Finite Group Databasebeta

← #13167508

Subgroups — #13167508

Order 1728.

482 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 61 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×#88942C212yesmaximal
2C4⋊#6296C211yesmaximal, characteristic, complemented
3C4×#6295C211yesmaximal, characteristic
4C3×C4⋊#622C211yesmaximal, characteristic, complemented
5#11722794C212yesmaximal, complemented
6#10593437—31nomaximal
7C2×C3×C4×C3⋊D3—21no—
8C2×#6296C2212yes—
9C2×#6295C2211yescharacteristic
10C2×C3×#622C2212yes—
11#88942—24no—
12C3×C4⋊(C3⋊Dic3)—21no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14#88815—21no—
15#88815C2211yescharacteristic
16#88814—21no—
17C2×#3775—32no—
18#63919—31no—
19#63918—61no—
20C4×#626—61no—
21C4×#625—31no—
22C4⋊#622D311yescharacteristic, complemented
23#63740—32no—
24C22×C3×C3⋊D3—21no—
25C2×C33×C4D411yescharacteristic
26C2×C3×C3⋊Dic3—22no—
27C2×C3×C3⋊Dic3—21no—
28C2×C3×C3⋊Dic3D411yescharacteristic
29C2×C3×C3⋊Dic3C2311yescharacteristic
30C3×C4×C3⋊D3—42no—
31#6296—22no—
32#6296—22no—
33#6295—21no—
34#6295D412yes—
35C3×#622—22no—
36#4841—91nomaximal
37C2×C4×C3⋊D3—21no—
38C2×C3×C4×D3—61no—
39C2×C3×C4×D3—61no—
40C2×C3×C4×D3—121no—
41C2×#627—32no—
42C2×#627—62no—
43C2×#626—61no—
44C2×#625—31no—
45C2×#622C2×D312yes—
46#3775—64no—
47C4⋊(C3⋊Dic3)—21no—
48C4×C3⋊Dic3C2×D311yescharacteristic
49C3×C4⋊Dic3—61no—
50C3×C4⋊Dic3—61no—
51C3×C4⋊Dic3—121no—
52C3×C4×Dic3—61no—
53C3×C4×Dic3—61no—
54#3724—31no—
55#3724—61no—
56Dic3⋊Dic3—61no—
57Dic3⋊Dic3—62no—
58#3722—61no—
59(C2×D3)⋊Dic3—61no—
60(C2×D3)⋊Dic3—61no—
61C22×C33C2×D411yescharacteristic
62C2×C3×C3⋊D3—22no—
63C2×C3×C3⋊D3—41no—
64C2×C3×C3⋊D3—41no—
65C33×C4SD1612yescomplemented
66C3×C3⋊Dic3—21no—
67C3×C3⋊Dic3—42no—
68C3×C3⋊Dic3—41no—
69C3×C3⋊Dic3C2×D411yescharacteristic
70C3×C3⋊Dic3#4111yescharacteristic
71C4×#67—181no—
72C2×#204—92no—
73#888—181no—
74C4×D12—91no—
75#869—91no—
76C3×C4⋊C8—91no—
77#816—92no—
78C22×C3⋊D3—21no—
79C22×C3×D3—61no—
80C22×C3×D3—61no—
81C22×C3×D3—121no—
82C2×C32×C4—21no—
83C2×C32×C4—21no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C32×C4D1211yescharacteristic
87C2×C3⋊Dic3—21no—
88C2×C3⋊Dic3—22no—
89C2×C3⋊Dic3D1211yescharacteristic
90C2×C3⋊Dic3C22×D311yescharacteristic
91C4×C3⋊D3—42no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—61no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—121no—
101C2×C3×Dic3—122no—
102C3×C4×D3—122no—
103C3×C4×D3—124no—
104C3×C4×D3—122no—
105#627—62no—
106#627—62no—
107#627—122no—
108#627—122no—
109#626—62no—
110#626—62no—
111#625—32no—
112#625—61no—
113#622—22no—
114#461—271noSylow
115C2×C33#12111yescharacteristic
116C2×C33C2×SD1611yescharacteristic
117C2×C33#13711yescharacteristic
118C3×C3⋊D3—42no—
119C22×C3×C4—181no—
120C2×#67—181no—
121C2×D12—91no—
122C2×C4×D3—61no—
123C2×C4×D3—61no—
124C2×C4×D3—121no—
125C2×Dic6—92no—
126C2×Dic6—182no—
127C2×C3×C8—92no—
128C3×C4⋊C4—181no—
129C3×C42—91no—
130#217—181no—
131(C2×D3)⋊C4—181no—
132(C2×D3)⋊C4—181no—
133C4⋊Dic3—61no—
134C4⋊Dic3—61no—
135C4⋊Dic3—91no—
136C4⋊Dic3—121no—
137C4⋊Dic3—181no—
138Dic3⋊C4—181no—
139Dic3⋊C4—182no—
140C4×Dic3—61no—
141C4×Dic3—61no—
142#204—184no—
143C22×C32—21no—
144C22×C32—21no—
145C22×C32—41no—
146C22×C32—41no—
147C22×C32C2×D1211yescharacteristic
148C2×C3⋊D3—22no—
149C2×C3⋊D3—41no—
150C2×C3⋊D3—41no—
151C2×C3×D3—62no—
152C2×C3×D3—62no—
153C2×C3×D3—122no—
154C2×C3×D3—121no—
155C2×C3×D3—121no—
156C2×C3×D3—122no—
157C2×C3×D3—122no—
158C2×C3×D3—121no—
159C2×C3×D3—121no—
160C32×C4—22no—
161C32×C4—22no—
162C32×C4—42no—
163C32×C4—42no—
164C32×C4#20412yescomplemented
165C3⋊Dic3—21no—
166C3⋊Dic3—41no—
