Finite Group Databasebeta

← #13155735

Subgroups — #13155735

Order 1728.

586 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 102 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#11723660C211yesmaximal, characteristic
2C3×#63817C212yesmaximal, complemented
3#11722045C212yesmaximal, complemented
4#11722018C211yesmaximal, characteristic, complemented
5C3×#63578C211yesmaximal, characteristic, complemented
6#10590995—32nomaximal
7#10590921—31nomaximal
8C3×#3828C2212yes—
9C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
11C32×#208C2212yes—
12C2×C32×C3⋊C8C2211yescharacteristic
13#88812C411yescharacteristic, complemented
14#88811C412yescomplemented
15#88810C411yescharacteristic, complemented
16C3×#3712—22no—
17C3×#3710—22no—
18C3×#907—32no—
19C3×#901—32no—
20#63829—32no—
21#63822—31no—
22#63817D312yescomplemented
23#63652—32no—
24C8×C3⋊Dic3—31no—
25C3×#822—32no—
26#63578D311yescharacteristic, complemented
27#63568—32no—
28#63568—32no—
29#63568—62no—
30#63553—32no—
31#63553—61no—
32C2×C33×C4C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34C3×#638—22no—
35C3×C4⋊(C3⋊D3)C2×C412yes—
36C3×C4×C3⋊D3—22no—
37C3×#634C2×C412yes—
38C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
39C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
40C32×C3⋊C8—22no—
41C32×C3⋊C8D411yescharacteristic
42C32×C3⋊C8D411yescharacteristic
43#5144—92no—
44#4895—91no—
45#3828C2×D312yes—
46C3×#235—32no—
47C3×#235—32no—
48C3×#235—62no—
49C32×M16—32no—
50C2×C32×C8—31no—
51C4×C3⋊Dic3—32no—
52C4×C3⋊Dic3—31no—
53C4×C3⋊Dic3—61no—
54C4×C3⋊Dic3—62no—
55C4×C3⋊Dic3—121no—
56C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
57C3×#208—22no—
58C3×#208C2×D312yes—
59C3×#208C2×D312yes—
60C2×C3×C3⋊C8—21no—
61C2×C3×C3⋊C8—32no—
62C2×C3×C3⋊C8—61no—
63C2×C3×C3⋊C8C2×D312yes—
64C3×ΓL(1,𝔽25)—62no—
65C3×ΓL(1,𝔽25)—62no—
66C3×D3×C8—62no—
67C3×D3×C8—62no—
68#3719—32no—
69#3718—31no—
70#3717—32no—
71#3716—32no—
72#3715—32no—
73#3714—32no—
74#3713—31no—
75#3712—22no—
76#3710—22no—
77C22×C33#4111yescharacteristic
78C2×C3×C3⋊D3—22no—
79C33×C4C22×C411yescharacteristic
80C33×C4C2×D411yescharacteristic
81C3⋊(C3⋊Dic3)—22no—
82C3×C3⋊Dic3—22no—
83C3×#131—92no—
84#950—92no—
85#907—32no—
86#904—91no—
87#901—32no—
88C3×#108—91no—
89#837—92no—
90#825—92no—
91#825—92no—
92#825—182no—
93#822—32no—
94C8×Dic3—92no—
95C8×Dic3—91no—
96C8×Dic3—181no—
97C2×C32×C4—21no—
98C2×C32×C4—22no—
99C2×C32×C4—41no—
100C2×C32×C4C22×D311yescharacteristic
101C2×C32×C4C22×D312yes—
102#638—22no—
103C2×C3⋊Dic3—61no—
104C2×C3⋊Dic3—62no—
105C2×C3⋊Dic3—62no—
