Finite Group Databasebeta

← #13161155

Subgroups — #13161155

Order 1728.

1004 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 196 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
1C2×#88862C211yesmaximal, characteristic
2C32×Q8×Dic3C211yesmaximal, characteristic, complemented
3#11723484C211yesmaximal, characteristic
4C3×#63662C211yesmaximal, characteristic, complemented
5#11722053C211yesmaximal, characteristic, complemented
6#11722035C211yesmaximal, characteristic
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C4×#3755—31nomaximal
9#10591536—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×#6289—21no—
13#88862C414yescomplemented
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3—21no—
20C32×C4×Dic3C2211yescharacteristic
21#88804—21no—
22C32×C4×Q8—31no—
23C2×#3755—31no—
24C2×#3755—34no—
25C2×#3755—64no—
26C4×C4⋊(C3⋊D3)—31no—
27C3×Q8×Dic3D314yescomplemented
28#63894—34no—
29#63662D311yescharacteristic, complemented
30#63636—31no—
31#63635—31no—
32C4×C3⋊(C3⋊C8)—31no—
33C3×#835—34no—
34#63574—34no—
35#63564—34no—
36Dic3×C3⋊C8—34no—
37C22×C3⋊(C3⋊D3)—21no—
38C33×Q8—21no—
39C33×Q8C2×C412yescomplemented
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))C2×C412yes—
44C2×C3×C3⋊Dic3—21no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3D411yescharacteristic
47C2×C32×Dic3D411yescharacteristic
48#6289—42no—
49C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
50C3×C3⋊(C3⋊C8)—21no—
51C4×#215—94no—
52#4788—91no—
53C2×C32×Q8—24no—
54C2×C32×Q8C2×D314yes—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4⋊(C3⋊D3)—34no—
58C2×C4⋊(C3⋊D3)—64no—
59C2×C4×C3⋊D3—61no—
60C2×#624—64no—
61C32×C4⋊C4—31no—
62C32×C4⋊C4—61no—
63C32×C42—31no—
64C32×C42—61no—
65#3755—34no—
66#3755—316no—
67#3755—616no—
68#3752—61no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)—31no—
71C2×C3⋊(C3⋊C8)—34no—
72C2×C3⋊(C3⋊C8)—64no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C3×C4⋊Dic3—24no—
75C3×C4⋊Dic3—34no—
76C3×C4⋊Dic3C2×D314yes—
77C3×C4×Dic3—24no—
78C3×C4×Dic3C2×D314yes—
79C2×C3×C3⋊C8—34no—
80#3722—64no—
81C22×C33C2×D411yescharacteristic
82C2×C3⋊(C3⋊D3)—22no—
83C2×C3⋊(C3⋊D3)—41no—
84C2×C3⋊(C3⋊D3)—41no—
85C33×C4—22no—
86C33×C4C22×C411yescharacteristic
87C33×C4#4111yescharacteristic
88C3×C3⋊Dic3—41no—
89C32×Dic3—21no—
90C32×Dic3—41no—
91C32×Dic3SD1612yescomplemented
92C3×C4×Q8—34no—
93Q8×Dic3C3⋊D311yescharacteristic, complemented
94C2×#215—94no—
95C2×#215—91no—
96C2×#215—184no—
97#893—91no—
98C4×D12—94no—
99C3×#103—91no—
100#835—34no—
101#821—91no—
102C8×Dic3—91no—
103#809—94no—
104#808—94no—
105#808—91no—
106C4×C3⋊C8—94no—
107C22×C3⋊D3—61no—
108C22×C3⋊D3—64no—
109C22×C3⋊D3—124no—
110C32×Q8—21no—
111C32×Q8—28no—
112C32×Q8—24no—
113C32×Q8—44no—
114C32×Q8C4×D318yescomplemented
115C32×Q8C4×D312yescomplemented