167C3⋊Dic3—42no—
168C3⋊Dic3D3×D411yescharacteristic
169C3⋊Dic3#23911yescharacteristic
170C3×Dic3—61no—
171C3×Dic3—61no—
172C3×Dic3—61no—
173C3×Dic3—61no—
174C3×Dic3—62no—
175C3×Dic3—121no—
176C3×Dic3—122no—
177C3×Dic3—121no—
178C3×Dic3—122no—
179C3×Dic3—122no—
180C3×Dic3—121no—
181C3×Dic3—242no—
182C2×SD16—272no—
183C4⋊Q8—271no—
184C4×D4—271no—
185C4⋊C8—91no—
186#103—272no—
187C33#46111yescharacteristic, complemented, Sylow
188C23×C3—181no—
189C22×D3—61no—
190C22×D3—61no—
191C22×D3—121no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—91no—
197C2×C3×C4—91no—
198C2×C3×C4—181no—
199C2×C3×C4—182no—
200C2×C3×C4—362no—
201C2×C3×C4GO+(4,𝔽2)11yescharacteristic
202#67—182no—
203#67—182no—
204C2×Dic3—61no—
205C2×Dic3—61no—
206C2×Dic3—61no—
207C2×Dic3—61no—
208C2×Dic3—62no—
209C2×Dic3—61no—
210C2×Dic3—62no—
211C2×Dic3—61no—
212C2×Dic3—121no—
213C2×Dic3—122no—
214D12—92no—
215D12—181no—
216C4×D3—122no—
217C4×D3—124no—
218C4×D3—122no—
219Dic6—182no—
220Dic6—182no—
221Dic6—362no—
222Dic6—362no—
223C3×C8—182no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—41no—
235C2×C32—41no—
236C2×C32#89511yescharacteristic
237C2×C32C2×#20411yescharacteristic
238C2×C32#90911yescharacteristic
239C3⋊D3—42no—
240C3×D3—122no—
241C3×D3—124no—
242C3×D3—122no—
243C2×Q8—272no—
244C2×D4—271no—
245C22×C4—181no—
246SD16—544no—
247C2×C8—92no—
248C4⋊C4—181no—
249C4⋊C4—271no—
250C4⋊C4—541no—
251C22⋊C4—541no—
252C42—91no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—182no—
258C22×C3—361no—
259C22×C3—361no—
260C22×C3C2×GO+(4,𝔽2)11yescharacteristic
261C2×D3—62no—
262C2×D3—62no—
263C2×D3—122no—
264C2×D3—121no—
265C2×D3—121no—
266C2×D3—122no—
267C2×D3—122no—
268C2×D3—121no—
269C2×D3—121no—
270C3×C4—22no—
271C3×C4—22no—
272C3×C4—42no—
273C3×C4—42no—
274C3×C4—91no—
275C3×C4—91no—
276C3×C4—181no—
277C3×C4—362no—
278C3×C4—361no—
279C3×C4#377512yescomplemented
280Dic3—61no—
281Dic3—61no—
282Dic3—61no—
283Dic3—61no—
284Dic3—62no—
285Dic3—121no—
286Dic3—122no—
287Dic3—121no—
288Dic3—122no—
289Dic3—122no—
290Dic3—121no—
291Dic3—242no—
292C32—21no—
293C32—21no—
294C32—41no—
295C32—41no—
296C32#484111yescharacteristic, complemented
297C23—181no—
298Q8—542no—
299Q8—542no—
300D4—272no—
301D4—541no—
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—181no—
305C2×C4—182no—
306C2×C4—362no—
307C2×C4#157⋊D311yescharacteristic
308C8—182no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3—41no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3—41no—
321C2×C3—362no—
322C2×C3#6422611yescharacteristic
323C2×C3#6423011yescharacteristic
324C2×C3C2×#377511yescharacteristic
325D3—122no—
326D3—124no—
327D3—122no—
328C22—182no—
329C22—361no—
330C22—361no—
331C22C2×#157⋊D311yescharacteristic
332C4—91no—
333C4—91no—
334C4—181no—
335C4—362no—
336C4—361no—
337C4#8894212yescomplemented
338C3—21no—
339C3—21no—
340C3—41no—
341C3—41no—
342C3#1059343711yescharacteristic, complemented
343C2—362no—
344C2#1172433811yescharacteristic
345C2#1172434311yescharacteristic
346C2C2×#8894211yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×#6295C22
hypercenterC2×C4#157⋊D3
hypocenterC33#461
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4⋊#6296C2semidirect
4. C3×C4⋊#622C2semidirect
5. #11722794C2semidirect
22. C4⋊#622D3semidirect
65. C33×C4SD16semidirect
164. C32×C4#204semidirect
187. C33#461semidirect
279. C3×C4#3775semidirect
296. C32#4841semidirect
337. C4#88942semidirect
342. C3#10593437semidirect

Maximal subgroups

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

subgroupindex
C2×#889422
C4⋊#62962
C4×#62952
C3×C4⋊#6222
#117227942
#105934373
#48419