Finite Group Databasebeta

← #13159325

Subgroups — #13159325

Order 1728.

988 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 988 conjugacy classes of subgroups — too many to draw. Only its 170 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#11723933C211yesmaximal, characteristic
2#11723413C211yesmaximal, characteristic
3C2×#88785C211yesmaximal, characteristic
4C3×#63634C211yesmaximal, characteristic, complemented
5C32×#808C211yesmaximal, characteristic, complemented
6#11722057C211yesmaximal, characteristic, complemented
7#11722052C211yesmaximal, characteristic, complemented
8#10592559—31nomaximal
9#10590574—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C33×C4⋊C4C2211yescharacteristic
12#88859—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20Dic3×C3⋊Dic3—21no—
21#88785C414yescomplemented
22C2×C8×C3⋊D3—31no—
23#64101—31no—
24#64101—31no—
25#64101—34no—
26#64101—64no—
27#63849—34no—
28C2×#3710—34no—
29C32×#106—31no—
30#63644—31no—
31#63634—31no—
32#63634D311yescharacteristic, complemented
33C3×#808D314yescomplemented
34C3×#807—34no—
35#63575—34no—
36#63573—34no—
37C22×C3⋊(C3⋊D3)D411yescharacteristic
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C4×C3⋊(C3⋊D3)C2×C412yes—
43C4×C3⋊(C3⋊D3)Q812yes—
44C2×C3×C3⋊Dic3—21no—
45C2×C32×Dic3—21no—
46C3×C3⋊(C3⋊C8)C2×C412yes—
47C32×C3⋊C8C2×C412yes—
48#4880—91no—
49#4860—94no—
50C2×C4×C3⋊D3—31no—
51C2×C4×C3⋊D3—34no—
52C2×C4×C3⋊D3—64no—
53C2×C32×C8—31no—
54C32×C4⋊C4—24no—
55C32×C4⋊C4—31no—
56C32×C4⋊C4C2×D314yes—
57C32×C4⋊C4C2×D311yescharacteristic
58#3752—61no—
59#3752—61no—
60#3752—64no—
61#3752—68no—
62C4⋊(C3⋊Dic3)—31no—
63C4⋊(C3⋊Dic3)—34no—
64C4⋊(C3⋊Dic3)—64no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C4×C3⋊Dic3—61no—
67C4×C3⋊Dic3—61no—
68C4×C3⋊Dic3—64no—
69C4×C3⋊Dic3—68no—
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C8×C3⋊D3—34no—
72C3×C4⋊Dic3—34no—
73C3×C4⋊Dic3C2×D314yes—
74C2×C3×C3⋊C8—34no—
75C2×C3×C3⋊C8C2×D314yes—
76#3722—64no—
77Dic32—64no—
78#3710—316no—
79C22×C33C2×D411yescharacteristic
80C2×C3⋊(C3⋊D3)—21no—
81C2×C3⋊(C3⋊D3)—21no—
82C2×C3⋊(C3⋊D3)C4⋊C412yes—
83C33×C4—41no—
84C33×C4C22×C411yescharacteristic
85C33×C4C2×Q811yescharacteristic
86C3⋊(C3⋊Dic3)—41no—
87C3⋊(C3⋊Dic3)C4⋊C412yes—
88C3×C3⋊Dic3—41no—
89C32×Dic3—41no—
90C2×D3×C8—94no—
91C2×D3×C8—91no—
92#892—94no—
93#892—94no—
94#892—91no—
95#892—91no—
96#892—184no—
97C3×#107—91no—
98C3×#106—34no—
99#818—91no—
100#817—94no—
101#808—91no—
102#808C3⋊D311yescharacteristic, complemented
103#807—34no—
104#807—94no—
105C22×C3⋊D3—31no—
106C22×C3⋊D3—34no—
107C22×C3⋊D3—64no—
108C2×C32×C4—28no—
