Finite Group Databasebeta

← #13159346

Subgroups — #13159346

Order 1728.

1012 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1012 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#11723503C211yesmaximal, characteristic
3C2×#88788C211yesmaximal, characteristic
4C3×#63635C211yesmaximal, characteristic, complemented
5C32×#808C211yesmaximal, characteristic, complemented
6#11722057C211yesmaximal, characteristic, complemented
7#11722055C211yesmaximal, characteristic, complemented
8#10592560—31nomaximal
9#10590584—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×#6289—21no—
12C33×C4⋊C4C2211yescharacteristic
13#88859—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21#88788C414yescomplemented
22C2×#3743—31no—
23#64101—31no—
24#64101—34no—
25#64101—64no—
26C4⋊C4×C3⋊D3—31no—
27#63900—34no—
28C2×#3712—34no—
29C32×#106—31no—
30#63645—31no—
31#63635D311yescharacteristic, complemented
32#63634—31no—
33C3×#808—34no—
34C3×#808D314yescomplemented
35#63575—34no—
36#63575—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—
46#6289—42no—
47C3×C3⋊(C3⋊C8)C2×C412yescomplemented
48C32×C3⋊C8C2×C412yescomplemented
49#4861—94no—
50#4861—91no—
51C2×C4×C3⋊D3—31no—
52C2×C4×C3⋊D3—34no—
53C2×C4×C3⋊D3—64no—
54C2×C4×C3⋊D3—61no—
55C2×#624—64no—
56C2×C32×C8—31no—
57C32×C4⋊C4—24no—
58C32×C4⋊C4—31no—
59C32×C4⋊C4C2×D314yes—
60C32×C4⋊C4C2×D311yescharacteristic
61#3752—61no—
62#3752—64no—
63#3752—68no—
64C4⋊(C3⋊Dic3)—31no—
65C4⋊(C3⋊Dic3)—34no—
66C4⋊(C3⋊Dic3)—64no—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68#3750—61no—
69C4×C3⋊Dic3—61no—
70C4×C3⋊Dic3—64no—
71C4×C3⋊Dic3—68no—
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73#3743—34no—
74C3×C4⋊Dic3—34no—
75C3×C4⋊Dic3C2×D314yes—
76C2×C3×C3⋊C8—34no—
77C2×C3×C3⋊C8C2×D314yes—
78#3724—64no—
79#3712—316no—
80C22×C33C2×D411yescharacteristic
81C2×C3⋊(C3⋊D3)—21no—
82C2×C3⋊(C3⋊D3)—21no—
83C2×C3⋊(C3⋊D3)C4⋊C412yes—
84C33×C4—41no—
85C33×C4C22×C411yescharacteristic
86C33×C4C2×Q811yescharacteristic
87C3⋊(C3⋊Dic3)—41no—
88C3⋊(C3⋊Dic3)C4⋊C412yes—
89C3×C3⋊Dic3—41no—
90C32×Dic3—41no—
91C2×ΓL(1,𝔽25)—94no—
92C2×ΓL(1,𝔽25)—91no—
93#892—94no—
94#892—91no—
95#892—184no—
96D3×C4⋊C4—94no—
97D3×C4⋊C4—91no—
98C3×#106—34no—
99C3×#106—91no—
100#818—94no—
101#818—91no—
102#808—34no—
103#808C3⋊D311yescharacteristic, complemented
104#807—94no—
105#807—91no—
106C22×C3⋊D3—31no—
107C22×C3⋊D3—34no—
108C22×C3⋊D3—64no—
109C2×C32×C4—28no—
110C2×C32×C4—24no—
111C2×C32×C4—21no—
112C2×C32×C4—24no—
113C2×C32×C4—61no—
114C2×C32×C4C22×D314yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—31no—
118C2×C3⋊Dic3—34no—
119C2×C3⋊Dic3—64no—
