Finite Group Databasebeta

← #13158982

Subgroups — #13158982

Order 1728.

880 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 186 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
1C3×C4×#634C211yesmaximal, characteristic, complemented
2C32×C4×D12C211yesmaximal, characteristic
3C2×#88792C211yesmaximal, characteristic
4#11722173C211yesmaximal, characteristic, complemented
5#11722056C211yesmaximal, characteristic, complemented
6#11722039C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic
8#10592717—31nomaximal
9#10590358—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×C32×C4×D3—21no—
13C33×C42C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3750—21no—
18C3×C4×C3⋊Dic3—21no—
19C32×(C2×D3)⋊C4—21no—
20C32×C4⋊Dic3C2211yescharacteristic
21#88792C414yescomplemented
22C32×C4×D4—31no—
23C2×#3754—31no—
24C4×#634D311yescharacteristic, complemented
25C3×C4×D12D314yes—
26C3×C4×Dic6—34no—
27C2×#3715—34no—
28#63659—31no—
29#63637—31no—
30#63634—31no—
31#63630—31no—
32#63630—34no—
33#63630—64no—
34#63576—34no—
35#63565—34no—
36#63562—34no—
37C22×C32×D3—21no—
38C2×C33×C4D411yescharacteristic
39C2×C33×C4D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3×C3⋊Dic3—21no—
42C2×C3×C3⋊Dic3—21no—
43C3×#634—21no—
44C3×#634C2×C412yescomplemented
45C2×C32×Dic3—21no—
46C32×D12—21no—
47C32×D12C2×C412yes—
48C32×C4×D3—42no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
51#5026—91no—
52#5018—94no—
53C2×C32×D4—31no—
54C22×C32×C4—61no—
55C2×#634C2×D311yescharacteristic
56C2×C3×D12C2×D314yes—
57C2×C3×C4×D3—24no—
58C2×C3×Dic6—34no—
59C32×C4⋊C4—31no—
60C32×C22⋊C4—61no—
61C32×C42—24no—
62C32×C42C2×D314yes—
63C32×C42C2×D311yescharacteristic
64#3754—34no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66#3750—21no—
67C4×C3⋊Dic3—21no—
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—31no—
71C2×C3⋊(C3⋊C8)—34no—
72C2×C3⋊(C3⋊C8)—64no—
73C2×C3⋊(C3⋊C8)—64no—
74C3×(C2×D3)⋊C4—24no—
75C3×C4⋊Dic3—34no—
76C3×C4⋊Dic3C2×D314yes—
77C3×Dic3⋊C4—64no—
78C3×C4×Dic3—64no—
79#3715—316no—
80C22×C33C2×D411yescharacteristic
81C2×C32×D3—22no—
82C2×C32×D3—41no—
83C2×C32×D3—41no—
84C33×C4—21no—
85C33×C4—21no—
86C33×C4C22×C411yescharacteristic
87C33×C4#4111yescharacteristic
88C3×C3⋊Dic3—22no—
89C3×C3⋊Dic3—41no—
90C32×Dic3—41no—
91C3×C4×Q8—91no—
92C3×C4×D4—34no—
93C2×#215—91no—
94C2×#214—94no—
95C4×D12C3⋊D311yescharacteristic
96C4×Dic6—34no—
97#835—91no—
98#832—94no—
99#810—94no—
100#809—91no—
101#807—94no—
102#807—91no—
103#803—94no—
104#803—91no—
105#803—184no—
106C23×C32—61no—
107C22×C3×D3—24no—
108C32×D4—32no—
109C32×D4—61no—
110C2×C32×C4—24no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—61no—
114C2×C32×C4—122no—
115C2×C32×C4#6714yes—
116C2×C32×C4#6714yes—
117C2×C32×C4#6711yescharacteristic
118C2×C32×C4#6711yescharacteristic
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—21no—
123#634—21no—
124#634C4×D312yescomplemented
125C2×C3×Dic3—24no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—64no—
128C3×D12—24no—
129C3×D12C4×D318yes—
130C3×C4×D3—48no—
131C3×Dic6—38no—
132C3×Dic6—64no—
133C3⋊(C3⋊C8)—32no—
134C3⋊(C3⋊C8)—38no—
135C3⋊(C3⋊C8)—68no—
136C3⋊(C3⋊C8)—68no—
137C3⋊(C3⋊C8)—64no—
138C3⋊(C3⋊C8)—61no—
139#440—271noSylow
140C2×C33C4×D411yescharacteristic
141C2×C33AGL(1,ℤ/8)11yescharacteristic
142C2×C33#13711yescharacteristic
143C32×D3—42no—
144C2×C3×Q8—91no—
145C2×C3×D4—34no—
146C22×C3×C4—64no—
147C2×D12C2×C3⋊D311yescharacteristic
148C2×C4×D3—21no—
149C2×Dic6—34no—
150C3×C4⋊C4—34no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—181no—
153C3×C22⋊C4—64no—
154C3×C42—24no—
155C3×C42—181no—
156C3×C42D3214yes—
157C3×C42C2×C3⋊D311yescharacteristic
158#215—94no—
159#214—916no—
160(C2×D3)⋊C4—21no—
161C4⋊Dic3—34no—
162C4⋊Dic3C2×C3⋊D311yescharacteristic
163Dic3⋊C4—64no—
164C4×Dic3—64no—
165C2×C3⋊C8—94no—
166C2×C3⋊C8—91no—
167C2×C3⋊C8—94no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—184no—
170C2×C3⋊C8—184no—
171C22×C32—24no—
