Finite Group Databasebeta

← #13165671

Subgroups — #13165671

Order 1728.

358 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 50 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#11723015C211yesmaximal, characteristic
2#11722944C211yesmaximal, characteristic
3#11722794C211yesmaximal, characteristic
4#11722778C211yesmaximal, characteristic
5#11722774C211yesmaximal, characteristic
6C3×#63743C211yesmaximal, characteristic
7#11722732C211yesmaximal, characteristic
8#10593382—31nomaximal
9C2×#6296C2211yescharacteristic
10C2×C3⋊#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12#88815C2211yescharacteristic
13#88807—21no—
14#88807—21no—
15Dic3×C3⋊Dic3C2211yescharacteristic
16C3×#3724C2211yescharacteristic
17C3×#3724C2211yescharacteristic
18C3×Dic3⋊Dic3—21no—
19C3×Dic32—21no—
20C4×#627—31no—
21#63870—31no—
22#63846—61no—
23#63837—61no—
24#63779—31no—
25#63743D311yescharacteristic
26#63742—31no—
27#63740—31no—
28#63740—31no—
29C2×C3⋊(C3⋊Dic3)D411yescharacteristic
30C2×C3×C3⋊Dic3—21no—
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C2×C32×Dic3—21no—
34C2×C32×Dic3—21no—
35C2×C32×Dic3D411yescharacteristic
36#6296—21no—
37#6296Q812yes—
38C3⋊#622—21no—
39C3×#622—21no—
40#4877—91nomaximal
41C2×#627—31no—
42C2×#627—61no—
43C2×#622—31no—
44C2×#622C2×D311yescharacteristic
45#3750—61no—
46#3750—61no—
47C4×C3⋊Dic3—31no—
48C3×C4⋊Dic3—61no—
49C3×Dic3⋊C4—61no—
50C3×Dic3⋊C4—61no—
51C3×Dic3⋊C4—61no—
52C3×C4×Dic3—61no—
53C3×C4×Dic3—61no—
54#3724—31no—
55#3724C2×D311yescharacteristic
56#3724C2×D311yescharacteristic
57Dic3⋊Dic3—21no—
58Dic3⋊Dic3—61no—
59Dic3⋊Dic3—61no—
60Dic3⋊Dic3—61no—
61Dic3⋊Dic3—61no—
62Dic3⋊Dic3—61no—
63Dic3⋊Dic3—121no—
64Dic3⋊Dic3—121no—
65Dic32—21no—
66Dic32—61no—
67Dic32—61no—
68C22×C33C2×D411yescharacteristic
69C3⋊(C3⋊Dic3)—21no—
70C3×C3⋊Dic3—22no—
71C3×C3⋊Dic3—41no—
72C3×C3⋊Dic3C2×Q811yescharacteristic
73C3×C3⋊Dic3#4111yescharacteristic
74C32×Dic3—21no—
75C32×Dic3—41no—
76C32×Dic3—41no—
77#890—91no—
78#889—181no—
79#887—91no—
80C4×Dic6—181no—
81C3×#107—91no—
82#816—91no—
83Dic3⋊C8—91no—
84#810—91no—
85#808—91no—
86C2×C32×C4—31no—
87C2×C32×C4—61no—
88C2×C32×C4—61no—
89C2×C3⋊Dic3—21no—
90C2×C3⋊Dic3—21no—
91C2×C3⋊Dic3—31no—
92C2×C3⋊Dic3—61no—
93C2×C3⋊Dic3—61no—
94C2×C3⋊Dic3—121no—
95C2×C3⋊Dic3—121no—
96C2×C3⋊Dic3C22×D311yescharacteristic
97C2×C3×Dic3—21no—
98C2×C3×Dic3—21no—
99C2×C3×Dic3—21no—
100C2×C3×Dic3—21no—
101C2×C3×Dic3—21no—
102C2×C3×Dic3—21no—
103C2×C3×Dic3—41no—
104C2×C3×Dic3—41no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—121no—
112C2×C3×Dic3—121no—
113#627—32no—
114#627—62no—
115#627—62no—
116#627—61no—
117#622—21no—
118#622—61no—
119#479—271noSylow
120C2×C33C22⋊Q811yescharacteristic
121C2×C33#13511yescharacteristic
122C2×C33#13711yescharacteristic
123C2×Dic6—91no—
124C2×Dic6—181no—
125C2×C3×C8—91no—
126C3×C4⋊C4—91no—
127C3×C4⋊C4—91no—
128C3×C4⋊C4—181no—
129C3×C42—181no—
130C4⋊Dic3—61no—
131C4⋊Dic3—91no—
132C4⋊Dic3—181no—
133C4⋊Dic3—181no—
134Dic3⋊C4—61no—
135Dic3⋊C4—61no—
136Dic3⋊C4—61no—
137Dic3⋊C4—181no—
138Dic3⋊C4—181no—
139Dic3⋊C4—181no—
140Dic3⋊C4—181no—
141Dic3⋊C4—181no—
142Dic3⋊C4—361no—
143Dic3⋊C4—361no—
144C4×Dic3—61no—
145C4×Dic3—61no—
146C4×Dic3—91no—
147C4×Dic3—181no—
148C4×Dic3—181no—
149C2×C3⋊C8—91no—
150C22×C32—21no—
151C22×C32—21no—
152C22×C32—41no—
153C22×C32—41no—
154C22×C32D3×D411yescharacteristic
155C32×C4—61no—
156C32×C4—121no—
