Finite Group Databasebeta

← #13165745

Subgroups — #13165745

Order 1728.

356 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 54 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#11723032C211yesmaximal, characteristic
2#11723015C211yesmaximal, characteristic
3#11722797C211yesmaximal, characteristic
4#11722779C211yesmaximal, characteristic
5#11722778C211yesmaximal, characteristic
6C3×#63742C211yesmaximal, characteristic
7#11722724C211yesmaximal, characteristic
8#10593392—31nomaximal
9C2×C3×#627—21no—
10C2×C3⋊#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12#88815C2211yescharacteristic
13#88815C2211yescharacteristic
14#88807—21no—
15#88806—21no—
16Dic3×C3⋊Dic3C2211yescharacteristic
17C3×#3724C2211yescharacteristic
18C3×#3724C2211yescharacteristic
19C3×Dic3⋊Dic3—21no—
20#63873—31no—
21#63870—31no—
22#63846—61no—
23#63839—61no—
24#63777—31no—
25#63743—31no—
26#63743—31no—
27#63742—31no—
28#63742D311yescharacteristic
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
36C3×#627—42no—
37C3⋊#622Q812yescomplemented
38C3×#622Q812yescomplemented
39#4856—91nomaximal
40C2×C3×Dic6—61no—
41C2×C3×Dic6—61no—
42C2×#627—21no—
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—
52#3724—31no—
53#3724—31no—
54#3724—61no—
55#3724—61no—
56#3724—121no—
57#3724C2×D311yescharacteristic
58#3724C2×D311yescharacteristic
59Dic3⋊Dic3—21no—
60Dic3⋊Dic3—61no—
61Dic3⋊Dic3—61no—
62Dic3⋊Dic3—61no—
63Dic3⋊Dic3—61no—
64Dic3⋊Dic3—121no—
65Dic32—61no—
66Dic32—61no—
67C22×C33C2×D411yescharacteristic
68C3⋊(C3⋊Dic3)—21no—
69C3×C3⋊Dic3—41no—
70C3×C3⋊Dic3—41no—
71C3×C3⋊Dic3C2×Q811yescharacteristic
72C3×C3⋊Dic3C2×Q811yescharacteristic
73C32×Dic3—21no—
74C32×Dic3—41no—
75C32×Dic3—41no—
76#944—181no—
77#890—91no—
78#889—181no—
79#888—91no—
80C3×#106—91no—
81#818—91no—
82#815—91no—
83#808—91no—
84#807—91no—
85C2×C32×C4—31no—
86C2×C32×C4—61no—
87C2×C32×C4—61no—
88C2×C3⋊Dic3—21no—
89C2×C3⋊Dic3—21no—
90C2×C3⋊Dic3—31no—
91C2×C3⋊Dic3—61no—
92C2×C3⋊Dic3—61no—
93C2×C3⋊Dic3—121no—
94C2×C3⋊Dic3—121no—
95C2×C3⋊Dic3C22×D311yescharacteristic
96C2×C3×Dic3—21no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—21no—
99C2×C3×Dic3—21no—
100C2×C3×Dic3—21no—
101C2×C3×Dic3—21no—
102C2×C3×Dic3—41no—
103C2×C3×Dic3—41no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—121no—
111C2×C3×Dic3—121no—
112C3×Dic6—122no—
113C3×Dic6—122no—
114#627—42no—
115#622—32no—
116#622Dic612yescomplemented
117#501—271noSylow
118C2×C33C4⋊Q811yescharacteristic
119C2×C33AGL(1,ℤ/8)11yescharacteristic
120C2×C33#13711yescharacteristic
121C2×C3×Q8—181no—
122C2×Dic6—61no—
123C2×Dic6—61no—
124C2×C3×C8—91no—
125C3×C4⋊C4—91no—
126C3×C4⋊C4—91no—
127C3×C4⋊C4—181no—
128C4⋊Dic3—61no—
129C4⋊Dic3—91no—
130C4⋊Dic3—91no—
131C4⋊Dic3—181no—
132Dic3⋊C4—61no—
133Dic3⋊C4—61no—
134Dic3⋊C4—61no—
135Dic3⋊C4—181no—
136Dic3⋊C4—181no—
137Dic3⋊C4—181no—
138Dic3⋊C4—181no—
139Dic3⋊C4—181no—
140Dic3⋊C4—181no—
141Dic3⋊C4—181no—
142Dic3⋊C4—361no—
143Dic3⋊C4—361no—
144C4×Dic3—91no—
145C4×Dic3—181no—
146C4×Dic3—181no—
147C2×C3⋊C8—91no—
148C22×C32—21no—
149C22×C32—21no—
150C22×C32—41no—
151C22×C32—41no—
152C22×C32D3×D411yescharacteristic
153C32×C4—61no—
154C32×C4—121no—
155C32×C4—121no—
156C3⋊Dic3—41no—
157C3⋊Dic3—41no—
158C3⋊Dic3—62no—
