Finite Group Databasebeta

← #13139280

Subgroups — #13139280

Order 1728.

347 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 347 conjugacy classes of subgroups — too many to draw. Only its 49 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
1Q16×#272C211yesmaximal, characteristic, complemented
2#11720844C211yesmaximal, characteristic
3#63656⋊C3C211yesmaximal, characteristic, complemented
4#63655⋊C3C211yesmaximal, characteristic, complemented
5#11720256C211yesmaximal, characteristic, complemented
6#11720255C211yesmaximal, characteristic, complemented
7#11720252C211yesmaximal, characteristic, complemented
8#10591047—31nomaximal
9#10591045—31nomaximal
10Q8×#272—21no—
11#88649—21no—
12Q16×H(3,𝔽3)C2211yescharacteristic
13#88550—21no—
14#88530C2211yescharacteristic
15C8×#272C2211yescharacteristic
16#3745⋊C3C2211yescharacteristic
17#3746⋊C3C2211yescharacteristic
18#88431—21no—
19(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic
20(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic
21C3×D3×Q16—31no—
22C3×D3×Q16—31no—
23C3×D3×Q16—61no—
24#63808—31no—
25#63807—31no—
26#63656D311yescharacteristic, complemented
27#63655D311yescharacteristic, complemented
28C3×#829—31no—
29C3×#828—31no—
30#63551—31no—
31#63550—31no—
32#63549—31no—
33#63547—31no—
34#63543—31no—
35#63543—31no—
36Q8×H(3,𝔽3)—21no—
37C4×#272—41no—
38C4×#272D411yescharacteristic
39#6229—21no—
40#6229—41no—
41C4⋊((C3⋊D3)⋊C3)—21no—
42#634⋊C3—21no—
43#6200—41no—
44((C3⋊D3)⋊C3)⋊C4—41no—
45C8×H(3,𝔽3)C2311yescharacteristic
46#6180D411yescharacteristic
47#5179—91no—
48#4918—91no—
49#4915—91no—
50C3×D3×Q8—61no—
51C3×D3×Q8—61no—
52C3×D3×Q8—62no—
53#3799—61no—
54#3795—61no—
55C32×Q16—21no—
56C32×Q16C2×D311yescharacteristic
57C32×Q16C2×D311yescharacteristic
58#3746C2×D311yescharacteristic
59#3745C2×D311yescharacteristic
60C3×#216—61no—
61C3×#216—61no—
62C3×#216—62no—
63C3×Dic12—31no—
64C3×Dic12—31no—
65C3×Dic12—31no—
66C3×Dic12—61no—
67C3×D24—31no—
68C3×D3×C8—31no—
69C3×D3×C8—31no—
70C3×D3×C8—61no—
71#3717—61no—
72#3716—61no—
73C3⋊(C3⋊C16)C2×D311yescharacteristic
74C3⋊(C3⋊C16)C2×D311yescharacteristic
75C3×C3⋊C16—31no—
76C3×C3⋊C16—31no—
77C2×#272D811yescharacteristic
78C2×(C3⋊D3)⋊C3—41no—
79C4×H(3,𝔽3)—41no—
80C4×H(3,𝔽3)C2×D411yescharacteristic
81#1084—41no—
82#1084D811yescharacteristic
83(C3⋊Dic3)⋊C3—41no—
84C2×C3×Q16—91no—
85#919—91no—
86D3×Q16—31no—
87D3×Q16—31no—
88D3×Q16—61no—
89#912—91no—
90#904—91no—
91C3×Q32—91no—
92C3×SD32—91no—
93#829—31no—
94#829—91no—
95#829—91no—
96#829—91no—
97#828—31no—
98#828—91no—
99#828—91no—
100#827—91no—
101#812—91no—
102Dic24—91no—
103#800—91no—
104#798—91no—
105#798—91no—
106C32×Q8—22no—
107C32×Q8—21no—
108C32×Q8—21no—
109C4⋊(C3⋊D3)—21no—
110#634—21no—
111C3×D12—61no—
112C3×C4×D3—31no—
113C3×C4×D3—31no—
114C3×C4×D3—61no—
115C3×C4×D3—121no—
116C3×C4×D3—121no—
117C3×C4×D3—122no—
118C3×Dic6—62no—
119C3×Dic6—61no—
120C3×Dic6—61no—
121C3×Dic6—61no—
122C3×Dic6—121no—
123C3×Dic6—121no—
124C3×Dic6—122no—
125#626—121no—
126#625—121no—
127#624—121no—
128D3×Dic3—121no—
129C32×C8—21no—
130C32×C8C22×D311yescharacteristic
131C32×C8C22×D311yescharacteristic
132C3×C3⋊C8—31no—
133C3×C3⋊C8—31no—
134C3×C3⋊C8—61no—
135#510—271noSylow
136C2×H(3,𝔽3)C2×D811yescharacteristic
137#272—21no—
138(C3⋊D3)⋊C3—81no—
139C2×C3×Q8—181no—
140#239—181no—
141D3×Q8—61no—
142D3×Q8—61no—
143D3×Q8—62no—
144#237—181no—
145#235—181no—
146C3×Q16—31no—
147C3×Q16—31no—
148C3×Q16—61no—
149C3×Q16—91no—
150C3×Q16—91no—
151C3×Q16—181no—
152C3×Q16D3211yescharacteristic
153C3×D8—91no—
154C2×C3×C8—91no—
155#216—61no—
156#216—61no—
157#216—62no—
158#215—181no—
159#214—181no—
160Dic12—31no—
161Dic12—31no—