106C2×C3⋊Dic3—122no—
107C2×C3⋊Dic3—122no—
108C4⋊(C3⋊D3)C4×D312yes—
109C4×C3⋊D3—22no—
110#634C4×D312yes—
111C3×#67—62no—
112C3×#67—62no—
113C3×#67—122no—
114C3×D12—32no—
115C3×D12—32no—
116C3×D12—62no—
117C3×C4×D3—62no—
118C3×C4×D3—62no—
119C3×C4×D3—122no—
120C3×Dic6—32no—
121C3×Dic6—32no—
122C3×Dic6—62no—
123C32×C8—31no—
124C32×C8—31no—
125C32×C8—62no—
126C3⋊(C3⋊C8)C4×D311yescharacteristic
127C3⋊(C3⋊C8)C4×D311yescharacteristic
128C3×C3⋊C8—22no—
129C3×C3⋊C8—22no—
130C3×C3⋊C8—21no—
131C3×C3⋊C8—21no—
132C3×C3⋊C8—32no—
133C3×C3⋊C8—32no—
134C3×C3⋊C8—42no—
135C3×C3⋊C8—61no—
136C3×C3⋊C8—61no—
137C3×C3⋊C8#6712yes—
138C3×C3⋊C8#6712yes—
139#443—271noSylow
140C2×C33—21no—
141C2×C33C4×D411yescharacteristic
142C3×C3⋊D3—42no—
143C3×#41—92no—
144#235—32no—
145#235—32no—
146#235—62no—
147C3×M16—32no—
148C3×M16—32no—
149C3×M16—62no—
150C3×M16—182no—
151C2×C3×C8—32no—
152C2×C3×C8—61no—
153C2×C3×C8—91no—
154C2×C3×C8—182no—
155#216—92no—
156#215—92no—
157#214—92no—
158#213—92no—
159C4×Dic3—91no—
160C4×Dic3—92no—
161C4×Dic3—181no—
162C4×Dic3—182no—
163C4×Dic3—361no—
164#208D3212yes—
165C2×C3⋊C8—32no—
166C2×C3⋊C8—61no—
167C2×C3⋊C8D3211yescharacteristic
168Dic12—91no—
169D24—91no—
170#204—92no—
171ΓL(1,𝔽25)—62no—
172ΓL(1,𝔽25)—62no—
173D3×C8—62no—
174D3×C8—62no—
175C22×C32—21no—
176C22×C32—22no—
177C22×C32—41no—
178C22×C32#23512yes—
179C22×C32#23911yescharacteristic
180C2×C3⋊D3—22no—
181C2×C3×D3—62no—
182C2×C3×D3—62no—
183C2×C3×D3—122no—
184C32×C4—21no—
185C32×C4—22no—
186C32×C4—21no—
187C32×C4—22no—
188C32×C4—41no—
189C32×C4—41no—
190C32×C4C2×C4×D311yescharacteristic
191C32×C4C2×C4×D312yes—
192C32×C4D3×D411yescharacteristic
193C32×C4C2×#6712yes—
194C3⋊Dic3—22no—
195C3⋊Dic3—62no—
196C3⋊Dic3—64no—
197C3⋊Dic3—64no—
198C3⋊Dic3—124no—
199C3⋊Dic3—124no—
200C3×Dic3—62no—
201C3×Dic3—62no—
202C3×Dic3—122no—
203#135—271no—
204#131—92no—
205#108—91no—
206#104—272no—
207C4×C8—271no—
208C33#44311yescharacteristic, complemented, Sylow
209C3×Q8—92no—
210C3×D4—92no—
211C3×D4—182no—
212C2×C3×C4—21no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—182no—
216C2×C3×C4C2×D3211yescharacteristic
217C2×C3×C4C2×D3212yes—
218#67—62no—
219#67—62no—
220#67—122no—
221C2×Dic3—182no—
222C2×Dic3—181no—
223C2×Dic3—182no—
224C2×Dic3—362no—
225C2×Dic3—362no—
226D12—32no—
227D12—32no—
228D12—62no—
229C4×D3—62no—
230C4×D3—62no—
231C4×D3—122no—
232Dic6—32no—
233Dic6—32no—
234Dic6—62no—
235C3×C8—32no—
236C3×C8—32no—
237C3×C8—61no—
238C3×C8—61no—
239C3×C8—62no—
240C3×C8—62no—
241C3×C8—91no—
242C3×C8—91no—
243C3×C8—122no—
244C3⋊C8—22no—
