Finite Group Databasebeta

← #13159326

Subgroups — #13159326

Order 1728.

1148 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 178 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⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2#11723503C211yesmaximal, characteristic
3C2×#88785C211yesmaximal, characteristic
4C3×#63635C211yesmaximal, characteristic, complemented
5C32×#808C211yesmaximal, characteristic, complemented
6#11722054C211yesmaximal, characteristic, complemented
7#11722052C211yesmaximal, characteristic, complemented
8C3⋊D3×#106—31nomaximal
9#10590575—34nomaximal
10C2×C4×C3⋊(C3⋊D3)—21no—
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×#6289—21no—
13C33×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15#88857—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21#88785C414yescomplemented
22C2×C8×C3⋊D3—31no—
23C4⋊C4×C3⋊D3—31no—
24C4⋊C4×C3⋊D3—31no—
25C4⋊C4×C3⋊D3—34no—
26C4⋊C4×C3⋊D3—64no—
27#63900—34no—
28C2×#3710—34no—
29C32×#106—31no—
30#63644—31no—
31#63635—31no—
32#63635D311yescharacteristic, complemented
33C3×#808—34no—
34C3×#808D314yescomplemented
35#63573—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)—42no—
43C4×C3⋊(C3⋊D3)C2×C412yes—
44C4×C3⋊(C3⋊D3)Q812yes—
45C2×C3×C3⋊Dic3—21no—
46C2×C32×Dic3—21no—
47#6289—42no—
48C3×C3⋊(C3⋊C8)C2×C412yescomplemented
49C32×C3⋊C8C2×C412yescomplemented
50D3×#106—94no—
51D3×#106—91no—
52C2×C4×C3⋊D3—31no—
53C2×C4×C3⋊D3—34no—
54C2×C4×C3⋊D3—64no—
55C2×C4×C3⋊D3—61no—
56C2×C4×C3⋊D3—61no—
57C2×C4×C3⋊D3—64no—
58C2×C4×C3⋊D3—68no—
59C2×#624—64no—
60C2×C32×C8—31no—
61C32×C4⋊C4—24no—
62C32×C4⋊C4—31no—
63C32×C4⋊C4C2×D314yes—
64C32×C4⋊C4C2×D311yescharacteristic
65C4⋊(C3⋊Dic3)—31no—
66C4⋊(C3⋊Dic3)—34no—
67C4⋊(C3⋊Dic3)—64no—
68C4⋊(C3⋊Dic3)C2×D311yescharacteristic
69#3750—61no—
70#3750—61no—
71#3750—64no—
72#3750—68no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C8×C3⋊D3—34no—
75C3×C4⋊Dic3—34no—
76C3×C4⋊Dic3C2×D314yes—
77C2×C3×C3⋊C8—34no—
78C2×C3×C3⋊C8C2×D314yes—
79#3724—64no—
80#3710—316no—
81C22×C33C2×D411yescharacteristic
82C2×C3⋊(C3⋊D3)C4⋊C412yes—
83C2×C3⋊(C3⋊D3)SD1612yes—
84C2×C3⋊(C3⋊D3)SD1612yes—
85C33×C4—41no—
86C33×C4C22×C411yescharacteristic
87C33×C4C2×Q811yescharacteristic
88C3⋊(C3⋊Dic3)—41no—
89C3⋊(C3⋊Dic3)C4⋊C412yes—
90C3×C3⋊Dic3—41no—
91C32×Dic3—41no—
92C2×D3×C8—94no—
93C2×D3×C8—91no—
94D3×C4⋊C4—94no—
95D3×C4⋊C4—94no—
96D3×C4⋊C4—91no—
97D3×C4⋊C4—91no—
98D3×C4⋊C4—184no—
99C3×#106—34no—
100C3×#106—91no—
101#817—94no—
102#817—91no—
103#808—34no—
104#808—94no—
105#808—91no—
106#808C3⋊D311yescharacteristic, complemented
107C22×C3⋊D3—31no—
108C22×C3⋊D3—34no—
109C22×C3⋊D3—64no—
110C2×C32×C4—28no—
111C2×C32×C4—24no—
112C2×C32×C4—21no—
113C2×C32×C4—24no—
114C2×C32×C4—61no—