162Dic12—31no—
163Dic12—61no—
164Dic12—91no—
165D24—31no—
166D24—91no—
167#204—181no—
168D3×C8—31no—
169D3×C8—31no—
170D3×C8—61no—
171C3×C16—91no—
172C3×C16—91no—
173C3⋊C16—31no—
174C3⋊C16—31no—
175C3⋊C16—91no—
176C3⋊C16—91no—
177C2×C3⋊D3—41no—
178C2×C3×D3—31no—
179C2×C3×D3—31no—
180C2×C3×D3—61no—
181C2×C3×D3—121no—
182C32×C4—21no—
183C32×C4—42no—
184C32×C4—41no—
185C32×C4—41no—
186C32×C4D3×D411yescharacteristic
187C32×C4D3×D411yescharacteristic
188C3⋊Dic3—41no—
189C3×Dic3—31no—
190C3×Dic3—31no—
191C3×Dic3—61no—
192C3×Dic3—122no—
193C3×Dic3—121no—
194C3×Dic3—121no—
195C3×Dic3—121no—
196#135—271no—
197C2×Q16—91no—
198Q32—91no—
199Q32—271no—
200SD32—91no—
201SD32—271no—
202M32—271no—
203H(3,𝔽3)#51011yescharacteristic, complemented, Sylow
204C3×Q8—21no—
205C3×Q8—61no—
206C3×Q8—61no—
207C3×Q8—62no—
208C3×Q8—181no—
209C3×Q8—181no—
210C3×Q8—361no—
211C3×D4—181no—
212C2×C3×C4—91no—
213C2×C3×C4—361no—
214#67—361no—
215#67—361no—
216C2×Dic3—361no—
217D12—61no—
218D12—181no—
219D12—361no—
220C4×D3—31no—
221C4×D3—31no—
222C4×D3—61no—
223C4×D3—121no—
224C4×D3—121no—
225C4×D3—122no—
226C4×D3—361no—
227C4×D3—361no—
228Dic6—61no—
229Dic6—61no—
230Dic6—62no—
231Dic6—61no—
232Dic6—121no—
233Dic6—121no—
234Dic6—122no—
235Dic6—181no—
236C3×C8—31no—
237C3×C8—31no—
238C3×C8—61no—
239C3×C8—91no—
240C3×C8C2×D3211yescharacteristic
241C3⋊C8—31no—
242C3⋊C8—31no—
243C3⋊C8—61no—
244C2×C32—21no—
245C2×C32D3×D811yescharacteristic
246C2×C32D3×D811yescharacteristic
247C3⋊D3—81no—
248C3×D3—62no—
249C3×D3—61no—
250C3×D3—61no—
251C3×D3—241no—
252#41—541no—
253C2×Q8—181no—
254Q16—91no—
255Q16—91no—
256Q16—181no—
257Q16((C3⋊D3)⋊C3)⋊C211yescharacteristic
258SD16—541no—
259D8—91no—
260C2×C8—91no—
261C16—91no—
262C16—91no—
263C22×C3—91no—
264C22×C3—361no—
265C2×D3—31no—
266C2×D3—31no—
267C2×D3—61no—
268C2×D3—121no—
269C2×D3—361no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—41no—
273C3×C4—61no—
274C3×C4—91no—
275C3×C4—121no—
276C3×C4—121no—
277C3×C4—122no—
278C3×C4—361no—
279C3×C4—361no—
280C3×C4#381111yescharacteristic
281Dic3—31no—
282Dic3—31no—
283Dic3—61no—
284Dic3—121no—
285Dic3—121no—
286Dic3—122no—
287Dic3—121no—
288Dic3—361no—
289C32—21no—
290C32#491511yescharacteristic, complemented
291C32#491811yescharacteristic, complemented
292Q8—21no—
293Q8—181no—
294Q8—181no—
295Q8—361no—
296D4—181no—
297D4—1081no—
298C2×C4—91no—
299C2×C4—361no—
300C2×C4—1081no—
301C8—91no—
302C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—61no—
306C2×C3—181no—
307C2×C3—721no—
308C2×C3#6392611yescharacteristic
309D3—61no—
310D3—61no—
311D3—62no—
312D3—241no—
313D3—721no—
314C22—91no—
315C22—361no—
316C4—41no—
317C4—91no—
318C4—361no—
319C4—361no—
320C4#8867711yescharacteristic
321C3—31no—
322C3—31no—
323C3—61no—
324C3#1059105011yescharacteristic
325C2—181no—
326C2—721no—
327C2#1172119011yescharacteristic
subgroupG/H
centerC2#11721190
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingQ16×H(3,𝔽3)C22
socleC2×C3#63926
prime radical#11720844C2
hypercenterQ16((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#510
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16

Complements

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

NH ≅ G/Ntype
1. Q16×#272C2semidirect
3. #63656⋊C3C2semidirect
4. #63655⋊C3C2semidirect
5. #11720256C2semidirect
6. #11720255C2semidirect
7. #11720252C2semidirect
26. #63656D3semidirect
27. #63655D3semidirect
203. H(3,𝔽3)#510semidirect
290. C32#4915semidirect
291. C32#4918semidirect

Maximal subgroups

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

subgroupindex
Q16×#2722
#117208442
#63656⋊C32
#63655⋊C32
#117202562
#117202552
#117202522
#105910473
#105910453