Finite Group Databasebeta

← #13165510

Subgroups — #13165510

Order 1728.

412 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 64 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×#88938C211yesmaximal, characteristic
2#11722944C211yesmaximal, characteristic
3Dic3×#627C211yesmaximal, characteristic
4#11722794C211yesmaximal, characteristic
5#11722779C211yesmaximal, characteristic
6C3×#63740C211yesmaximal, characteristic
7Dic3×#622C211yesmaximal, characteristic
8C4×#3776—31nomaximal
9C2×#6296C2211yescharacteristic
10C2×C3×#627C2211yescharacteristic
11C2×C3⋊#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13#88938C414yescomplemented
14#88815C2211yescharacteristic
15#88807—21no—
16Dic3⋊(C3⋊Dic3)—21no—
17Dic3×C3⋊Dic3C2211yescharacteristic
18C3×#3724C2211yescharacteristic
19C3×Dic32—21no—
20C3×Dic32—21no—
21C2×#3776—31no—
22C4×#627—31no—
23C4×#627—31no—
24Dic3×Dic6—61no—
25#63837—61no—
26C4×#622—31no—
27#63743—31no—
28#63740—31no—
29#63740D311yescharacteristic
30C2×C3⋊(C3⋊Dic3)D411yescharacteristic
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C2×C32×Dic3—21no—
35C2×C32×Dic3—21no—
36C2×C32×Dic3D411yescharacteristic
37#6296—21no—
38#6296C2×C412yes—
39C3×#627—21no—
40C3×#627C2×C412yes—
41C3⋊#622C2×C412yes—
42C3×#622—21no—
43#4790—91nomaximal
44C2×C3×Dic6—61no—
45C2×#627—31no—
46C2×#627—61no—
47C2×#627C2×D311yescharacteristic
48C2×#622—31no—
49C2×#622C2×D311yescharacteristic
50#3776—34no—
51C4⋊(C3⋊Dic3)—61no—
52#3750—61no—
53C4×C3⋊Dic3—31no—
54C3×Dic3⋊C4—61no—
55C3×C4×Dic3—61no—
56C3×C4×Dic3—61no—
57C3×C4×Dic3—61no—
58C3×C4×Dic3—61no—
59#3724—31no—
60#3724C2×D311yescharacteristic
61Dic3⋊Dic3—61no—
62Dic3⋊Dic3—61no—
63Dic3⋊Dic3—61no—
64Dic3⋊Dic3—61no—
65Dic3⋊Dic3—61no—
66Dic3⋊Dic3—121no—
67Dic3⋊Dic3—121no—
68Dic32—21no—
69Dic32—21no—
70Dic32—61no—
71Dic32—61no—
72C22×C33C2×D411yescharacteristic
73C3⋊(C3⋊Dic3)—21no—
74C3×C3⋊Dic3—22no—
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3C22×C411yescharacteristic
77C3×C3⋊Dic3#4111yescharacteristic
78C32×Dic3—22no—
79C32×Dic3—41no—
80C32×Dic3Q1612yescomplemented
81Q8×Dic3—91no—
82C2×#216—91no—
83#887—91no—
84C4×Dic6—181no—
85C4×Dic6—181no—
86C3×#103—91no—
87#816—91no—
88C8×Dic3—91no—
89#807—91no—
90C2×C32×C4—31no—
91C2×C32×C4—61no—
92C2×C32×C4—61no—
93C2×C3⋊Dic3—21no—
94C2×C3⋊Dic3—21no—
95C2×C3⋊Dic3—31no—
96C2×C3⋊Dic3—61no—
97C2×C3⋊Dic3—61no—
98C2×C3⋊Dic3—121no—
99C2×C3⋊Dic3—121no—
100C2×C3⋊Dic3C22×D311yescharacteristic
101C2×C3×Dic3—21no—
102C2×C3×Dic3—21no—
103C2×C3×Dic3—21no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—21no—
107C2×C3×Dic3—41no—
108C2×C3×Dic3—41no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—121no—
116C2×C3×Dic3—121no—
117C3×Dic6—62no—
118C3×Dic6—62no—
119#627—21no—
120#627—32no—
121#627—62no—
122#627—62no—
123#627—61no—
124#627C4×D312yes—
125#622—21no—
126#622—32no—
127C4×Q16—271noSylow
128C2×C33C4×D411yescharacteristic
129C2×C33C2×Q1611yescharacteristic
130C2×C33#13511yescharacteristic
131C2×C3×Q8—91no—
132C2×Dic6—61no—
133C2×Dic6—91no—
134C2×Dic6—181no—
135C2×C3×C8—91no—
136C3×C4⋊C4—91no—
137C3×C42—181no—
138C3×C42—181no—
139#216—94no—
140C4⋊Dic3—91no—
141C4⋊Dic3—181no—
142C4⋊Dic3—181no—
143C4⋊Dic3—181no—
144C4⋊Dic3—361no—
145Dic3⋊C4—61no—
146Dic3⋊C4—181no—
147Dic3⋊C4—181no—
148Dic3⋊C4—181no—
149Dic3⋊C4—181no—
150Dic3⋊C4—361no—
151C4×Dic3—61no—
152C4×Dic3—61no—
153C4×Dic3—61no—
154C4×Dic3—61no—
155C4×Dic3—91no—
156C4×Dic3—181no—
157C4×Dic3—181no—
158C2×C3⋊C8—91no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32—41no—
162C22×C32—41no—
163C22×C32D3×D411yescharacteristic
164C32×C4—32no—
165C32×C4—62no—
