Finite Group Databasebeta

← #13161205

Subgroups — #13161205

Order 1728.

828 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 156 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×#64155C211yesmaximal, characteristic
2C2×C32×#216C211yesmaximal, characteristic, complemented
3#11723435C211yesmaximal, characteristic
4C2×#88787C211yesmaximal, characteristic
5#11722205C211yesmaximal, characteristic, complemented
6#11722052C211yesmaximal, characteristic, complemented
7#11722037C211yesmaximal, characteristic, complemented
8#10592888—31nomaximal
9#10591638—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3750—21no—
18C32×#216—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21#88803—21no—
22#88787—22no—
23C2×C32×Q16—31no—
24#64155—31no—
25#64155D311yescharacteristic
26C2×#3743—31no—
27C3×#946—34no—
28C2×C3×#216D314yescomplemented
29#63856—34no—
30C2×#3711—34no—
31#63662—31no—
32#63662—31no—
33#63662—34no—
34#63662—64no—
35#63644—31no—
36#63573—34no—
37#63565—34no—
38C22×C3×C3⋊D3D411yescharacteristic
39C33×Q8—21no—
40C33×Q8—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3D411yescharacteristic
46C3×C4×C3⋊D3—21no—
47C3×C4×C3⋊D3—21no—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51C3⋊(C3⋊(C3⋊C8))—21no—
52C32×C3⋊C8D412yes—
53#5189—94no—
54#5052—91no—
55C2×C32×Q8—24no—
56C2×C32×Q8—31no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C4×C3⋊D3C2×D311yescharacteristic
60C2×C3×C4×D3—34no—
61C2×C3×Dic6C2×D314yes—
62C32×Q16—62no—
63C2×C32×C8—31no—
64#3752—21no—
65#3752—61no—
66C4⋊(C3⋊Dic3)—31no—
67C4⋊(C3⋊Dic3)—34no—
68C4⋊(C3⋊Dic3)—64no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70#3750—21no—
71#3750—61no—
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—34no—
74C2×C3⋊(C3⋊C8)—64no—
75#3743—62no—
76C3×#216—28no—
77C3×(C2×D3)⋊C4—64no—
78C3×C4⋊Dic3—34no—
79C3×Dic3⋊C4—64no—
80C2×C3×C3⋊C8C2×D314yes—
81#3724—64no—
82(C2×D3)⋊Dic3—64no—
83#3711—68no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—21no—
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3—21no—
88C33×C4—41no—
89C33×C4C2×D411yescharacteristic
90C33×C4#4111yescharacteristic
91C3⋊(C3⋊Dic3)—41no—
92C3×C3⋊Dic3—21no—
93C3×C3⋊Dic3—41no—
94C32×Dic3—41no—
95C2×C3×Q16—34no—
96C3×C22⋊Q8—91no—
97#946—34no—
98#946—94no—
99C2×#216C3⋊D311yescharacteristic, complemented
100#924—91no—
101C2×#208—91no—
102C2×ΓL(1,𝔽25)—94no—
103#835—94no—
104#835—94no—
105#835—91no—
106#835—184no—
107#817—94no—
108#810—91no—
109#808—91no—
110C22×C3⋊D3#6711yescharacteristic
111C22×C3×D3—34no—
112C32×Q8—28no—
113C32×Q8—21no—
114C32×Q8—24no—
115C32×Q8—21no—
116C32×Q8—24no—
117C32×Q8—44no—
118C32×Q8—61no—
119C32×Q8—61no—
120C2×C32×C4—21no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—44no—
124C2×C32×C4—61no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—61no—
129C2×C3⋊Dic3—64no—
130C2×C3⋊Dic3—124no—
131C2×C3⋊Dic3#6711yescharacteristic
132C4×C3⋊D3—21no—
133C4×C3⋊D3—21no—
134C2×C3×Dic3—24no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—64no—
137C3×C4×D3—64no—
138C3×C4×D3—64no—
139C3×Dic6—24no—
140C3×Dic6—24no—
141C32×C8—32no—
142C3⋊(C3⋊C8)—61no—
143C3⋊(C3⋊C8)—64no—
144C3⋊(C3⋊C8)—124no—
145C3×C3⋊C8#6718yes—
146#470—271noSylow
147C2×C33#12111yescharacteristic
148C2×C33#13711yescharacteristic
149C2×C33#13711yescharacteristic
150C3×C3⋊D3—41no—
151C2×C3×Q8—24no—
152C2×C3×Q8—34no—
153C2×C3×Q8D3214yes—
154C2×C3×Q8C2×C3⋊D311yescharacteristic
155C22×C3×C4—91no—
156C2×C4×D3—34no—
157C2×Dic6C2×C3⋊D311yescharacteristic
158C3×Q16—68no—
159C2×C3×C8—34no—
160C3×C4⋊C4—91no—
161C3×C4⋊C4—181no—
162C3×C22⋊C4—181no—
163#217—181no—
164#216—22no—
165(C2×D3)⋊C4—64no—
166(C2×D3)⋊C4—184no—
167C4⋊Dic3—34no—
168C4⋊Dic3—94no—
169C4⋊Dic3—91no—
170C4⋊Dic3—184no—
171Dic3⋊C4—64no—
172Dic3⋊C4—181no—
173Dic3⋊C4—184no—
174#208—182no—
175C2×C3⋊C8—94no—
176C2×C3⋊C8—91no—
177C2×C3⋊C8—184no—
178C2×C3⋊C8C2×C3⋊D311yescharacteristic
