Finite Group Databasebeta

← #13139279

Subgroups — #13139279

Order 1728.

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

normal subgroups only

This group has 356 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
1#11721808C211yesmaximal, characteristic, complemented
2#11720844C211yesmaximal, characteristic, complemented
3#63656⋊C3C211yesmaximal, characteristic, complemented
4#63655⋊C3C211yesmaximal, characteristic, complemented
5#11720254C211yesmaximal, characteristic, complemented
6#11720253C211yesmaximal, characteristic, complemented
7#11720251C211yesmaximal, characteristic, complemented
8#10591048—31nomaximal
9#10591046—31nomaximal
10#88748—21no—
11#88649—21no—
12Q16×H(3,𝔽3)C2211yescharacteristic
13#88549—21no—
14#88529C2211yescharacteristic
15C8×#272C2211yescharacteristic
16#3745⋊C3C2211yescharacteristic
17#3746⋊C3C2211yescharacteristic
18#88431—21no—
19(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic
20(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic
21C3×#919—31no—
22C3×#919—31no—
23C3×#919—61no—
24#63808—31no—
25#63807—31no—
26#63656D311yescharacteristic, complemented
27#63655D311yescharacteristic, complemented
28C3×#829—31no—
29C3×#828—31no—
30#63548—31no—
31#63547—31no—
32#63546—31no—
33#63545—31no—
34D3×C3⋊C16—31no—
35D3×C3⋊C16—31no—
36Q8×H(3,𝔽3)—21no—
37C4⋊#272—21no—
38C4⋊#272—41no—
39C4×#272—41no—
40C4×#272D411yescharacteristic
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#5194—91no—
48#4919—91no—
49#4916—91no—
50C3×#239—61no—
51C3×#239—61no—
52C3×#239—62no—
53#3799—61no—
54#3795—61no—
55C32×Q16—21no—
56C32×Q16C2×D311yescharacteristic
57C32×Q16C2×D311yescharacteristic
58#3746C2×D311yescharacteristic
59#3745C2×D311yescharacteristic
60C3×#215—61no—
61C3×#215—61no—
62C3×#215—62no—
63C3×Dic12—31no—
64C3×D24—31no—
65C3×D24—31no—
66C3×D24—31no—
67C3×D24—61no—
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×#272—41no—
78C2×#272D811yescharacteristic
79C2×(C3⋊D3)⋊C3—41no—
80C4×H(3,𝔽3)—41no—
81C4×H(3,𝔽3)C2×D411yescharacteristic
82#1084D811yescharacteristic
83(C3⋊Dic3)⋊C3—41no—
84C3×#135—91no—
85#919—31no—
86#919—31no—
87#919—61no—
88#919—91no—
89#912—91no—
90#904—91no—
91C3×Q32—91no—
92C3×SD32—91no—
93#829—31no—
94#829—91no—
95#828—31no—
96#828—91no—
97#828—91no—
98#827—91no—
99#826—91no—
100#826—91no—
101C2×C3⋊C16—91no—
102#800—91no—
103D48—91no—
104D3×C16—91no—
105D3×C16—91no—
106C32×Q8—22no—
107C32×Q8—21no—
108C32×Q8—21no—
109C4⋊(C3⋊D3)—21no—
110#634—21no—
111C3×D12—62no—
112C3×D12—61no—
113C3×D12—61no—
114C3×D12—61no—
115C3×D12—121no—
116C3×D12—121no—
117C3×D12—122no—
118C3×C4×D3—31no—
119C3×C4×D3—31no—
120C3×C4×D3—61no—
121C3×C4×D3—121no—
122C3×C4×D3—121no—
123C3×C4×D3—122no—
124C3×Dic6—61no—
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#508—271noSylow
136C2×H(3,𝔽3)C2×D811yescharacteristic
137#272—21no—
138#272—81no—
139(C3⋊D3)⋊C3—81no—
140C3×#41—181no—
141#239—61no—
142#239—61no—
143#239—62no—
144#239—181no—
145#237—181no—
146#235—181no—
147C3×Q16—31no—
148C3×Q16—31no—
149C3×Q16—61no—
150C3×Q16—91no—
151C3×Q16D3211yescharacteristic
152C3×SD16—181no—
153C3×D8—91no—
154C3×D8—91no—
155C2×C3×C8—91no—
156#215—61no—
157#215—61no—
158#215—62no—
159#215—181no—
160#214—181no—
161Dic12—31no—
162Dic12—91no—
163D24—31no—
164D24—31no—
165D24—31no—
166D24—61no—
167D24—91no—
168#204—181no—
169D3×C8—31no—
170D3×C8—31no—
171D3×C8—61no—
172C3×C16—91no—
173C3×C16—91no—
174C3⋊C16—31no—
175C3⋊C16—31no—
176C3⋊C16—91no—
177C3⋊C16—91no—
178C2×C3⋊D3—41no—