245C3⋊C8—32no—
246C3⋊C8—32no—
247C3⋊C8—61no—
248C3⋊C8—61no—
249C3⋊C8#62511yescharacteristic
250C3⋊C8#62511yescharacteristic
251C2×C32—21no—
252C2×C32—22no—
253C2×C32—22no—
254C2×C32—21no—
255C2×C32—22no—
256C2×C32—41no—
257C2×C32—42no—
258C2×C32—42no—
259C2×C32#89311yescharacteristic
260C2×C32C4×#6712yes—
261C3⋊D3—42no—
262C3×D3—122no—
263C3×D3—122no—
264C3×D3—242no—
265#41—92no—
266Q16—271no—
267SD16—272no—
268D8—271no—
269M16—32no—
270M16—182no—
271C2×C8—31no—
272C2×C8—91no—
273C2×C8—182no—
274C42—271no—
275C22×C3—21no—
276C22×C3—22no—
277C22×C3—41no—
278C22×C3—182no—
279C22×C3#379811yescharacteristic
280C22×C3#379912yes—
281C2×D3—62no—
282C2×D3—62no—
283C2×D3—122no—
284C3×C4—21no—
285C3×C4—22no—
286C3×C4—21no—
287C3×C4—22no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—182no—
291C3×C4C4×D3211yescharacteristic
292C3×C4C4×D3212yes—
293C3×C4C2×#62511yescharacteristic
294C3×C4D3×#6712yes—
295Dic3—62no—
296Dic3—62no—
297Dic3—122no—
298Dic3—182no—
299Dic3—184no—
300Dic3—184no—
301Dic3—364no—
302Dic3—364no—
303C32—22no—
304C32—21no—
305C32—41no—
306C32#489511yescharacteristic, complemented
307C32#514412yescomplemented
308Q8—92no—
309D4—92no—
310D4—182no—
311C2×C4—182no—
312C2×C4—541no—
313C2×C4D3311yescharacteristic
314C8—31no—
315C8—31no—
316C8—62no—
317C8—91no—
318C8—91no—
319C2×C3—22no—
320C2×C3—22no—
321C2×C3—21no—
322C2×C3—22no—
323C2×C3—21no—
324C2×C3—42no—
325C2×C3—42no—
326C2×C3—41no—
327C2×C3—362no—
328C2×C3C4×#62511yescharacteristic
329C2×C3#6390412yes—
330D3—122no—
331D3—122no—
332D3—242no—
333C22—182no—
334C22#8896511yescharacteristic
335C4—182no—
336C4—542no—
337C4#8894911yescharacteristic
338C4D3×#62511yescharacteristic
339C3—22no—
340C3—21no—
341C3—41no—
342C3#1059092111yescharacteristic, complemented
343C3#1059099512yescomplemented
344C2—21no—
345C2—362no—
346C2#1172293111yescharacteristic
subgroupG/H
centerC4D3×#625
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C4×D4
prime radical#11723660C2
hypocenterC33#443
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63817C2semidirect
3. #11722045C2semidirect
4. #11722018C2semidirect
5. C3×#63578C2semidirect
13. #88812C4semidirect
14. #88811C4semidirect
15. #88810C4semidirect
22. #63817D3semidirect
26. #63578D3semidirect
208. C33#443semidirect
306. C32#4895semidirect
307. C32#5144semidirect
342. C3#10590921semidirect
343. C3#10590995semidirect

Maximal subgroups

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

subgroupindex
#117236602
C3×#638172
#117220452
#117220182
C3×#635782
#105909953
#105909213