115C2×C32×C4C22×D314yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—31no—
119C2×C3⋊Dic3—34no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—68no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—61no—
124C4×C3⋊D3—32no—
125C4×C3⋊D3—32no—
126C4×C3⋊D3—38no—
127C4×C3⋊D3—38no—
128C4×C3⋊D3—68no—
129C4×C3⋊D3—68no—
130C4×C3⋊D3—122no—
131C4×C3⋊D3—122no—
132C4×C3⋊D3—128no—
133C4×C3⋊D3—1216no—
134C2×C3×Dic3—24no—
135C2×C3×Dic3—64no—
136#624—128no—
137C32×C8—32no—
138C3⋊(C3⋊C8)C4×D312yescomplemented
139C3×C3⋊C8—38no—
140C3×C3⋊C8C4×D318yescomplemented
141C2×#106—271noSylow
142C2×C33C2×C4⋊C411yescharacteristic
143C2×C33C2×SD1611yescharacteristic
144C2×C33C2×SD1611yescharacteristic
145C3⋊(C3⋊D3)#10614yescomplemented
146C2×C4×D3—94no—
147C2×C4×D3—91no—
148C2×C4×D3—184no—
149C2×C4×D3—181no—
150C2×C4×D3—184no—
151C2×C4×D3—184no—
152C2×C4×D3—181no—
153C2×C4×D3—188no—
154C2×C3×C8—34no—
155C2×C3×C8—91no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—34no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4D3214yes—
160C3×C4⋊C4C2×C3⋊D311yescharacteristic
161C4⋊Dic3—34no—
162C4⋊Dic3—94no—
163C4⋊Dic3—91no—
164C4⋊Dic3—184no—
165C4⋊Dic3C2×C3⋊D311yescharacteristic
166Dic3⋊C4—181no—
167Dic3⋊C4—184no—
168Dic3⋊C4—184no—
169Dic3⋊C4—181no—
170Dic3⋊C4—188no—
171C2×C3⋊C8—34no—
172C2×C3⋊C8C2×C3⋊D311yescharacteristic
173D3×C8—916no—
174D3×C8—94no—
175C22×C32—24no—
176C22×C32D3×D414yes—
177C22×C32D3×D411yescharacteristic
178C2×C3⋊D3—32no—
179C2×C3⋊D3—32no—
180C2×C3⋊D3—32no—
181C2×C3⋊D3—38no—
182C2×C3⋊D3—38no—
183C2×C3⋊D3—38no—
184C2×C3⋊D3—68no—
185C2×C3⋊D3—68no—
186C2×C3⋊D3—68no—
187C32×C4—24no—
188C32×C4—24no—
189C32×C4—48no—
190C32×C4—44no—
191C32×C4—41no—
192C32×C4—121no—
193C32×C4C2×C4×D314yes—
194C32×C4C2×C4×D311yescharacteristic
195C32×C4D3×Q814yes—
196C32×C4D3×Q811yescharacteristic
197C3⋊Dic3—32no—
198C3⋊Dic3—38no—
199C3⋊Dic3—41no—
200C3⋊Dic3—68no—
201C3⋊Dic3—128no—
202C3⋊Dic3—124no—
203C3⋊Dic3—121no—
204C3×Dic3—44no—
205C3×Dic3—124no—
206C22×C8—271no—
207C2×C4⋊C4—271no—
208C2×C4⋊C4—271no—
209#106—31no—
210#106—91no—
211#106—271no—
212#106—271no—
213C33C2×#10611yescharacteristic, complemented, Sylow
214C22×D3—94no—
215C22×D3—91no—
216C22×D3—184no—
217C2×C3×C4—28no—
218C2×C3×C4—24no—
219C2×C3×C4—21no—
220C2×C3×C4—24no—
221C2×C3×C4—64no—
222C2×C3×C4—181no—
223C2×C3×C4C2×D3214yes—
224C2×C3×C4C22×C3⋊D311yescharacteristic
225C2×Dic3—21no—
226C2×Dic3—64no—
227C2×Dic3—94no—
228C2×Dic3—91no—
229C2×Dic3—184no—
230C2×Dic3—188no—
231C2×Dic3—184no—
232C2×Dic3—181no—
233C4×D3—98no—
234C4×D3—98no—
235C4×D3—92no—
236C4×D3—92no—
237C4×D3—188no—
238C4×D3—188no—
239C4×D3—362no—
240C4×D3—368no—
241C4×D3—362no—
242C4×D3—368no—
243C4×D3—3616no—
244C3×C8—38no—
245C3×C8—92no—
246C3⋊C8—38no—
247C3⋊C8C4×C3⋊D312yescomplemented
248C2×C32—24no—
249C2×C32—24no—