109C2×C32×C4—24no—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—61no—
113C2×C32×C4C22×D314yes—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—31no—
117C2×C3⋊Dic3—34no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3—68no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—61no—
122C4×C3⋊D3—32no—
123C4×C3⋊D3—32no—
124C4×C3⋊D3—38no—
125C4×C3⋊D3—38no—
126C4×C3⋊D3—68no—
127C4×C3⋊D3—68no—
128C2×C3×Dic3—24no—
129C2×C3×Dic3—64no—
130C32×C8—32no—
131C3⋊(C3⋊C8)C4×D312yes—
132C3×C3⋊C8—38no—
133C3×C3⋊C8C4×D318yes—
134#427—271noSylow
135C2×C33C2×C4⋊C411yescharacteristic
136C2×C33#13511yescharacteristic
137C2×C33#13511yescharacteristic
138C3⋊(C3⋊D3)—22no—
139C2×C4×D3—94no—
140C2×C4×D3—91no—
141C2×C4×D3—184no—
142C2×C3×C8—34no—
143C2×C3×C8—91no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—34no—
146C3×C4⋊C4—91no—
147C3×C4⋊C4D3214yes—
148C3×C4⋊C4C2×C3⋊D311yescharacteristic
149(C2×D3)⋊C4—181no—
150(C2×D3)⋊C4—184no—
151(C2×D3)⋊C4—184no—
152(C2×D3)⋊C4—181no—
153(C2×D3)⋊C4—188no—
154C4⋊Dic3—34no—
155C4⋊Dic3—94no—
156C4⋊Dic3—91no—
157C4⋊Dic3—184no—
158C4⋊Dic3C2×C3⋊D311yescharacteristic
159C4×Dic3—181no—
160C4×Dic3—184no—
161C4×Dic3—184no—
162C4×Dic3—181no—
163C4×Dic3—188no—
164C2×C3⋊C8—34no—
165C2×C3⋊C8C2×C3⋊D311yescharacteristic
166D3×C8—916no—
167D3×C8—94no—
168C22×C32—24no—
169C22×C32D3×D414yes—
170C22×C32D3×D411yescharacteristic
171C2×C3⋊D3—32no—
172C2×C3⋊D3—38no—
173C2×C3⋊D3—68no—
174C2×C3⋊D3—61no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—64no—
177C2×C3⋊D3—64no—
178C2×C3⋊D3—124no—
179C2×C3⋊D3—124no—
180C32×C4—24no—
181C32×C4—24no—
182C32×C4—48no—
183C32×C4—41no—
184C32×C4—44no—
185C32×C4—121no—
186C32×C4C2×C4×D314yes—
187C32×C4C2×C4×D311yescharacteristic
188C32×C4D3×Q814yes—
189C32×C4D3×Q811yescharacteristic
190C3⋊Dic3—32no—
191C3⋊Dic3—38no—
192C3⋊Dic3—41no—
193C3⋊Dic3—68no—
194C3⋊Dic3—128no—
195C3⋊Dic3—124no—
196C3⋊Dic3—121no—
197C3×Dic3—44no—
198C3×Dic3—124no—
199C22×C8—271no—
200#117—271no—
201#117—271no—
202#107—91no—
203#107—271no—
204#106—31no—
205#106—271no—
206C33#42711yescharacteristic, complemented, Sylow
207C22×D3—94no—
208C22×D3—91no—
209C22×D3—184no—
210C2×C3×C4—28no—
211C2×C3×C4—24no—
212C2×C3×C4—21no—
213C2×C3×C4—24no—
214C2×C3×C4—64no—
215C2×C3×C4—181no—
216C2×C3×C4C2×D3214yes—
217C2×C3×C4C22×C3⋊D311yescharacteristic
218C2×Dic3—21no—
219C2×Dic3—64no—
220C2×Dic3—94no—
221C2×Dic3—91no—
222C2×Dic3—184no—
223C2×Dic3—188no—
224C2×Dic3—184no—
225C2×Dic3—181no—
226C4×D3—98no—
227C4×D3—98no—
228C4×D3—92no—
229C4×D3—92no—
230C4×D3—188no—
231C4×D3—188no—
232C3×C8—38no—
233C3×C8—92no—
234C3⋊C8—38no—