157C32×C4—121no—
158C3⋊Dic3—22no—
159C3⋊Dic3—41no—
160C3⋊Dic3—61no—
161C3⋊Dic3—121no—
162C3⋊Dic3—121no—
163C3⋊Dic3—241no—
164C3⋊Dic3—241no—
165C3⋊Dic3#23511yescharacteristic
166C3⋊Dic3D3×Q811yescharacteristic
167C3×Dic3—22no—
168C3×Dic3—41no—
169C3×Dic3—41no—
170C3×Dic3—41no—
171C3×Dic3—41no—
172C3×Dic3—41no—
173C3×Dic3—61no—
174C3×Dic3—61no—
175C3×Dic3—61no—
176C3×Dic3—61no—
177C3×Dic3—62no—
178C3×Dic3—62no—
179C3×Dic3—81no—
180C3×Dic3—81no—
181C3×Dic3—122no—
182C3×Dic3—121no—
183C3×Dic3—122no—
184C3×Dic3—121no—
185#125—271no—
186C4×Q8—271no—
187#107—91no—
188#106—271no—
189C4⋊C8—271no—
190#103—271no—
191#103—271no—
192C33#47911yescharacteristic, complemented, Sylow
193C2×C3×C4—61no—
194C2×C3×C4—61no—
195C2×C3×C4—61no—
196C2×C3×C4—61no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—91no—
200C2×C3×C4—121no—
201C2×C3×C4—121no—
202C2×C3×C4—181no—
203C2×C3×C4—181no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—91no—
213C2×Dic3—121no—
214C2×Dic3—121no—
215C2×Dic3—181no—
216C2×Dic3—181no—
217C2×Dic3—361no—
218C2×Dic3—361no—
219C2×Dic3GO+(4,𝔽2)11yescharacteristic
220Dic6—92no—
221Dic6—182no—
222Dic6—181no—
223Dic6—182no—
224C3×C8—181no—
225C3⋊C8—181no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—21no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—41no—
235C2×C32—41no—
236C2×C32—41no—
237C2×C32—41no—
238C2×C32#89611yescharacteristic
239C2×C32#91211yescharacteristic
240C2×C32#91811yescharacteristic
241C2×Q8—271no—
242C2×C8—91no—
243C2×C8—271no—
244C4⋊C4—91no—
245C4⋊C4—91no—
246C4⋊C4—181no—
247C4⋊C4—271no—
248C4⋊C4—541no—
249C4⋊C4—541no—
250C42—181no—
251C42—271no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3C2×GO+(4,𝔽2)11yescharacteristic
257C3×C4—62no—
258C3×C4—91no—
259C3×C4—91no—
260C3×C4—121no—
261C3×C4—121no—
262C3×C4—121no—
263C3×C4—121no—
264C3×C4—121no—
265C3×C4—182no—
266C3×C4—241no—
267C3×C4—241no—
268C3×C4—361no—
269Dic3—21no—
270Dic3—41no—
271Dic3—41no—
272Dic3—61no—
273Dic3—61no—
274Dic3—61no—
275Dic3—61no—
276Dic3—62no—
277Dic3—62no—
278Dic3—122no—
279Dic3—121no—
280Dic3—122no—
281Dic3—121no—
282Dic3—181no—
283Dic3—361no—
284Dic3—361no—
285Dic3—721no—
286Dic3—721no—
287C32—21no—
288C32—21no—
289C32—41no—
290C32—41no—
291C32#487711yescharacteristic, complemented
292Q8—272no—
293Q8—541no—
294C2×C4—31no—
295C2×C4—61no—
296C2×C4—61no—
297C2×C4—91no—
298C2×C4—181no—
299C2×C4—181no—
300C2×C4—271no—
301C8—181no—
302C8—541no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—41no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3D32⋊Q811yescharacteristic
316C2×C3#6422311yescharacteristic
317C2×C3#6423911yescharacteristic
318C22D3×GO+(4,𝔽2)11yescharacteristic
319C4—61no—
320C4—91no—
321C4—91no—
322C4—121no—
323C4—121no—
324C4—182no—
325C4—361no—
326C4—541no—
327C3—21no—
328C3—21no—
329C3—41no—
330C3—41no—
331C3#1059338211yescharacteristic, complemented
332C2#1172412011yescharacteristic
333C2#1172413511yescharacteristic
334C2#1172414311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socle, prime radicalC22×C33C2×D4
hypocenterC33#479
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
192. C33#479semidirect
291. C32#4877semidirect
331. C3#10593382semidirect

Maximal subgroups

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

subgroupindex
#117230152
#117229442
#117227942
#117227782
#117227742
C3×#637432
#117227322
#105933823
#48779