Finite Group Databasebeta

← #13165652

Subgroups — #13165652

Order 1728.

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

normal subgroups only

This group has 398 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#11723037C211yesmaximal, characteristic
2#11722941C211yesmaximal, characteristic
3#11722796C211yesmaximal, characteristic
4#11722778C211yesmaximal, characteristic
5#11722775C211yesmaximal, characteristic
6C3×#63740C211yesmaximal, characteristic
7#11722732C211yesmaximal, characteristic
8#10593385—31nomaximal
9C2×#6293—21no—
10C2×C3×#627C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13#88815C2211yescharacteristic
14#88815C2211yescharacteristic
15Dic3×C3⋊Dic3—21no—
16Dic3×C3⋊Dic3C2211yescharacteristic
17C3×#3724C2211yescharacteristic
18C3×Dic3⋊Dic3—21no—
19C3×Dic3⋊Dic3—21no—
20#63919—31no—
21#63843—61no—
22#63840—61no—
23#63838—31no—
24#63779—31no—
25#63742—31no—
26#63742—31no—
27#63740—31no—
28#63740D311yescharacteristic
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#6293—42no—
37C3×#627—21no—
38C3×#627Q812yescomplemented
39C3⋊#622—21no—
40C3×#622—21no—
41#4791—91nomaximal
42C2×#634—61no—
43C2×C3×Dic6—61no—
44C2×#627—61no—
45C2×#627—61no—
46C2×#627—121no—
47C2×#627C2×D311yescharacteristic
48C2×#622—31no—
49C2×#622C2×D311yescharacteristic
50C4×C3⋊Dic3—31no—
51C4×C3⋊Dic3—61no—
52C3×C4⋊Dic3—61no—
53C3×C4⋊Dic3—61no—
54C3×Dic3⋊C4—61no—
55C3×Dic3⋊C4—61no—
56C3×Dic3⋊C4—61no—
57#3724—31no—
58#3724—31no—
59#3724C2×D311yescharacteristic
60Dic3⋊Dic3—21no—
61Dic3⋊Dic3—21no—
62Dic3⋊Dic3—61no—
63Dic3⋊Dic3—61no—
64Dic32—61no—
65Dic32—61no—
66Dic32—61no—
67Dic32—61no—
68Dic32—121no—
69C22×C33C2×D411yescharacteristic
70C3⋊(C3⋊Dic3)SD1612yes—
71C3×C3⋊Dic3—41no—
72C3×C3⋊Dic3—41no—
73C3×C3⋊Dic3C2×Q811yescharacteristic
74C3×C3⋊Dic3#4111yescharacteristic
75C32×Dic3—22no—
76C32×Dic3—21no—
77C32×Dic3—41no—
78Q8×Dic3—91no—
79#888—91no—
80#888—181no—
81#887—181no—
82C3×#103—91no—
83#835—91no—
84#817—91no—
85Dic3⋊C8—91no—
86#808—91no—
87C2×C32×C4—31no—
88C2×C32×C4—61no—
89C2×C32×C4—61no—
90C2×C3⋊Dic3—21no—
91C2×C3⋊Dic3—21no—
92C2×C3⋊Dic3—31no—
93C2×C3⋊Dic3—61no—
94C2×C3⋊Dic3—61no—
95C2×C3⋊Dic3—121no—
96C2×C3⋊Dic3—121no—
97C2×C3⋊Dic3C22×D311yescharacteristic
98#634—122no—
99C2×C3×Dic3—21no—
100C2×C3×Dic3—21no—
101C2×C3×Dic3—21no—
102C2×C3×Dic3—21no—
103C2×C3×Dic3—21no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—41no—
106C2×C3×Dic3—41no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—121no—
114C2×C3×Dic3—121no—
115C3×Dic6—62no—
116C3×Dic6—62no—
117#627—21no—
118#627—122no—
119#627—122no—
120#627—124no—
121#627Dic612yescomplemented
122#622—21no—
123#622—61no—
124#475—271noSylow
125C2×C33C22⋊Q811yescharacteristic
126C2×C33C2×SD1611yescharacteristic
127C2×C33#13711yescharacteristic
128C2×C3×Q8—91no—
129C2×Dic6—61no—
130C2×Dic6—181no—
131C2×Dic6—181no—
132C2×Dic6—181no—
133C2×Dic6—361no—
134C2×C3×C8—91no—
135C3×C4⋊C4—91no—
136C3×C4⋊C4—181no—
137C3×C4⋊C4—181no—
138C4⋊Dic3—61no—
139C4⋊Dic3—61no—
140C4⋊Dic3—91no—
141C4⋊Dic3—91no—
142Dic3⋊C4—61no—
143Dic3⋊C4—61no—
144Dic3⋊C4—61no—
145Dic3⋊C4—181no—
146Dic3⋊C4—181no—
147C4×Dic3—91no—
148C4×Dic3—181no—
149C4×Dic3—181no—
150C4×Dic3—181no—
151C4×Dic3—181no—
152C4×Dic3—181no—
153C4×Dic3—361no—
154C2×C3⋊C8—91no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32—41no—
158C22×C32—41no—
159C22×C32D3×D411yescharacteristic
160C32×C4—62no—
161C32×C4—61no—
162C32×C4—121no—
163C3⋊Dic3—32no—
164C3⋊Dic3—41no—
165C3⋊Dic3—41no—
