Finite Group Databasebeta

← #13156549

Subgroups — #13156549

Order 1728.

630 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 98 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#11723440C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic, complemented
3#11722071C211yesmaximal, characteristic, complemented
4#11722035C212yesmaximal
5C3×#63565C212yesmaximal, complemented
6#10591632—31nomaximal
7#10590535—32nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×C32×Dic6C2211yescharacteristic
10C2×C3×#627—21no—
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4⋊Dic3C2212yes—
14#88804—21no—
15#88804—21no—
16C3×#3724—21no—
17#64156—31no—
18C3×#888—32no—
19#63919D311yescharacteristic, complemented
20#63864—32no—
21#63864—61no—
22#63636—32no—
23C3×#835—32no—
24C3×#810—32no—
25#63565D312yescomplemented
26#63564—32no—
27#63564—32no—
28#63564—62no—
29#63555—31no—
30#63554—32no—
31C22×C3⋊(C3⋊D3)—21no—
32C2×C33×C4C2311yescharacteristic
33C4⋊(C3⋊(C3⋊D3))—22no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C32×Dic3—21no—
37C2×C32×Dic3—22no—
38C32×Dic6—22no—
39C3×#627—42no—
40C3×C3⋊(C3⋊C8)—22no—
41#5089—91no—
42#4794—92no—
43C2×C32×Q8—31no—
44C2×C4⋊(C3⋊D3)—31no—
45C2×C4⋊(C3⋊D3)—32no—
46C2×C4⋊(C3⋊D3)—62no—
47C2×C4⋊(C3⋊D3)—61no—
48C2×C4⋊(C3⋊D3)—121no—
49C2×C3×Dic6—21no—
50C2×C3×Dic6—62no—
51C2×C3×Dic6C2×D312yes—
52C2×#627—21no—
53C32×C4⋊C4—32no—
54#3752—61no—
55#3752—61no—
56C4×C3⋊Dic3C2×D311yescharacteristic
57C2×C3⋊(C3⋊C8)C2×D312yes—
58C3×C4⋊Dic3—22no—
59C3×C4⋊Dic3C2×D312yes—
60C3×C4⋊Dic3C2×D312yes—
61C3×Dic3⋊C4—62no—
62C3×C4×Dic3—32no—
63C3×C4×Dic3—61no—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—62no—
67#3724—21no—
68#3722—62no—
69#3722—62no—
70#3722—62no—
71#3722—62no—
72C22×C33C2×D411yescharacteristic
73C2×C3⋊(C3⋊D3)—41no—
74C2×C3⋊(C3⋊D3)—41no—
75C2×C3⋊(C3⋊D3)—41no—
76C33×C4#4112yes—
77C3×C3⋊Dic3—21no—
78C3×C3⋊Dic3—21no—
79C32×Dic3—41no—
80C32×Dic3—42no—
81C3×C4⋊Q8—91no—
82#947—92no—
83#947—181no—
84#888—32no—
85#875—91no—
86C3×#103—92no—
87#835—32no—
88#821—92no—
89#815—92no—
90#810—32no—
91#809—92no—
92#809—92no—
93#809—182no—
94#803—91no—
95C22×C3⋊D3—61no—
96C22×C3⋊D3—62no—
97C22×C3⋊D3—62no—
98C22×C3⋊D3—122no—
99C22×C3⋊D3—122no—
100C32×Q8—62no—
101C2×C32×C4—22no—
102C2×C32×C4—21no—
103C2×C32×C4—41no—
104C2×C32×C4—61no—
105C2×C32×C4—62no—
106C2×C32×C4C22×D312yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3#6711yescharacteristic
109C2×C3⋊Dic3#6711yescharacteristic
110C4⋊(C3⋊D3)—62no—
111C4⋊(C3⋊D3)—64no—
112C4⋊(C3⋊D3)—62no—
113C4⋊(C3⋊D3)—62no—
114C4⋊(C3⋊D3)—122no—
115C4⋊(C3⋊D3)—122no—