120C2×C3⋊Dic3—68no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—61no—
123C4×C3⋊D3—32no—
124C4×C3⋊D3—32no—
125C4×C3⋊D3—38no—
126C4×C3⋊D3—38no—
127C4×C3⋊D3—68no—
128C4×C3⋊D3—68no—
129C4×C3⋊D3—122no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—64no—
132#624—128no—
133C32×C8—32no—
134C3⋊(C3⋊C8)C4×D312yescomplemented
135C3×C3⋊C8—38no—
136C3×C3⋊C8C4×D318yescomplemented
137#428—271noSylow
138C2×C33C2×C4⋊C411yescharacteristic
139C2×C33AGL(1,ℤ/8)11yescharacteristic
140C2×C33#13711yescharacteristic
141C3⋊(C3⋊D3)—22no—
142C2×C4×D3—94no—
143C2×C4×D3—91no—
144C2×C4×D3—184no—
145C2×C4×D3—181no—
146C2×C4×D3—184no—
147C2×C3×C8—34no—
148C2×C3×C8—91no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—34no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4D3214yes—
153C3×C4⋊C4C2×C3⋊D311yescharacteristic
154(C2×D3)⋊C4—184no—
155(C2×D3)⋊C4—181no—
156(C2×D3)⋊C4—188no—
157C4⋊Dic3—34no—
158C4⋊Dic3—94no—
159C4⋊Dic3—91no—
160C4⋊Dic3—184no—
161C4⋊Dic3C2×C3⋊D311yescharacteristic
162Dic3⋊C4—181no—
163Dic3⋊C4—184no—
164C4×Dic3—184no—
165C4×Dic3—181no—
166C4×Dic3—188no—
167C2×C3⋊C8—34no—
168C2×C3⋊C8C2×C3⋊D311yescharacteristic
169ΓL(1,𝔽25)—916no—
170ΓL(1,𝔽25)—94no—
171C22×C32—24no—
172C22×C32D3×D414yes—
173C22×C32D3×D411yescharacteristic
174C2×C3⋊D3—32no—
175C2×C3⋊D3—38no—
176C2×C3⋊D3—68no—
177C2×C3⋊D3—61no—
178C2×C3⋊D3—61no—
179C2×C3⋊D3—64no—
180C2×C3⋊D3—64no—
181C2×C3⋊D3—124no—
182C2×C3⋊D3—124no—
183C32×C4—24no—
184C32×C4—24no—
185C32×C4—48no—
186C32×C4—44no—
187C32×C4—41no—
188C32×C4—121no—
189C32×C4C2×C4×D314yes—
190C32×C4C2×C4×D311yescharacteristic
191C32×C4D3×Q814yes—
192C32×C4D3×Q811yescharacteristic
193C3⋊Dic3—32no—
194C3⋊Dic3—38no—
195C3⋊Dic3—41no—
196C3⋊Dic3—68no—
197C3⋊Dic3—128no—
198C3⋊Dic3—124no—
199C3⋊Dic3—121no—
200C3×Dic3—44no—
201C3×Dic3—124no—
202C2×M16—271no—
203#117—271no—
204C2×C4⋊C4—271no—
205#107—271no—
206#107—271no—
207#106—31no—
208#106—91no—
209C33#42811yescharacteristic, complemented, Sylow
210C22×D3—94no—
211C22×D3—91no—
212C22×D3—184no—
213C2×C3×C4—28no—
214C2×C3×C4—24no—
215C2×C3×C4—21no—
216C2×C3×C4—24no—
217C2×C3×C4—64no—
218C2×C3×C4—181no—
219C2×C3×C4C2×D3214yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221C2×Dic3—21no—
222C2×Dic3—64no—
223C2×Dic3—94no—
224C2×Dic3—91no—
225C2×Dic3—184no—
226C2×Dic3—188no—
227C2×Dic3—184no—
228C2×Dic3—181no—
229C4×D3—98no—
230C4×D3—98no—
231C4×D3—92no—
232C4×D3—92no—
233C4×D3—188no—
234C4×D3—188no—
235C4×D3—362no—
236C4×D3—368no—
237C3×C8—38no—
238C3×C8—92no—
239C3⋊C8—38no—
240C3⋊C8C4×C3⋊D312yescomplemented
241C2×C32—24no—
242C2×C32—24no—
243C2×C32—24no—
244C2×C32D3×C4⋊C414yes—
245C2×C32D3×C4⋊C411yescharacteristic
246C2×C32#91414yes—
