Finite Group Databasebeta

← #13156554

Subgroups — #13156554

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
1C4⋊#6296C211yesmaximal, characteristic, complemented
2#11723440C211yesmaximal, characteristic
3#11722036C212yesmaximal
4C3×#63566C212yesmaximal, complemented
5C3×#63555C211yesmaximal, characteristic, complemented
6#10591632—31nomaximal
7#10590893—32nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
9C2×C32×Dic6C2211yescharacteristic
10C2×#6296—21no—
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14#88815—21no—
15#88804—21no—
16#88804—21no—
17#64156—31no—
18#63919—31no—
19#63918—32no—
20#63864—32no—
21#63864—61no—
22#63648—32no—
23C3×#835—32no—
24C3×#816—32no—
25C3×#815—32no—
26#63566D312yescomplemented
27#63564—32no—
28#63564—32no—
29#63564—62no—
30#63555D311yescharacteristic, complemented
31C22×C3⋊(C3⋊D3)—21no—
32C2×C33×C4C2311yescharacteristic
33C4⋊(C3⋊(C3⋊D3))—22no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C32×Dic3—21no—
38C32×Dic6—22no—
39#6296—42no—
40C32×C3⋊C8—22no—
41#4794—92no—
42#4714—91no—
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×Dic6C2×D312yes—
51C2×#627—62no—
52C2×#627—61no—
53C2×C32×C8—32no—
54#3752—61no—
55#3752—61no—
56C4⋊(C3⋊Dic3)C2×D312yes—
57C4×C3⋊Dic3C2×D311yescharacteristic
58C3×C4⋊Dic3—32no—
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—62no—
61C3×C4×Dic3—32no—
62C3×C4×Dic3—61no—
63C2×C3×C3⋊C8—22no—
64C2×C3×C3⋊C8C2×D312yes—
65C2×C3×C3⋊C8C2×D312yes—
66#3724—61no—
67Dic3⋊Dic3—62no—
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—
79C3×C3⋊Dic3—42no—
80C32×Dic3—41no—
81#947—92no—
82#947—181no—
83#888—92no—
84#875—91no—
85#869—91no—
86C3×#103—92no—
87C3×#97—91no—
88#835—32no—
89#821—92no—
90#821—92no—
91#821—182no—
92#816—32no—
93#815—32no—
94#809—92no—
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×C4C22×D311yescharacteristic
106C2×C32×C4C22×D312yes—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3D1211yescharacteristic
109C2×C3⋊Dic3D1211yescharacteristic
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—32no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—122no—
126C3×Dic6—22no—
127C3×Dic6—22no—
128C3×Dic6—42no—
129#627—124no—
130#627—122no—
131C32×C8—62no—
132C3×C3⋊C8—22no—
133C3×C3⋊C8—22no—
134C3×C3⋊C8—42no—
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×D12—92no—
143C2×D12—91no—
144C2×D12—182no—
145C2×D12—181no—
146C2×D12—361no—
147C2×Dic6—182no—
148C2×Dic6—181no—
149C2×Dic6D3211yescharacteristic
150C2×C3×C8—32no—
151C2×C3×C8—32no—
152C2×C3×C8—62no—
153C3×C4⋊C4—92no—
154C3×C42—91no—
155(C2×D3)⋊C4—181no—
156(C2×D3)⋊C4—181no—
157(C2×D3)⋊C4—182no—
158(C2×D3)⋊C4—182no—
159(C2×D3)⋊C4—182no—
160(C2×D3)⋊C4—182no—
161C4⋊Dic3—32no—
162C4⋊Dic3—32no—
163C4⋊Dic3—62no—
164C4⋊Dic3—181no—
165Dic3⋊C4—182no—
166C4×Dic3—32no—
167C4×Dic3—61no—
168C2×C3⋊C8D3212yes—
169C22×C32—22no—
170C22×C32—21no—
171C22×C32—41no—
172C22×C32C2×D1211yescharacteristic
173C22×C32D3×D412yes—
174C2×C3⋊D3—124no—
175C2×C3⋊D3—122no—
176C2×C3⋊D3—124no—
177C2×C3⋊D3—121no—
178C2×C3⋊D3—122no—
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#23512yes—
195C32×C4#23512yes—
196C32×C4#23712yes—
