32—41no—
76C23×C32—41no—
77C23×C32#6711yescharacteristic
78C2×C32×C4—61no—
79C2×C32×C4—61no—
80C2×C3⋊Dic3—22no—
81C2×C3⋊Dic3—22no—
82C2×C3⋊Dic3—42no—
83C2×C3⋊Dic3#6711yescharacteristic
84C2×C3⋊Dic3C22×D311yescharacteristic
85C2×C3×Dic3—21no—
86C2×C3×Dic3—21no—
87C2×C3×Dic3—21no—
88C2×C3×Dic3—21no—
89C2×C3×Dic3—41no—
90C2×C3×Dic3—41no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—64no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—122no—
99C2×C3×Dic3—122no—
100C2×C3×Dic3—122no—
101C2×C3×Dic3—122no—
102C2×C3×Dic3—124no—
103C3×Dic6—122no—
104#627—22no—
105#622—62no—
106#484—271noSylow
107C2×C33—22no—
108C2×C33#12311yescharacteristic
109C2×C33C2×Q1611yescharacteristic
110C2×C33AGL(1,ℤ/8)11yescharacteristic
111C2×C3×Q8—91no—
112C22×C3×C4—91no—
113C22×C3×C4—181no—
114C22×Dic3—61no—
115C22×Dic3—61no—
116C22×Dic3—61no—
117C22×Dic3—61no—
118C22×Dic3—121no—
119C2×Dic6—61no—
120C3×C4⋊C4—91no—
121C3×C4⋊C4—181no—
122C3×C22⋊C4—61no—
123C3×C22⋊C4—61no—
124C3×C22⋊C4—121no—
125#217—21no—
126C4⋊Dic3—61no—
127C4⋊Dic3—92no—
128C4⋊Dic3—181no—
129Dic3⋊C4—61no—
130Dic3⋊C4—61no—
131Dic3⋊C4—182no—
132Dic3⋊C4—182no—
133C2×C3⋊C8—92no—
134C22×C32—22no—
135C22×C32—22no—
136C22×C32—21no—
137C22×C32—21no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32—22no—
141C22×C32—22no—
142C22×C32—42no—
143C22×C32—41no—
144C22×C32—41no—
145C22×C32—42no—
146C22×C32—41no—
147C22×C32—41no—
148C22×C32—42no—
149C22×C32—42no—
150C22×C32—81no—
151C22×C32—81no—
152C22×C32#21612yes—
153C22×C32C2×#6711yescharacteristic
154C32×C4—121no—
155C32×C4—121no—
156C3⋊Dic3—21no—
157C3⋊Dic3—42no—
158C3⋊Dic3#23712yes—
159C3×Dic3—41no—
160C3×Dic3—41no—
161C3×Dic3—41no—
162C3×Dic3—41no—
163C3×Dic3—62no—
164C3×Dic3—62no—
165C3×Dic3—62no—
166C3×Dic3—81no—
167C3×Dic3—81no—
168C3×Dic3—121no—
169C3×Dic3—122no—
170C3×Dic3—124no—
171C3×Dic3—122no—
172C22⋊Q8—91no—
173C2×C4⋊C4—271no—
174#107—272no—
175#103—272no—
176C22⋊C8—271no—
177C33#48411yescharacteristic, complemented, Sylow
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—41no—
181C23×C3—41no—
182C23×C3GO+(4,𝔽2)11yescharacteristic
183C3×Q8—182no—
184C2×C3×C4—61no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—61no—
188C2×C3×C4—91no—
189C2×C3×C4—91no—
190C2×C3×C4—121no—
191C2×C3×C4—121no—
192C2×C3×C4—182no—
193C2×C3×C4—182no—
194C2×C3×C4—362no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—61no—
198C2×Dic3—61no—
199C2×Dic3—61no—
200C2×Dic3—61no—
201C2×Dic3—64no—
202C2×Dic3—62no—
203C2×Dic3—62no—
204C2×Dic3—122no—
205C2×Dic3—122no—
206C2×Dic3—122no—
207C2×Dic3—122no—
208C2×Dic3—124no—
209Dic6—122no—
210C3⋊C8—182no—
211C2×C32—24no—
212C2×C32—22no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—41no—
220C2×C32—41no—
221C2×C32—41no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—42no—
226C2×C32—44no—
227C2×C32—82no—
228C2×C32#93511yescharacteristic
229C2×C32C2×#21611yescharacteristic
230C2×C32#94911yescharacteristic
231C2×Q8—91no—
232C22×C4—91no—
233C22×C4—181no—
234C2×C8—272no—
235C4⋊C4—91no—
236C4⋊C4—181no—
237C4⋊C4—272no—
238C4⋊C4—541no—
239C22⋊C4—61no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—41no—
249C22×C3—41no—
250C22×C3—42no—
251C22×C3—42no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3—42no—
255C22×C3—42no—
256C22×C3—81no—
257C22×C3—81no—
258C22×C3#377612yes—
259C22×C3C2×GO+(4,𝔽2)11yescharacteristic
260C3×C4—92no—
261C3×C4—121no—
262C3×C4—121no—
263C3×C4—121no—
264C3×C4—121no—
265C3×C4—181no—
266C3×C4—241no—
267C3×C4—241no—
268C3×C4—362no—
269Dic3—41no—
270Dic3—41no—
271Dic3—62no—
272Dic3—62no—
273Dic3—62no—
274Dic3—121no—
275Dic3—122no—
276Dic3—124no—
277Dic3—122no—
278C32—21no—
279C32—21no—
280C32—41no—
281C32—41no—
282C32#504411yescharacteristic, complemented
283C23D32⋊D311yescharacteristic
284Q8—182no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—91no—
288C2×C4—91no—
289C2×C4—182no—
290C2×C4—182no—
291C2×C4—362no—
292C8—542no—
293C2×C3—24no—
294C2×C3—22no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—42no—
302C2×C3—44no—
303C2×C3—41no—
304C2×C3—41no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—82no—
310C2×C3#6423311yescharacteristic
311C2×C3#6423611yescharacteristic
312C2×C3C2×#377611yescharacteristic
313C22—21no—
314C22—21no—
315C22#8893812yes—
316C22C2×D32⋊D311yescharacteristic
317C4—92no—
318C4—121no—
319C4—121no—
320C4—181no—
321C4—362no—
322C3—21no—
323C3—21no—
324C3—41no—
325C3—41no—
326C3#1059350011yescharacteristic, complemented
327C2—22no—
328C2#1172431911yescharacteristic
329C2#1172432211yescharacteristic
330C2C2×#8893811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, prime radicalC23×C33D4
socleC22×C33C2×D4
hypercenterC23D32⋊D3
hypocenterC33#484
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722779C2semidirect
5. #11722775C2semidirect
177. C33#484semidirect
282. C32#5044semidirect
326. C3#10593500semidirect

Maximal subgroups

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

subgroupindex
C2×#888152
C3×#639822
#117231122
#117227792
#117227752
#105935003
#50449