247C2×C32#91411yescharacteristic
248C2×C32#91514yes—
249C2×C32#91511yescharacteristic
250C3⋊D3—62no—
251C3⋊D3—68no—
252C3⋊D3—128no—
253C22×C4—271no—
254C22×C4—541no—
255M16—274no—
256C2×C8—31no—
257C2×C8—91no—
258C4⋊C4—31no—
259C4⋊C4—91no—
260C4⋊C4—271no—
261C4⋊C4—541no—
262C4⋊C4D3×C3⋊D311yescharacteristic
263C22⋊C4—541no—
264C42—541no—
265C22×C3—24no—
266C22×C3#381114yes—
267C22×C3D4×C3⋊D311yescharacteristic
268C2×D3—92no—
269C2×D3—98no—
270C2×D3—188no—
271C2×D3—181no—
272C2×D3—181no—
273C2×D3—184no—
274C2×D3—184no—
275C2×D3—364no—
276C2×D3—364no—
277C3×C4—24no—
278C3×C4—24no—
279C3×C4—48no—
280C3×C4—41no—
281C3×C4—44no—
282C3×C4—124no—
283C3×C4—361no—
284C3×C4#379714yes—
285C3×C4C4×D3214yes—
286C3×C4C2×C4×C3⋊D311yescharacteristic
287C3×C4Q8×C3⋊D311yescharacteristic
288Dic3—41no—
289Dic3—92no—
290Dic3—98no—
291Dic3—124no—
292Dic3—188no—
293Dic3—368no—
294Dic3—361no—
295Dic3—364no—
296C32—24no—
297C32#486114yescomplemented
298C32#486111yescharacteristic, complemented
299C23—271no—
300C2×C4—21no—
301C2×C4—61no—
302C2×C4—181no—
303C2×C4—271no—
304C2×C4—272no—
305C2×C4—272no—
306C2×C4—541no—
307C2×C4—1082no—
308C2×C4C2×D3×C3⋊D311yescharacteristic
309C8—32no—
310C8—92no—
311C2×C3—24no—
312C2×C3—24no—
313C2×C3—24no—
314C2×C3#6388314yes—
315C2×C3#6393114yes—
316C2×C3#6394114yes—
317C2×C3C4⋊C4×C3⋊D311yescharacteristic
318C2×C3#6412311yescharacteristic
319C2×C3#6412411yescharacteristic
320D3—182no—
321D3—188no—
322D3—368no—
323C22—272no—
324C22—541no—
325C22—541no—
326C22#8903911yescharacteristic
327C4—41no—
328C4—121no—
329C4—272no—
330C4—361no—
331C4—1081no—
332C4#8901911yescharacteristic
333C4C4×D3×C3⋊D311yescharacteristic
334C3—24no—
335C3#1059058414yescomplemented
336C3#1059256011yescharacteristic, complemented
337C2—542no—
338C2#1172347011yescharacteristic
339C2#1172355411yescharacteristic
340C2#1172357211yescharacteristic
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#428
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×#63635C2semidirect
5. C32×#808C2semidirect
6. #11722057C2semidirect
7. #11722055C2semidirect
21. #88788C4semidirect
31. #63635D3semidirect
34. C3×#808D3semidirect
47. C3×C3⋊(C3⋊C8)C2×C4semidirect
48. C32×C3⋊C8C2×C4semidirect
103. #808C3⋊D3semidirect
134. C3⋊(C3⋊C8)C4×D3semidirect
136. C3×C3⋊C8C4×D3semidirect
209. C33#428semidirect
240. C3⋊C8C4×C3⋊D3semidirect
297. C32#4861semidirect
298. C32#4861semidirect
335. C3#10590584semidirect
336. C3#10592560semidirect

Maximal subgroups

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

subgroupindex
#117239332
#117235032
C2×#887882
C3×#636352
C32×#8082
#117220572
#117220552
#105925603
#105905843