116C4⋊(C3⋊D3)—124no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—32no—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—42no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C3×Dic6—22no—
127C3×Dic6—22no—
128C3×Dic6—42no—
129C3×Dic6—124no—
130#627—42no—
131C3⋊(C3⋊C8)—22no—
132C3×C3⋊C8—62no—
133C3×C3⋊C8—62no—
134C3×C3⋊C8—122no—
135#489—271noSylow
136C2×C33#12411yescharacteristic
137C2×C33AGL(1,ℤ/8)11yescharacteristic
138C2×C33#13711yescharacteristic
139C3⋊(C3⋊D3)—81no—
140C2×C3×Q8—32no—
141C2×C3×Q8—61no—
142C2×C3×Q8—181no—
143C2×D12—91no—
144C2×D12—92no—
145C2×D12—182no—
146C2×D12—181no—
147C2×D12—361no—
148C2×Dic6—62no—
149C2×Dic6D3211yescharacteristic
150C2×C3×C8—92no—
151C3×C4⋊C4—32no—
152C3×C4⋊C4—32no—
153C3×C4⋊C4—62no—
154C3×C4⋊C4—181no—
155C3×C42—91no—
156(C2×D3)⋊C4—181no—
157(C2×D3)⋊C4—181no—
158(C2×D3)⋊C4—182no—
159(C2×D3)⋊C4—182no—
160(C2×D3)⋊C4—182no—
161(C2×D3)⋊C4—182no—
162C4⋊Dic3D3212yes—
163Dic3⋊C4—62no—
164C4×Dic3—32no—
165C4×Dic3—61no—
166C2×C3⋊C8—32no—
167C2×C3⋊C8—32no—
168C2×C3⋊C8—62no—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—41no—
172C22×C32D3×D412yes—
173C22×C32C2×#6711yescharacteristic
174C2×C3⋊D3—124no—
175C2×C3⋊D3—122no—
176C2×C3⋊D3—124no—
177C2×C3⋊D3—122no—
178C2×C3⋊D3—121no—
179C2×C3⋊D3—121no—
180C2×C3⋊D3—121no—
181C2×C3⋊D3—122no—
182C2×C3⋊D3—122no—
183C2×C3⋊D3—122no—
184C2×C3⋊D3—122no—
185C2×C3⋊D3—242no—
186C2×C3⋊D3—242no—
187C2×C3⋊D3—242no—
188C2×C3⋊D3—242no—
189C32×C4—22no—
190C32×C4—22no—
191C32×C4—22no—
192C32×C4—42no—
193C32×C4—121no—
194C32×C4—122no—
195C32×C4#23512yes—
196C32×C4#23712yes—
197C32×C4#23912yes—
198C3⋊Dic3—21no—
199C3⋊Dic3—21no—
200C3×Dic3—42no—
201C3×Dic3—42no—
202C3×Dic3—42no—
203C3×Dic3—42no—
204C3×Dic3—62no—
205C3×Dic3—62no—
206C3×Dic3—82no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C4⋊Q8—91no—
210#124—271no—
211#103—92no—
212#102—272no—
213#97—271no—
214C33#48911yescharacteristic, complemented, Sylow
215C22×D3—182no—
216C22×D3—182no—
217C22×D3—181no—
218C22×D3—362no—
219C22×D3—362no—
220C3×Q8—62no—
221C3×Q8—62no—
222C3×Q8—122no—
223C3×Q8—362no—
224C2×C3×C4—22no—
225C2×C3×C4—21no—
226C2×C3×C4—41no—
227C2×C3×C4—62no—
228C2×C3×C4—62no—
229C2×C3×C4—62no—
230C2×C3×C4—62no—
231C2×C3×C4—91no—
232C2×C3×C4—91no—
233C2×C3×C4—122no—
234C2×C3×C4C2×D3212yes—
235C2×C3×C4C2×D3211yescharacteristic
236C2×Dic3—22no—
237C2×Dic3—21no—
238C2×Dic3—32no—
239C2×Dic3—32no—
240C2×Dic3—61no—
241C2×Dic3—61no—
242D12—184no—
243D12—182no—
244D12—182no—
245D12—182no—
246D12—362no—
247D12—362no—
248D12—364no—
249Dic6—22no—
250Dic6—124no—
251C3×C8—182no—
252C3⋊C8—62no—
253C3⋊C8—62no—
254C3⋊C8—122no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—21no—
258C2×C32—21no—
259C2×C32—22no—