197C3⋊Dic3—21no—
198C3⋊Dic3—21no—
199C3⋊Dic3—42no—
200C3×Dic3—42no—
201C3×Dic3—42no—
202C3×Dic3—62no—
203C3×Dic3—62no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3—122no—
207C3×Dic3—122no—
208C3×Dic3—242no—
209C4⋊Q8—271no—
210#124—271no—
211#103—92no—
212#102—272no—
213#97—91no—
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—
223C2×C3×C4—21no—
224C2×C3×C4—22no—
225C2×C3×C4—41no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—91no—
229C2×C3×C4—91no—
230C2×C3×C4—182no—
231C2×C3×C4C2×D3212yes—
232C2×C3×C4C2×D3211yescharacteristic
233C2×Dic3—21no—
234C2×Dic3—32no—
235C2×Dic3—32no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—62no—
239C2×Dic3—62no—
240C2×Dic3—122no—
241D12—184no—
242D12—182no—
243D12—182no—
244D12—182no—
245D12—362no—
246D12—362no—
247D12—364no—
248Dic6—22no—
249Dic6—364no—
250Dic6—362no—
251C3×C8—62no—
252C3×C8—62no—
253C3×C8—122no—
254C3⋊C8—22no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32#87511yescharacteristic
265C2×C32#88512yes—
266C2×C32#90811yescharacteristic
267C2×C32#90911yescharacteristic
268C2×C32#91412yes—
269C2×C32#91812yes—
270C3⋊D3—242no—
271C3⋊D3—244no—
272C3⋊D3—244no—
273C3⋊D3—241no—
274C3⋊D3—242no—
275C2×Q8—31no—
276C2×Q8—541no—
277C2×D4—271no—
278C2×C8—32no—
279C4⋊C4—92no—
280C4⋊C4—541no—
281C22⋊C4—541no—
282C22⋊C4—541no—
283C42—91no—
284C22×C3—21no—
285C22×C3—22no—
286C22×C3—41no—
287C22×C3D3×D1212yes—
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—362no—
297C2×D3—361no—
298C2×D3—361no—
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—181no—
311C3×C4—181no—
312C3×C4—362no—
313C3×C4#379812yes—
314C3×C4#380412yes—
315C3×C4#380412yes—
316Dic3—41no—
317Dic3—62no—
318Dic3—62no—
319Dic3—121no—
320Dic3—121no—
321Dic3—122no—
322Dic3—122no—
323Dic3—242no—
324C32—21no—
325C32—22no—
326C32—41no—
327C32#471411yescharacteristic, complemented
328C32#479412yescomplemented
329C23—541no—
330Q8—62no—
331Q8—1082no—
332D4—542no—
333C2×C4—61no—
334C2×C4—91no—
335C2×C4—91no—
336C2×C4—182no—
337C2×C4D3311yescharacteristic
338C8—62no—
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#6379412yes—
349C2×C3#6380112yes—
350C2×C3#6385212yes—
351C2×C3#6394111yescharacteristic
352C2×C3#6394511yescharacteristic
353C2×C3#6395311yescharacteristic
354D3—722no—
355D3—724no—
356D3—724no—
357D3—721no—
358D3—722no—
359C22—1081no—
360C22—1081no—
361C22—1081no—
362C22#8895511yescharacteristic
363C4—121no—
364C4—181no—
365C4—181no—
366C4—362no—
367C4#8895912yes—
368C3—21no—
369C3—22no—
370C3—41no—
371C3#1059089312yescomplemented
372C3#1059163211yescharacteristic, complemented
373C2—2161no—
374C2#1172288111yescharacteristic
375C2#1172289711yescharacteristic
376C2#1172300811yescharacteristic
subgroupG/H
centerC22#88955
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
1. C4⋊#6296C2semidirect
4. C3×#63566C2semidirect
5. C3×#63555C2semidirect
26. #63566D3semidirect
30. #63555D3semidirect
214. C33#489semidirect
327. C32#4714semidirect
328. C32#4794semidirect
371. C3#10590893semidirect
372. C3#10591632semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
#117234402
#117220362
C3×#635662
C3×#635552
#105916323
#105908933