179ΓL(1,𝔽25)—188no—
180C22×C32—24no—
181C22×C32D3×D414yes—
182C22×C32C2×#6711yescharacteristic
183C2×C3⋊D3—21no—
184C2×C3⋊D3—21no—
185C2×C3⋊D3—21no—
186C2×C3×D3—64no—
187C2×C3×D3—64no—
188C2×C3×D3—64no—
189C32×C4—24no—
190C32×C4—24no—
191C32×C4—48no—
192C32×C4—44no—
193C32×C4—41no—
194C32×C4—121no—
195C32×C4D3×D411yescharacteristic
196C32×C4#23714yes—
197C32×C4#23711yescharacteristic
198C32×C4C2×#6714yes—
199C3⋊Dic3—21no—
200C3⋊Dic3—41no—
201C3⋊Dic3—121no—
202C3⋊Dic3—124no—
203C3⋊Dic3—244no—
204C3×Dic3—44no—
205C3×Dic3—64no—
206C3×Dic3—124no—
207C2×Q16—31no—
208C2×M16—271no—
209C22⋊Q8—91no—
210C22⋊Q8—271no—
211#106—271no—
212#103—271no—
213#103—271no—
214C33#47011yescharacteristic, complemented, Sylow
215C23×C3—91no—
216C22×D3—34no—
217C3×Q8—28no—
218C3×Q8—24no—
219C3×Q8—24no—
220C3×Q8—21no—
221C3×Q8—21no—
222C3×Q8—44no—
223C3×Q8—64no—
224C3×Q8—64no—
225C2×C3×C4—24no—
226C2×C3×C4—21no—
227C2×C3×C4—24no—
228C2×C3×C4—44no—
229C2×C3×C4—64no—
230C2×C3×C4—91no—
231C2×C3×C4—181no—
232C2×C3×C4—181no—
233C2×C3×C4—181no—
234C2×C3×C4C2×D3214yes—
235C2×C3×C4C22×C3⋊D311yescharacteristic
236C2×Dic3—21no—
237C2×Dic3—34no—
238C2×Dic3—64no—
239C2×Dic3—184no—
240C2×Dic3—181no—
241C2×Dic3—364no—
242C4×D3—64no—
243C4×D3—64no—
244Dic6—21no—
245Dic6—21no—
246C3×C8—38no—
247C3⋊C8—184no—
248C3⋊C8—181no—
249C3⋊C8—364no—
250C3⋊C8#63812yes—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32#91814yes—
255C2×C32#91814yes—
256C2×C32#93814yes—
257C2×C32#93911yescharacteristic
258C2×C32#94211yescharacteristic
259C2×C32#95111yescharacteristic
260C3⋊D3—41no—
261C3×D3—124no—
262C2×Q8—31no—
263C2×Q8D3×C3⋊D311yescharacteristic
264C22×C4—91no—
265Q16—62no—
266M16—542no—
267C2×C8—31no—
268C2×C8—271no—
269C4⋊C4—91no—
270C4⋊C4—181no—
271C4⋊C4—271no—
272C4⋊C4—541no—
273C22⋊C4—181no—
274C22⋊C4—541no—
275C22×C3—24no—
276C22×C3—181no—
277C22×C3—181no—
278C22×C3—181no—
279C22×C3D3×#6714yes—
280C22×C3D4×C3⋊D311yescharacteristic
281C2×D3—64no—
282C2×D3—64no—
283C2×D3—64no—
284C3×C4—24no—
285C3×C4—24no—
286C3×C4—48no—
287C3×C4—41no—
288C3×C4—44no—
289C3×C4—124no—
290C3×C4—181no—
291C3×C4—361no—
292C3×C4#380514yes—
293C3×C4D3×#6714yes—
294C3×C4#383011yescharacteristic
295C3×C4C2×#63811yescharacteristic
296Dic3—41no—
297Dic3—64no—
298Dic3—124no—
299Dic3—361no—
300Dic3—364no—
301Dic3—724no—
302C32—24no—
303C32#505211yescharacteristic, complemented
304C32#518914yescomplemented
305C23—91no—
306Q8—21no—
307Q8—21no—
308Q8—61no—
309Q8—61no—
310C2×C4—21no—
311C2×C4—61no—
312C2×C4—91no—
313C2×C4—181no—
314C2×C4—181no—
315C2×C4—181no—
316C2×C4—541no—
317C2×C4C2×D3×C3⋊D311yescharacteristic
318C8—32no—
319C8—541no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3—24no—
323C2×C3—361no—
324C2×C3#6394314yes—
325C2×C3#6394814yes—
326C2×C3#6397014yes—
327C2×C3#6412711yescharacteristic
328C2×C3#6412711yescharacteristic
329C2×C3#6414711yescharacteristic
330D3—124no—
331C22—181no—
332C22—181no—
333C22—181no—
334C22C3⋊D3×#6711yescharacteristic
335C4—41no—
336C4—121no—
337C4—181no—
338C4—361no—
339C4—1081no—
340C4#8902911yescharacteristic
341C4D3×#63811yescharacteristic
342C3—24no—
343C3#1059163814yescomplemented
344C3#1059288811yescharacteristic, complemented
345C2—361no—
346C2#1172357611yescharacteristic
347C2#1172358611yescharacteristic
348C2#1172363111yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#470
LCSG ▷ C2×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. C2×C32×#216C2semidirect
5. #11722205C2semidirect
6. #11722052C2semidirect
7. #11722037C2semidirect
28. C2×C3×#216D3semidirect
99. C2×#216C3⋊D3semidirect
214. C33#470semidirect
303. C32#5052semidirect
304. C32#5189semidirect
343. C3#10591638semidirect
344. C3#10592888semidirect

Maximal subgroups

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

subgroupindex
C3×#641552
C2×C32×#2162
#117234352
C2×#887872
#117222052
#117220522
#117220372
#105928883
#105916383