260C2×C32—21no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32#88512yes—
265C2×C32#91412yes—
266C2×C32#91812yes—
267C2×C32#93211yescharacteristic
268C2×C32#94211yescharacteristic
269C2×C32#94711yescharacteristic
270C3⋊D3—242no—
271C3⋊D3—244no—
272C3⋊D3—244no—
273C3⋊D3—242no—
274C3⋊D3—241no—
275C2×Q8—31no—
276C2×Q8—181no—
277C2×D4—271no—
278C2×C8—92no—
279C4⋊C4—32no—
280C4⋊C4—181no—
281C22⋊C4—541no—
282C22⋊C4—541no—
283C42—91no—
284C22×C3—21no—
285C22×C3—22no—
286C22×C3—41no—
287C22×C3D3×#6712yes—
288C22×C3#381111yescharacteristic
289C2×D3—364no—
290C2×D3—362no—
291C2×D3—362no—
292C2×D3—362no—
293C2×D3—364no—
294C2×D3—362no—
295C2×D3—362no—
296C2×D3—361no—
297C2×D3—361no—
298C2×D3—362no—
299C2×D3—361no—
300C2×D3—722no—
301C2×D3—722no—
302C2×D3—722no—
303C2×D3—722no—
304C3×C4—22no—
305C3×C4—22no—
306C3×C4—22no—
307C3×C4—42no—
308C3×C4—122no—
309C3×C4—122no—
310C3×C4—122no—
311C3×C4—122no—
312C3×C4—181no—
313C3×C4—181no—
314C3×C4—242no—
315C3×C4#379612yes—
316C3×C4#379912yes—
317C3×C4#380412yes—
318Dic3—42no—
319Dic3—41no—
320Dic3—62no—
321Dic3—62no—
322Dic3—121no—
323Dic3—121no—
324C32—22no—
325C32—21no—
326C32—41no—
327C32#479412yescomplemented
328C32#508911yescharacteristic, complemented
329C23—541no—
330Q8—62no—
331Q8—362no—
332D4—542no—
333C2×C4—61no—
334C2×C4—62no—
335C2×C4—91no—
336C2×C4—91no—
337C2×C4D3311yescharacteristic
338C8—182no—
339C2×C3—21no—
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—22no—
343C2×C3—21no—
344C2×C3—22no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3#6390712yes—
349C2×C3#6393412yes—
350C2×C3#6394111yescharacteristic
351C2×C3#6394511yescharacteristic
352C2×C3#6394812yes—
353C2×C3#6395311yescharacteristic
354D3—722no—
355D3—724no—
356D3—724no—
357D3—722no—
358D3—721no—
359C22—1081no—
360C22—1081no—
361C22—1081no—
362C22#8895211yescharacteristic
363C4—122no—
364C4—121no—
365C4—181no—
366C4—181no—
367C4#8896212yes—
368C3—22no—
369C3—21no—
370C3—41no—
371C3#1059053512yescomplemented
372C3#1059163211yescharacteristic, complemented
373C2—2161no—
374C2#1172287311yescharacteristic
375C2#1172289011yescharacteristic
376C2#1172300511yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypocenterC33#489
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63919C2semidirect
3. #11722071C2semidirect
5. C3×#63565C2semidirect
19. #63919D3semidirect
25. #63565D3semidirect
214. C33#489semidirect
327. C32#4794semidirect
328. C32#5089semidirect
371. C3#10590535semidirect
372. C3#10591632semidirect

Maximal subgroups

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

subgroupindex
#117234402
C3×#639192
#117220712
#117220352
C3×#635652
#105916323
#105905353