116C2×C32×C4—21no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—31no—
120C2×C32×C4—31no—
121C2×C32×C4—44no—
122C2×C32×C4—61no—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—21no—
126C4⋊(C3⋊D3)—32no—
127C4⋊(C3⋊D3)—38no—
128C4⋊(C3⋊D3)—68no—
129C4⋊(C3⋊D3)—61no—
130C4⋊(C3⋊D3)—64no—
131C4⋊(C3⋊D3)—124no—
132C4×C3⋊D3—122no—
133C2×C3×Dic3—24no—
134C2×C3×Dic3—64no—
135C2×C3×Dic3#6714yes—
136C2×C3×Dic3#6714yes—
137#624—128no—
138C3⋊(C3⋊C8)—21no—
139C3⋊(C3⋊C8)—32no—
140C3⋊(C3⋊C8)—38no—
141C3⋊(C3⋊C8)—68no—
142C3×C3⋊C8—64no—
143C4×SD16—271noSylow
144C2×C33C4×D411yescharacteristic
145C2×C33C2×SD1611yescharacteristic
146C2×C33#13511yescharacteristic
147C3⋊(C3⋊D3)—42no—
148C2×C3×Q8—24no—
149C2×C3×Q8D3214yes—
150C2×C3×Q8C2×C3⋊D311yescharacteristic
151C2×D12—94no—
152C2×D12—91no—
153C2×D12—184no—
154C2×C4×D3—184no—
155C2×C4×D3—181no—
156C2×C3×C8—91no—
157C3×C4⋊C4—34no—
158C3×C4⋊C4—64no—
159C3×C4⋊C4—91no—
160C3×C42—34no—
161C3×C42—64no—
162#215—916no—
163#215—94no—
164#215—1816no—
165(C2×D3)⋊C4—184no—
166(C2×D3)⋊C4—181no—
167C4⋊Dic3—21no—
168C4⋊Dic3—34no—
169C4⋊Dic3C2×C3⋊D311yescharacteristic
170C4×Dic3—21no—
171C4×Dic3C2×C3⋊D311yescharacteristic
172C2×C3⋊C8—34no—
173C2×C3⋊C8—94no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—184no—
176C22×C32—24no—
177C22×C32D3×D411yescharacteristic
178C22×C32C2×#6714yes—
179C2×C3⋊D3—62no—
180C2×C3⋊D3—68no—
181C2×C3⋊D3—128no—
182C2×C3⋊D3—121no—
183C2×C3⋊D3—121no—
184C2×C3⋊D3—124no—
185C2×C3⋊D3—124no—
186C2×C3⋊D3—244no—
187C2×C3⋊D3—244no—
188C32×C4—22no—
189C32×C4—24no—
190C32×C4—24no—
191C32×C4—28no—
192C32×C4—32no—
193C32×C4—48no—
194C32×C4—61no—
195C32×C4—121no—
196C32×C4C2×C4×D314yes—
197C32×C4C2×C4×D311yescharacteristic
198C32×C4#23514yes—
199C32×C4#23711yescharacteristic
200C3⋊Dic3—41no—
201C3×Dic3—24no—
202C3×Dic3—44no—
203C3×Dic3—124no—
204C3×Dic3#21518yescomplemented
205C2×SD16—271no—
206C4×Q8—31no—
207C4×D4—271no—
208#106—271no—
209#103—91no—
210#102—271no—
211C4×C8—271no—
212C33C4×SD1611yescharacteristic, complemented, Sylow
213C22×D3—184no—
214C22×D3—181no—
215C22×D3—364no—
216C3×Q8—24no—
217C3×Q8—21no—
218C3×Q8—28no—
219C3×Q8—44no—
220C3×Q8#62418yescomplemented
221C3×Q8C4×C3⋊D312yescomplemented
222C2×C3×C4—24no—
223C2×C3×C4—21no—
224C2×C3×C4—24no—
225C2×C3×C4—34no—
226C2×C3×C4—34no—
227C2×C3×C4—44no—
228C2×C3×C4—64no—
229C2×C3×C4—181no—
230C2×C3×C4C2×D3214yes—
231C2×C3×C4C22×C3⋊D311yescharacteristic
232C2×Dic3—21no—
233C2×Dic3—64no—
234C2×Dic3#63811yescharacteristic
235C2×Dic3#63811yescharacteristic
236D12—98no—
237D12—92no—
238D12—188no—
239D12—184no—
240D12—181no—
241D12—364no—
242C4×D3—368no—
243C4×D3—362no—
244C3×C8—181no—
245C3⋊C8—64no—
246C3⋊C8—98no—
247C3⋊C8—92no—
248C3⋊C8—188no—
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32#67⋊C411yescharacteristic
253C2×C32D3×SD1611yescharacteristic
254C2×C32#91911yescharacteristic