172C22×C32—62no—
173C22×C32—121no—
174C22×C32—121no—
175C22×C32C2×#6714yes—
176C22×C32C2×#6711yescharacteristic
177C2×C3×D3—28no—
178C2×C3×D3—44no—
179C2×C3×D3—44no—
180C32×C4—21no—
181C32×C4—21no—
182C32×C4—24no—
183C32×C4—24no—
184C32×C4—24no—
185C32×C4—24no—
186C32×C4—44no—
187C32×C4—44no—
188C32×C4—121no—
189C32×C4C2×C4×D314yes—
190C32×C4C2×C4×D311yescharacteristic
191C32×C4#23514yes—
192C32×C4#23511yescharacteristic
193C3⋊Dic3—22no—
194C3⋊Dic3—41no—
195C3×Dic3—44no—
196C3×Dic3—68no—
197C3×Dic3—124no—
198C2×SD16—271no—
199C4×Q8—91no—
200C4×D4—31no—
201#107—271no—
202#103—271no—
203#102—271no—
204#97—271no—
205C33#44011yescharacteristic, complemented, Sylow
206C23×C3—64no—
207C22×D3—21no—
208C3×Q8—92no—
209C3×Q8—181no—
210C3×D4—38no—
211C3×D4—64no—
212C2×C3×C4—24no—
213C2×C3×C4—24no—
214C2×C3×C4—24no—
215C2×C3×C4—64no—
216C2×C3×C4—128no—
217C2×C3×C4—181no—
218C2×C3×C4—181no—
219C2×C3×C4#62514yes—
220C2×C3×C4#62514yes—
221C2×C3×C4#63811yescharacteristic
222C2×C3×C4#63811yescharacteristic
223C2×C3×C4C2×D3214yes—
224C2×C3×C4C22×C3⋊D311yescharacteristic
225C2×Dic3—21no—
226C2×Dic3—64no—
227C2×Dic3—64no—
228D12—21no—
229D12C4×C3⋊D312yes—
230C4×D3—42no—
231Dic6—38no—
232Dic6—64no—
233C3⋊C8—98no—
234C3⋊C8—92no—
235C3⋊C8—188no—
236C3⋊C8—188no—
237C3⋊C8—184no—
238C3⋊C8—181no—
239C2×C32—24no—
240C2×C32—24no—
241C2×C32—24no—
242C2×C32—122no—
243C2×C32C4×#6714yes—
244C2×C32C4×#6711yescharacteristic
245C2×C32#93214yes—
246C2×C32#94211yescharacteristic
247C2×C32#94911yescharacteristic
248C2×C32#95114yes—
249C3×D3—48no—
250C2×Q8—91no—
251C2×D4—31no—
252C22×C4—61no—
253SD16—274no—
254C2×C8—271no—
255C2×C8—271no—
256C4⋊C4—31no—
257C4⋊C4—91no—
258C4⋊C4—181no—
259C22⋊C4—61no—
260C42—181no—
261C42D3×C3⋊D311yescharacteristic
262C22×C3—24no—
263C22×C3—68no—
264C22×C3—124no—
265C22×C3—124no—
266C22×C3C2×#62514yes—
267C22×C3C2×#63811yescharacteristic
268C2×D3—22no—
269C2×D3—41no—
270C2×D3—41no—
271C3×C4—24no—
272C3×C4—24no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—24no—
276C3×C4—24no—
277C3×C4—44no—
278C3×C4—44no—
279C3×C4—124no—
280C3×C4—182no—
281C3×C4—361no—
282C3×C4#379814yes—
283C3×C4C4×D3214yes—
284C3×C4C2×C4×C3⋊D311yescharacteristic
285C3×C4#382811yescharacteristic
286Dic3—41no—
287Dic3—68no—
288Dic3—124no—
289C32—24no—
290C32#501814yescomplemented
291C32#502611yescharacteristic, complemented
292C23—61no—
293Q8—92no—
294Q8—181no—
295D4—32no—
296D4—61no—
297C2×C4—61no—
298C2×C4—122no—
299C2×C4—181no—
300C2×C4—181no—
301C2×C4#629011yescharacteristic
302C2×C4#629011yescharacteristic
303C2×C4C2×D3×C3⋊D311yescharacteristic
304C8—272no—
305C8—541no—
306C2×C3—24no—
307C2×C3—24no—
308C2×C3—24no—
309C2×C3—128no—
310C2×C3#6382314yes—
311C2×C3#6382714yes—
312C2×C3C4×#62514yes—
313C2×C3C4×#63811yescharacteristic
314C2×C3#6414111yescharacteristic
315C2×C3#6416011yescharacteristic
316D3—42no—
317C22—62no—
318C22—121no—
319C22—121no—
320C22C2×#629011yescharacteristic
321C4—21no—
322C4—21no—
323C4—121no—
324C4—182no—
325C4—361no—
326C4#8902011yescharacteristic
327C4C4×D3×C3⋊D311yescharacteristic
328C3—24no—
329C3#1059035814yescomplemented
330C3#1059271711yescharacteristic, complemented
331C2—122no—
332C2#1172336411yescharacteristic
333C2#1172337311yescharacteristic
334C2C4×#629011yescharacteristic
subgroupG/H
centerC22C2×#6290
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#440
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C3×C4×#634C2semidirect
4. #11722173C2semidirect
5. #11722056C2semidirect
6. #11722039C2semidirect
21. #88792C4semidirect
24. C4×#634D3semidirect
44. C3×#634C2×C4semidirect
50. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
124. #634C4×D3semidirect
205. C33#440semidirect
290. C32#5018semidirect
291. C32#5026semidirect
329. C3#10590358semidirect
330. C3#10592717semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#6342
C32×C4×D122
C2×#887922
#117221732
#117220562
#117220392
#117220312
#105927173
#105903583