Finite Group Databasebeta

← #13167519

Subgroups — #13167519

Order 1728.

412 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 63 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×#88943C211yesmaximal, characteristic
2C4×#6296C211yesmaximal, characteristic
3#11723665C211yesmaximal, characteristic
4C3×#63777C211yesmaximal, characteristic
5#11722796C211yesmaximal, characteristic
6#11722794C211yesmaximal, characteristic
7#11722794C211yesmaximal, characteristic
8#10593369—31nomaximal
9C2×#6296C2211yescharacteristic
10C2×#6296C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13#88943C414yescomplemented
14C3×#3750—21no—
15C3×C4×C3⋊Dic3—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17#88815—21no—
18#88815C2211yescharacteristic
19#88815C2211yescharacteristic
20#88813—21no—
21C2×#3776—31no—
22C4×#627—31no—
23C4×#627—61no—
24#63838—31no—
25#63837—61no—
26#63777D311yescharacteristic
27#63742—31no—
28#63740—31no—
29#63740—31no—
30C2×C33×C4D411yescharacteristic
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3C2311yescharacteristic
37#6296—21no—
38#6296—21no—
39#6296C2×C412yes—
40#6296C2×C412yes—
41C3×#622—21no—
42C3×#622C2×C412yes—
43#4716—91nomaximal
44C2×#627—31no—
45C2×#627—31no—
46C2×#627—61no—
47C2×#627—61no—
48C2×#622C2×D311yescharacteristic
49C2×#622C2×D311yescharacteristic
50#3776—34no—
51#3750—21no—
52C4×C3⋊Dic3—21no—
53C4×C3⋊Dic3C2×D311yescharacteristic
54C3×Dic3⋊C4—61no—
55C3×Dic3⋊C4—61no—
56C3×Dic3⋊C4—121no—
57C3×C4×Dic3—61no—
58C3×C4×Dic3—61no—
59C3×C4×Dic3—61no—
60C3×C4×Dic3—61no—
61C3×C4×Dic3—121no—
62#3724—31no—
63#3724—31no—
64#3724—61no—
65Dic3⋊Dic3—61no—
66Dic3⋊Dic3—61no—
67Dic3⋊Dic3—61no—
68Dic3⋊Dic3—61no—
69Dic32—61no—
70Dic32—61no—
71Dic32—61no—
72C22×C33C2×D411yescharacteristic
73C33×C4—21no—
74C3×C3⋊Dic3—22no—
75C3×C3⋊Dic3—22no—
76C3×C3⋊Dic3—21no—
77C3×C3⋊Dic3—41no—
78C3×C3⋊Dic3—41no—
79C3×C3⋊Dic3C22×C411yescharacteristic
80C3×C3⋊Dic3#4111yescharacteristic
81C2×Dic12—91no—
82#887—181no—
83C4×Dic6—91no—
84C4×Dic6—91no—
85C4×Dic6—181no—
86C3×#97—91no—
87#817—91no—
88#816—91no—
89#816—91no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4—41no—
93C2×C32×C4—41no—
94C2×C32×C4D1211yescharacteristic
95C2×C3⋊Dic3—21no—
96C2×C3⋊Dic3—21no—
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—21no—
99C2×C3⋊Dic3D1211yescharacteristic
100C2×C3⋊Dic3C22×D311yescharacteristic
101C2×C3×Dic3—61no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—61no—
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—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—121no—
114C2×C3×Dic3—121no—
115C2×C3×Dic3—121no—
116C2×C3×Dic3—121no—
117#627—32no—
118#627—32no—
119#627—62no—
120#627—62no—
121#627—62no—
122#627—62no—
123#627—61no—
124#627—61no—
125#622—21no—
126#622C4×D312yes—
127#441—271noSylow
128C2×C33C4×D411yescharacteristic
129C2×C33#13711yescharacteristic
130C2×C33#13711yescharacteristic
131C2×Dic6—91no—
132C2×Dic6—91no—
133C2×Dic6—181no—
134C2×Dic6—181no—
135C2×C3×C8—91no—
136C2×C3×C8—91no—
137C3×C4⋊C4—181no—
138C3×C42—91no—
139C3×C42—181no—
140C4⋊Dic3—91no—
141C4⋊Dic3—91no—
142C4⋊Dic3—181no—
143Dic3⋊C4—61no—
144Dic3⋊C4—61no—
145Dic3⋊C4—121no—
146Dic3⋊C4—181no—
147Dic3⋊C4—181no—
148Dic3⋊C4—181no—
149Dic3⋊C4—181no—
150C4×Dic3—61no—
151C4×Dic3—61no—
152C4×Dic3—61no—
153C4×Dic3—61no—
154C4×Dic3—121no—
155C4×Dic3—181no—
156C4×Dic3—181no—
157C4×Dic3—181no—
158Dic12—94no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32—41no—
162C22×C32—41no—
163C22×C32C2×D1211yescharacteristic
164C32×C4—22no—
165C32×C4—21no—
166C32×C4—41no—
167C32×C4—42no—