250C2×C32—24no—
251C2×C32D3×C4⋊C414yes—
252C2×C32D3×C4⋊C411yescharacteristic
253C2×C32D3×SD1614yes—
254C2×C32D3×SD1614yes—
255C2×C32D3×SD1611yescharacteristic
256C2×C32D3×SD1611yescharacteristic
257C3⋊D3—34no—
258C3⋊D3—316no—
259C3⋊D3—616no—
260C22×C4—271no—
261C22×C4—541no—
262C22×C4—541no—
263C2×C8—31no—
264C2×C8—91no—
265C2×C8—274no—
266C4⋊C4—31no—
267C4⋊C4—91no—
268C4⋊C4—271no—
269C4⋊C4—541no—
270C4⋊C4—541no—
271C4⋊C4D3×C3⋊D311yescharacteristic
272C22×C3—24no—
273C22×C3#381114yes—
274C22×C3D4×C3⋊D311yescharacteristic
275C2×D3—92no—
276C2×D3—92no—
277C2×D3—92no—
278C2×D3—98no—
279C2×D3—98no—
280C2×D3—98no—
281C2×D3—188no—
282C2×D3—188no—
283C2×D3—188no—
284C3×C4—24no—
285C3×C4—24no—
286C3×C4—48no—
287C3×C4—41no—
288C3×C4—44no—
289C3×C4—124no—
290C3×C4—361no—
291C3×C4#379714yes—
292C3×C4C4×D3214yes—
293C3×C4C2×C4×C3⋊D311yescharacteristic
294C3×C4Q8×C3⋊D311yescharacteristic
295Dic3—41no—
296Dic3—92no—
297Dic3—98no—
298Dic3—124no—
299Dic3—188no—
300Dic3—368no—
301Dic3—361no—
302Dic3—364no—
303C32—24no—
304C32D3×#10614yescomplemented
305C32D3×#10611yescharacteristic, complemented
306C23—271no—
307C2×C4—21no—
308C2×C4—61no—
309C2×C4—181no—
310C2×C4—271no—
311C2×C4—272no—
312C2×C4—272no—
313C2×C4—541no—
314C2×C4—1082no—
315C2×C4—1082no—
316C2×C4C2×D3×C3⋊D311yescharacteristic
317C8—32no—
318C8—92no—
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3#6388314yes—
323C2×C3#6393014yes—
324C2×C3#6394014yes—
325C2×C3C4⋊C4×C3⋊D311yescharacteristic
326C2×C3SD16×C3⋊D311yescharacteristic
327C2×C3SD16×C3⋊D311yescharacteristic
328D3—94no—
329D3—916no—
330D3—1816no—
331C22—272no—
332C22—272no—
333C22—272no—
334C22#8903911yescharacteristic
335C4—41no—
336C4—121no—
337C4—272no—
338C4—361no—
339C4—1081no—
340C4#8901911yescharacteristic
341C4C4×D3×C3⋊D311yescharacteristic
342C3—24no—
343C3#1059057514yescomplemented
344C3C3⋊D3×#10611yescharacteristic, complemented
345C2—274no—
346C2#1172347011yescharacteristic
347C2#1172355311yescharacteristic
348C2#1172357111yescharacteristic
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
hypocenterC33C2×#106
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. #11722054C2semidirect
7. #11722052C2semidirect
21. #88785C4semidirect
32. #63635D3semidirect
34. C3×#808D3semidirect
48. C3×C3⋊(C3⋊C8)C2×C4semidirect
49. C32×C3⋊C8C2×C4semidirect
106. #808C3⋊D3semidirect
138. C3⋊(C3⋊C8)C4×D3semidirect
140. C3×C3⋊C8C4×D3semidirect
145. C3⋊(C3⋊D3)#106semidirect
213. C33C2×#106semidirect
247. C3⋊C8C4×C3⋊D3semidirect
304. C32D3×#106semidirect
305. C32D3×#106semidirect
343. C3#10590575semidirect
344. C3C3⋊D3×#106semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×C3⋊(C3⋊D3)2
#117235032
C2×#887852
C3×#636352
C32×#8082
#117220542
#117220522
C3⋊D3×#1063
#105905753