179C2×C3×D3—31no—
180C2×C3×D3—31no—
181C2×C3×D3—61no—
182C2×C3×D3—122no—
183C2×C3×D3—121no—
184C2×C3×D3—121no—
185C2×C3×D3—121no—
186C32×C4—21no—
187C32×C4—42no—
188C32×C4—41no—
189C32×C4—41no—
190C32×C4D3×D411yescharacteristic
191C32×C4D3×D411yescharacteristic
192C3⋊Dic3—41no—
193C3×Dic3—31no—
194C3×Dic3—31no—
195C3×Dic3—61no—
196C3×Dic3—121no—
197#135—91no—
198#135—271no—
199Q32—91no—
200SD32—91no—
201SD32—271no—
202D16—271no—
203C2×C16—271no—
204H(3,𝔽3)#50811yescharacteristic, complemented, Sylow
205C3×Q8—21no—
206C3×Q8—61no—
207C3×Q8—61no—
208C3×Q8—62no—
209C3×Q8—181no—
210C3×D4—181no—
211C3×D4—181no—
212C3×D4—361no—
213C2×C3×C4—91no—
214C2×C3×C4—361no—
215#67—361no—
216#67—361no—
217C2×Dic3—361no—
218D12—61no—
219D12—61no—
220D12—62no—
221D12—61no—
222D12—121no—
223D12—121no—
224D12—122no—
225D12—181no—
226D12—361no—
227C4×D3—31no—
228C4×D3—31no—
229C4×D3—61no—
230C4×D3—121no—
231C4×D3—121no—
232C4×D3—122no—
233C4×D3—361no—
234C4×D3—361no—
235Dic6—61no—
236Dic6—181no—
237C3×C8—31no—
238C3×C8—31no—
239C3×C8—61no—
240C3×C8—91no—
241C3×C8C2×D3211yescharacteristic
242C3⋊C8—31no—
243C3⋊C8—31no—
244C3⋊C8—61no—
245C2×C32—21no—
246C2×C32D3×D811yescharacteristic
247C2×C32D3×D811yescharacteristic
248C3⋊D3—81no—
249C3×D3—61no—
250C3×D3—61no—
251C3×D3—121no—
252C3×D3—242no—
253C3×D3—241no—
254C3×D3—241no—
255C3×D3—241no—
256#41—181no—
257#41—541no—
258Q16—91no—
259Q16((C3⋊D3)⋊C3)⋊C211yescharacteristic
260SD16—181no—
261SD16—541no—
262D8—91no—
263D8—91no—
264C2×C8—91no—
265C16—91no—
266C16—91no—
267C22×C3—91no—
268C22×C3—361no—
269C22×C3—361no—
270C2×D3—31no—
271C2×D3—31no—
272C2×D3—61no—
273C2×D3—121no—
274C2×D3—121no—
275C2×D3—122no—
276C2×D3—121no—
277C2×D3—361no—
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—41no—
281C3×C4—61no—
282C3×C4—91no—
283C3×C4—121no—
284C3×C4—121no—
285C3×C4—122no—
286C3×C4—361no—
287C3×C4#381111yescharacteristic
288Dic3—31no—
289Dic3—31no—
290Dic3—61no—
291Dic3—121no—
292Dic3—361no—
293C32—21no—
294C32#491611yescharacteristic, complemented
295C32#491911yescharacteristic, complemented
296Q8—21no—
297Q8—181no—
298D4—181no—
299D4—181no—
300D4—361no—
301D4—1081no—
302C2×C4—91no—
303C2×C4—361no—
304C2×C4—1081no—
305C8—91no—
306C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
307C2×C3—31no—
308C2×C3—31no—
309C2×C3—61no—
310C2×C3—181no—
311C2×C3—721no—
312C2×C3—721no—
313C2×C3#6392611yescharacteristic
314D3—61no—
315D3—61no—
316D3—121no—
317D3—241no—
318D3—241no—
319D3—242no—
320D3—241no—
321D3—721no—
322C22—91no—
323C22—361no—
324C22—361no—
325C4—41no—
326C4—91no—
327C4—361no—
328C4#8867711yescharacteristic
329C3—31no—
330C3—31no—
331C3—61no—
332C3#1059104911yescharacteristic
333C2—181no—
334C2—721no—
335C2—721no—
336C2#1172119011yescharacteristic
subgroupG/H
centerC2#11721190
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingQ16×H(3,𝔽3)C22
socleC2×C3#63926
hypercenterQ16((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#508
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. #11721808C2semidirect
2. #11720844C2semidirect
3. #63656⋊C3C2semidirect
4. #63655⋊C3C2semidirect
5. #11720254C2semidirect
6. #11720253C2semidirect
7. #11720251C2semidirect
26. #63656D3semidirect
27. #63655D3semidirect
204. H(3,𝔽3)#508semidirect
294. C32#4916semidirect
295. C32#4919semidirect

Maximal subgroups

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

subgroupindex
#117218082
#117208442
#63656⋊C32
#63655⋊C32
#117202542
#117202532
#117202512
#105910483
#105910463