168C32×C4—81no—
169C3⋊Dic3—22no—
170C3⋊Dic3—22no—
171C3⋊Dic3—21no—
172C3⋊Dic3—41no—
173C3⋊Dic3—41no—
174C3⋊Dic3C2×C4×D311yescharacteristic
175C3⋊Dic3#23511yescharacteristic
176C3×Dic3—61no—
177C3×Dic3—61no—
178C3×Dic3—61no—
179C3×Dic3—61no—
180C3×Dic3—62no—
181C3×Dic3—62no—
182C3×Dic3—62no—
183C3×Dic3—62no—
184C3×Dic3—62no—
185C3×Dic3—121no—
186C3×Dic3—122no—
187C3×Dic3—122no—
188C3×Dic3—121no—
189C3×Dic3—121no—
190C3×Dic3—122no—
191C3×Dic3—121no—
192C3×Dic3—121no—
193C3×Dic3—241no—
194C2×Q16—271no—
195C4×Q8—271no—
196C4×Q8—271no—
197#106—271no—
198#103—271no—
199#103—271no—
200#97—91no—
201C33#44111yescharacteristic, complemented, Sylow
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—91no—
207C2×C3×C4—91no—
208C2×C3×C4—181no—
209C2×C3×C4—181no—
210C2×C3×C4—181no—
211C2×C3×C4—181no—
212C2×C3×C4GO+(4,𝔽2)11yescharacteristic
213C2×Dic3—61no—
214C2×Dic3—61no—
215C2×Dic3—61no—
216C2×Dic3—61no—
217C2×Dic3—61no—
218C2×Dic3—61no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—61no—
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225C2×Dic3—121no—
226C2×Dic3—121no—
227C2×Dic3—121no—
228C2×Dic3—121no—
229Dic6—92no—
230Dic6—92no—
231Dic6—182no—
232Dic6—182no—
233Dic6—181no—
234Dic6—181no—
235Dic6—182no—
236Dic6—182no—
237C3×C8—92no—
238C3×C8—181no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—41no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—41no—
251C2×C32C4×D1211yescharacteristic
252C2×C32#90911yescharacteristic
253C2×C32#90911yescharacteristic
254C2×Q8—271no—
255C2×Q8—271no—
256Q16—274no—
257C2×C8—91no—
258C2×C8—91no—
259C4⋊C4—181no—
260C4⋊C4—271no—
261C4⋊C4—271no—
262C4⋊C4—541no—
263C42—91no—
264C42—181no—
265C42—541no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3C2×GO+(4,𝔽2)11yescharacteristic
271C3×C4—22no—
272C3×C4—21no—
273C3×C4—41no—
274C3×C4—42no—
275C3×C4—81no—
276C3×C4—91no—
277C3×C4—91no—
278C3×C4—182no—
279C3×C4—182no—
280C3×C4—181no—
281C3×C4—361no—
282C3×C4—361no—
283Dic3—61no—
284Dic3—61no—
285Dic3—61no—
286Dic3—61no—
287Dic3—62no—
288Dic3—62no—
289Dic3—62no—
290Dic3—62no—
291Dic3—62no—
292Dic3—121no—
293Dic3—122no—
294Dic3—122no—
295Dic3—121no—
296Dic3—121no—
297Dic3—122no—
298Dic3—121no—
299Dic3—121no—
300Dic3—241no—
301C32—21no—
302C32—21no—
303C32—41no—
304C32—41no—
305C32#471611yescharacteristic, complemented
306Q8—272no—
307Q8—272no—
308Q8—541no—
309Q8—541no—
310C2×C4—91no—
311C2×C4—91no—
312C2×C4—181no—
313C2×C4—181no—
314C2×C4—181no—
315C2×C4—181no—
316C2×C4#157⋊D311yescharacteristic
317C8—92no—
318C8—181no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—41no—
326C2×C3—41no—
327C2×C3—41no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3#6422611yescharacteristic
332C2×C3C4×GO+(4,𝔽2)11yescharacteristic
333C2×C3#6423911yescharacteristic
334C22C2×#157⋊D311yescharacteristic
335C4—21no—
336C4—91no—
337C4—91no—
338C4—181no—
339C4—182no—
340C4—182no—
341C4—361no—
342C4—361no—
343C3—21no—
344C3—21no—
345C3—41no—
346C3—41no—
347C3#1059336911yescharacteristic, complemented
348C2#1172433811yescharacteristic
349C2C4×#157⋊D311yescharacteristic
350C2#1172435211yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC2×C33×C4D4
socle, prime radicalC22×C33C2×D4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#441
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
13. #88943C4semidirect
201. C33#441semidirect
305. C32#4716semidirect
347. C3#10593369semidirect

Maximal subgroups

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

subgroupindex
C2×#889432
C4×#62962
#117236652
C3×#637772
#117227962
#117227942
#105933693
#47169