166C3⋊Dic3—62no—
167C3⋊Dic3—62no—
168C3⋊Dic3—122no—
169C3⋊Dic3—122no—
170C3⋊Dic3C2×Dic611yescharacteristic
171C3⋊Dic3#23711yescharacteristic
172C3×Dic3—22no—
173C3×Dic3—22no—
174C3×Dic3—22no—
175C3×Dic3—41no—
176C3×Dic3—42no—
177C3×Dic3—41no—
178C3×Dic3—42no—
179C3×Dic3—41no—
180C3×Dic3—61no—
181C3×Dic3—61no—
182C3×Dic3—61no—
183C3×Dic3—61no—
184C3×Dic3—121no—
185C3×Dic3—121no—
186C3×Dic3—122no—
187C3×Dic3—122no—
188C3×Dic3—121no—
189C3×Dic3—121no—
190C4⋊Q8—271no—
191C4×Q8—271no—
192#106—271no—
193#106—271no—
194C4⋊C8—271no—
195#103—91no—
196#103—271no—
197C33#47511yescharacteristic, complemented, Sylow
198C3×Q8—92no—
199C3×Q8—181no—
200C2×C3×C4—61no—
201C2×C3×C4—61no—
202C2×C3×C4—61no—
203C2×C3×C4—61no—
204C2×C3×C4—61no—
205C2×C3×C4—61no—
206C2×C3×C4—91no—
207C2×C3×C4—121no—
208C2×C3×C4—121no—
209C2×C3×C4—181no—
210C2×C3×C4—181no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—61no—
214C2×Dic3—61no—
215C2×Dic3—61no—
216C2×Dic3—61no—
217C2×Dic3—61no—
218C2×Dic3—61no—
219C2×Dic3—91no—
220C2×Dic3—121no—
221C2×Dic3—121no—
222C2×Dic3—181no—
223C2×Dic3—181no—
224C2×Dic3—361no—
225C2×Dic3—361no—
226C2×Dic3GO+(4,𝔽2)11yescharacteristic
227Dic6—62no—
228Dic6—62no—
229Dic6—362no—
230Dic6—362no—
231Dic6—362no—
232Dic6—364no—
233C3×C8—181no—
234C3⋊C8—181no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—41no—
242C2×C32—41no—
243C2×C32—41no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32—41no—
247C2×C32#87811yescharacteristic
248C2×C32D3×SD1611yescharacteristic
249C2×C32#91811yescharacteristic
250C2×Q8—91no—
251C2×Q8—541no—
252C2×C8—91no—
253C2×C8—271no—
254C4⋊C4—91no—
255C4⋊C4—181no—
256C4⋊C4—181no—
257C4⋊C4—271no—
258C4⋊C4—271no—
259C42—271no—
260C42—541no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3C2×GO+(4,𝔽2)11yescharacteristic
266C3×C4—62no—
267C3×C4—62no—
268C3×C4—62no—
269C3×C4—91no—
270C3×C4—91no—
271C3×C4—121no—
272C3×C4—122no—
273C3×C4—121no—
274C3×C4—122no—
275C3×C4—121no—
276C3×C4—361no—
277C3×C4—361no—
278Dic3—22no—
279Dic3—21no—
280Dic3—41no—
281Dic3—61no—
282Dic3—61no—
283Dic3—61no—
284Dic3—61no—
285Dic3—92no—
286Dic3—121no—
287Dic3—121no—
288Dic3—122no—
289Dic3—122no—
290Dic3—121no—
291Dic3—121no—
292Dic3—182no—
293Dic3—182no—
294Dic3—362no—
295Dic3—362no—
296C32—21no—
297C32—21no—
298C32—41no—
299C32—41no—
300C32#479111yescharacteristic, complemented
301Q8—92no—
302Q8—181no—
303Q8—1082no—
304C2×C4—31no—
305C2×C4—61no—
306C2×C4—61no—
307C2×C4—91no—
308C2×C4—181no—
309C2×C4—181no—
310C2×C4—271no—
311C8—181no—
312C8—541no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—41no—
325C2×C3D32⋊Q811yescharacteristic
326C2×C3#6422711yescharacteristic
327C2×C3#6423911yescharacteristic
328C22D3×GO+(4,𝔽2)11yescharacteristic
329C4—62no—
330C4—61no—
331C4—91no—
332C4—91no—
333C4—121no—
334C4—272no—
335C4—361no—
336C4—361no—
337C3—21no—
338C3—21no—
339C3—41no—
340C3—41no—
341C3#1059338511yescharacteristic, complemented
342C2#1172411911yescharacteristic
343C2#1172412911yescharacteristic
344C2#1172414311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊Dic3C23
Fitting, socle, prime radicalC22×C33C2×D4
hypocenterC33#475
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
38. C3×#627Q8semidirect
121. #627Dic6semidirect
197. C33#475semidirect
300. C32#4791semidirect
341. C3#10593385semidirect

Maximal subgroups

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

subgroupindex
#117230372
#117229412
#117227962
#117227782
#117227752
C3×#637402
#117227322
#105933853
#47919