255C2×C32C4×#6714yes—
256C2×C32C2×#21514yes—
257C2×C32#95014yes—
258C3⋊D3—122no—
259C3⋊D3—128no—
260C3⋊D3—248no—
261C2×Q8D3×C3⋊D311yescharacteristic
262C2×D4—271no—
263C22×C4—541no—
264SD16—274no—
265C2×C8—91no—
266C2×C8—271no—
267C4⋊C4—31no—
268C4⋊C4—61no—
269C4⋊C4—91no—
270C22⋊C4—541no—
271C42—31no—
272C42—61no—
273C22×C3—24no—
274C22×C3D3×#6714yes—
275C22×C3C2×#63811yescharacteristic
276C2×D3—182no—
277C2×D3—188no—
278C2×D3—368no—
279C2×D3—361no—
280C2×D3—361no—
281C2×D3—364no—
282C2×D3—364no—
283C2×D3—724no—
284C2×D3—724no—
285C3×C4—22no—
286C3×C4—28no—
287C3×C4—24no—
288C3×C4—24no—
289C3×C4—38no—
290C3×C4—48no—
291C3×C4—64no—
292C3×C4—124no—
293C3×C4—361no—
294C3×C4#380414yes—
295C3×C4C2×#62414yes—
296C3×C4C2×C4×C3⋊D311yescharacteristic
297C3×C4#382811yescharacteristic
298Dic3—21no—
299Dic3—41no—
300Dic3—124no—
301Dic3#375512yescomplemented
302C32—24no—
303C32#478811yescharacteristic, complemented
304C32C4×#21514yescomplemented
305C23—541no—
306Q8—21no—
307Q8#628912yescomplemented
308D4—272no—
309D4—541no—
310C2×C4—21no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—61no—
314C2×C4—181no—
315C2×C4—1082no—
316C2×C4C2×D3×C3⋊D311yescharacteristic
317C8—181no—
318C8—272no—
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3D3×#21514yes—
323C2×C3#6394914yes—
324C2×C3#6395214yes—
325C2×C3C4×#63811yescharacteristic
326C2×C3C2×#375511yescharacteristic
327C2×C3#6415911yescharacteristic
328D3—368no—
329D3—362no—
330D3—728no—
331C22—542no—
332C22—1081no—
333C22—1081no—
334C22D3×#63811yescharacteristic
335C4—22no—
336C4—32no—
337C4—61no—
338C4—121no—
339C4—361no—
340C4#8903111yescharacteristic
341C4C2×#628911yescharacteristic
342C3—24no—
343C3#1059153614yescomplemented
344C3C4×#375511yescharacteristic, complemented
345C2—1082no—
346C2D3×#375511yescharacteristic
347C2#1172358211yescharacteristic
348C2#1172361811yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33C4×SD16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C32×Q8×Dic3C2semidirect
4. C3×#63662C2semidirect
5. #11722053C2semidirect
7. Dic3×C3⋊(C3⋊C8)C2semidirect
13. #88862C4semidirect
27. C3×Q8×Dic3D3semidirect
29. #63662D3semidirect
39. C33×Q8C2×C4semidirect
49. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
91. C32×Dic3SD16semidirect
93. Q8×Dic3C3⋊D3semidirect
114. C32×Q8C4×D3semidirect
115. C32×Q8C4×D3semidirect
204. C3×Dic3#215semidirect
212. C33C4×SD16semidirect
220. C3×Q8#624semidirect
221. C3×Q8C4×C3⋊D3semidirect
301. Dic3#3755semidirect
303. C32#4788semidirect
304. C32C4×#215semidirect
307. Q8#6289semidirect
343. C3#10591536semidirect
344. C3C4×#3755semidirect

Maximal subgroups

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

subgroupindex
C2×#888622
C32×Q8×Dic32
#117234842
C3×#636622
#117220532
#117220352
Dic3×C3⋊(C3⋊C8)2
C4×#37553
#105915363