235C3⋊C8C4×C3⋊D312yes—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32D3×C4⋊C414yes—
240C2×C32D3×C4⋊C411yescharacteristic
241C2×C32#91211yescharacteristic
242C2×C32#91614yes—
243C2×C32#91614yes—
244C2×C32#91911yescharacteristic
245C3⋊D3—62no—
246C3⋊D3—68no—
247C3⋊D3—128no—
248C22×C4—271no—
249C2×C8—31no—
250C2×C8—91no—
251C2×C8—274no—
252C4⋊C4—31no—
253C4⋊C4—91no—
254C4⋊C4—271no—
255C4⋊C4D3×C3⋊D311yescharacteristic
256C22⋊C4—541no—
257C22⋊C4—541no—
258C42—541no—
259C42—541no—
260C22×C3—24no—
261C22×C3#381114yes—
262C22×C3D4×C3⋊D311yescharacteristic
263C2×D3—92no—
264C2×D3—98no—
265C2×D3—188no—
266C2×D3—181no—
267C2×D3—181no—
268C2×D3—184no—
269C2×D3—184no—
270C2×D3—364no—
271C2×D3—364no—
272C3×C4—24no—
273C3×C4—24no—
274C3×C4—48no—
275C3×C4—41no—
276C3×C4—44no—
277C3×C4—124no—
278C3×C4—361no—
279C3×C4#379714yes—
280C3×C4C4×D3214yes—
281C3×C4C2×C4×C3⋊D311yescharacteristic
282C3×C4Q8×C3⋊D311yescharacteristic
283Dic3—41no—
284Dic3—92no—
285Dic3—98no—
286Dic3—124no—
287Dic3—188no—
288Dic3—368no—
289Dic3—361no—
290Dic3—364no—
291C32—24no—
292C32#486014yescomplemented
293C32#488011yescharacteristic, complemented
294C23—271no—
295C2×C4—21no—
296C2×C4—61no—
297C2×C4—181no—
298C2×C4—271no—
299C2×C4—272no—
300C2×C4—272no—
301C2×C4—541no—
302C2×C4C2×D3×C3⋊D311yescharacteristic
303C8—32no—
304C8—92no—
305C2×C3—24no—
306C2×C3—24no—
307C2×C3—24no—
308C2×C3#6388314yes—
309C2×C3#6393614yes—
310C2×C3#6395014yes—
311C2×C3C4⋊C4×C3⋊D311yescharacteristic
312C2×C3#6412511yescharacteristic
313C2×C3#6412511yescharacteristic
314D3—182no—
315D3—188no—
316D3—368no—
317C22—272no—
318C22—541no—
319C22—541no—
320C22#8903911yescharacteristic
321C4—41no—
322C4—121no—
323C4—272no—
324C4—361no—
325C4—1081no—
326C4#8901911yescharacteristic
327C4C4×D3×C3⋊D311yescharacteristic
328C3—24no—
329C3#1059057414yescomplemented
330C3#1059255911yescharacteristic, complemented
331C2—542no—
332C2#1172347011yescharacteristic
333C2#1172356511yescharacteristic
334C2#1172358711yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#427
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C3×#63634C2semidirect
5. C32×#808C2semidirect
6. #11722057C2semidirect
7. #11722052C2semidirect
21. #88785C4semidirect
32. #63634D3semidirect
33. C3×#808D3semidirect
102. #808C3⋊D3semidirect
206. C33#427semidirect
292. C32#4860semidirect
293. C32#4880semidirect
329. C3#10590574semidirect
330. C3#10592559semidirect

Maximal subgroups

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

subgroupindex
#117239332
#117234132
C2×#887852
C3×#636342
C32×#8082
#117220572
#117220522
#105925593
#105905743