166C32×C4—121no—
167C3⋊Dic3—22no—
168C3⋊Dic3—41no—
169C3⋊Dic3—61no—
170C3⋊Dic3—121no—
171C3⋊Dic3—121no—
172C3⋊Dic3—241no—
173C3⋊Dic3—241no—
174C3⋊Dic3C2×C4×D311yescharacteristic
175C3⋊Dic3#23711yescharacteristic
176C3×Dic3—22no—
177C3×Dic3—22no—
178C3×Dic3—22no—
179C3×Dic3—22no—
180C3×Dic3—42no—
181C3×Dic3—41no—
182C3×Dic3—41no—
183C3×Dic3—61no—
184C3×Dic3—61no—
185C3×Dic3—61no—
186C3×Dic3—61no—
187C3×Dic3—62no—
188C3×Dic3—62no—
189C3×Dic3—81no—
190C3×Dic3—122no—
191C3×Dic3—121no—
192C3×Dic3—121no—
193C3×Dic3—241no—
194C2×Q16—271no—
195C4×Q8—271no—
196C4×Q8—271no—
197#107—271no—
198#103—91no—
199#103—271no—
200C4×C8—271no—
201C33C4×Q1611yescharacteristic, complemented, Sylow
202C3×Q8—92no—
203C3×Q8—181no—
204C2×C3×C4—61no—
205C2×C3×C4—61no—
206C2×C3×C4—61no—
207C2×C3×C4—61no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—91no—
211C2×C3×C4—121no—
212C2×C3×C4—121no—
213C2×C3×C4—181no—
214C2×C3×C4—181no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—61no—
218C2×Dic3—61no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—61no—
222C2×Dic3—61no—
223C2×Dic3—91no—
224C2×Dic3—121no—
225C2×Dic3—121no—
226C2×Dic3—181no—
227C2×Dic3—181no—
228C2×Dic3—361no—
229C2×Dic3—361no—
230C2×Dic3GO+(4,𝔽2)11yescharacteristic
231Dic6—62no—
232Dic6—62no—
233Dic6—92no—
234Dic6—182no—
235Dic6—181no—
236Dic6—182no—
237C3×C8—181no—
238C3⋊C8—92no—
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×C32#67⋊C411yescharacteristic
252C2×C32#91611yescharacteristic
253C2×C32D3×Q1611yescharacteristic
254C2×Q8—91no—
255C2×Q8—271no—
256Q16—274no—
257C2×C8—91no—
258C2×C8—271no—
259C4⋊C4—91no—
260C4⋊C4—271no—
261C4⋊C4—541no—
262C4⋊C4—541no—
263C42—181no—
264C42—181no—
265C42—271no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3C2×GO+(4,𝔽2)11yescharacteristic
271C3×C4—62no—
272C3×C4—62no—
273C3×C4—62no—
274C3×C4—62no—
275C3×C4—91no—
276C3×C4—91no—
277C3×C4—122no—
278C3×C4—121no—
279C3×C4—121no—
280C3×C4—182no—
281C3×C4—241no—
282C3×C4—361no—
283Dic3—22no—
284Dic3—41no—
285Dic3—61no—
286Dic3—61no—
287Dic3—61no—
288Dic3—61no—
289Dic3—62no—
290Dic3—62no—
291Dic3—122no—
292Dic3—121no—
293Dic3—121no—
294Dic3—181no—
295Dic3—241no—
296Dic3—361no—
297Dic3—361no—
298Dic3—721no—
299Dic3—721no—
300Dic3#377612yescomplemented
301C32—21no—
302C32—21no—
303C32—41no—
304C32—41no—
305C32#479011yescharacteristic, complemented
306Q8—92no—
307Q8—181no—
308Q8—272no—
309Q8—541no—
310C2×C4—31no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—91no—
314C2×C4—181no—
315C2×C4—181no—
316C2×C4—271no—
317C8—181no—
318C8—272no—
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#6422311yescharacteristic
332C2×C3C4×GO+(4,𝔽2)11yescharacteristic
333C2×C3C2×#377611yescharacteristic
334C22D3×GO+(4,𝔽2)11yescharacteristic
335C4—32no—
336C4—62no—
337C4—91no—
338C4—91no—
339C4—121no—
340C4—182no—
341C4—361no—
342C4—541no—
343C3—21no—
344C3—21no—
345C3—41no—
346C3—41no—
347C3C4×#377611yescharacteristic, complemented
348C2(D32⋊D3)⋊C411yescharacteristic
349C2D3×#377611yescharacteristic
350C2#1172413611yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC3×C3⋊Dic3C22×C4
Fitting, socle, prime radicalC22×C33C2×D4
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33C4×Q16
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
13. #88938C4semidirect
80. C32×Dic3Q16semidirect
201. C33C4×Q16semidirect
300. Dic3#3776semidirect
305. C32#4790semidirect
347. C3C4×#3776semidirect

Maximal subgroups

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

subgroupindex
C2×#889382
#117229442
Dic3×#6272
#117227942
#117227792
C3×#637402
Dic3×#6222
C4×#37763
#47909