159C3⋊Dic3—61no—
160C3⋊Dic3—121no—
161C3⋊Dic3—122no—
162C3⋊Dic3—241no—
163C3⋊Dic3C2×Dic611yescharacteristic
164C3⋊Dic3D3×Q811yescharacteristic
165C3×Dic3—22no—
166C3×Dic3—41no—
167C3×Dic3—41no—
168C3×Dic3—41no—
169C3×Dic3—41no—
170C3×Dic3—41no—
171C3×Dic3—61no—
172C3×Dic3—61no—
173C3×Dic3—61no—
174C3×Dic3—61no—
175C3×Dic3—81no—
176C3×Dic3—81no—
177C3×Dic3—121no—
178C3×Dic3—121no—
179C3×Dic3—122no—
180C3×Dic3—121no—
181C3×Dic3—121no—
182C3×Dic3—241no—
183C4⋊Q8—271no—
184#125—271no—
185#107—271no—
186#107—271no—
187#106—91no—
188#106—271no—
189#97—271no—
190C33#50111yescharacteristic, complemented, Sylow
191C3×Q8—362no—
192C2×C3×C4—61no—
193C2×C3×C4—61no—
194C2×C3×C4—61no—
195C2×C3×C4—61no—
196C2×C3×C4—61no—
197C2×C3×C4—61no—
198C2×C3×C4—91no—
199C2×C3×C4—121no—
200C2×C3×C4—121no—
201C2×C3×C4—181no—
202C2×C3×C4—181no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—61no—
210C2×Dic3—61no—
211C2×Dic3—91no—
212C2×Dic3—121no—
213C2×Dic3—121no—
214C2×Dic3—181no—
215C2×Dic3—181no—
216C2×Dic3—361no—
217C2×Dic3—361no—
218C2×Dic3GO+(4,𝔽2)11yescharacteristic
219Dic6—122no—
220Dic6—122no—
221C3×C8—92no—
222C3⋊C8—92no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—41no—
235C2×C32#88811yescharacteristic
236C2×C32#91411yescharacteristic
237C2×C32#91511yescharacteristic
238C2×Q8—181no—
239C2×C8—91no—
240C2×C8—271no—
241C4⋊C4—91no—
242C4⋊C4—91no—
243C4⋊C4—181no—
244C4⋊C4—271no—
245C4⋊C4—271no—
246C4⋊C4—541no—
247C4⋊C4—541no—
248C42—271no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3C2×GO+(4,𝔽2)11yescharacteristic
254C3×C4—62no—
255C3×C4—91no—
256C3×C4—91no—
257C3×C4—121no—
258C3×C4—121no—
259C3×C4—121no—
260C3×C4—121no—
261C3×C4—121no—
262C3×C4—241no—
263C3×C4—241no—
264C3×C4—361no—
265C3×C4—361no—
266Dic3—21no—
267Dic3—41no—
268Dic3—41no—
269Dic3—61no—
270Dic3—61no—
271Dic3—61no—
272Dic3—61no—
273Dic3—121no—
274Dic3—121no—
275Dic3—122no—
276Dic3—121no—
277Dic3—121no—
278Dic3—182no—
279Dic3—181no—
280Dic3—241no—
281Dic3—361no—
282Dic3—362no—
283Dic3—721no—
284C32—21no—
285C32—21no—
286C32—41no—
287C32—41no—
288C32#485611yescharacteristic, complemented
289Q8—362no—
290C2×C4—31no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—91no—
294C2×C4—181no—
295C2×C4—181no—
296C2×C4—271no—
297C8—92no—
298C8—272no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3#6422211yescharacteristic
312C2×C3#6422611yescharacteristic
313C2×C3#6423611yescharacteristic
314C22D3×GO+(4,𝔽2)11yescharacteristic
315C4—61no—
316C4—91no—
317C4—91no—
318C4—121no—
319C4—121no—
320C4—361no—
321C4—361no—
322C4—541no—
323C3—21no—
324C3—21no—
325C3—41no—
326C3—41no—
327C3#1059339211yescharacteristic, complemented
328C2#1172412511yescharacteristic
329C2#1172413911yescharacteristic
330C2#1172414411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socle, prime radicalC22×C33C2×D4
hypocenterC33#501
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
37. C3⋊#622Q8semidirect
38. C3×#622Q8semidirect
116. #622Dic6semidirect
190. C33#501semidirect
288. C32#4856semidirect
327. C3#10593392semidirect

Maximal subgroups

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

subgroupindex
#117230322
#117230152
#117227972
#117227792
#117227782
C3×#637422
#117227242
#105933923
#48569