Finite Group Databasebeta

← #13155729

Subgroups — #13155729

Order 1728.

614 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 108 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#11723350C212yesmaximal, complemented
2C3×#63822C211yesmaximal, characteristic
3#11722081C211yesmaximal, characteristic, complemented
4Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C3×#63569C212yesmaximal, complemented
6#10591675—32nomaximal
7#10590389—31nomaximal
8C32×#235C2212yes—
9C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
10C3×#3748C2212yes—
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×C4×Dic3C2211yescharacteristic
13#88786—22no—
14D3×C3⋊(C3⋊C8)—22no—
15C3×#3718C411yescharacteristic, complemented
16C3×#3715C412yescomplemented
17C3×#3713C411yescharacteristic, complemented
18#64157—32no—
19C3×#950—32no—
20#63822D311yescharacteristic
21#63815—32no—
22#63815—32no—
23#63815—62no—
24C4×C3⋊(C3⋊C8)—31no—
25C3×#837—32no—
26C3×#805—32no—
27#63578—31no—
28#63577—32no—
29#63569D312yescomplemented
30Dic3×C3⋊C8—32no—
31Dic3×C3⋊C8—61no—
32C2×C33×C4C2311yescharacteristic
33C32×#67—22no—
34C2×C32×Dic3—21no—
35C32×D12C2×C412yes—
36C32×C4×D3—22no—
37C32×Dic6C2×C412yes—
38C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
39C3⋊(C3⋊(C3⋊C8))D411yescharacteristic
40C3×C3⋊(C3⋊C8)—22no—
41C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
42C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
43#5196—91no—
44#4824—92no—
45C32×#41—32no—
46C3×#235—22no—
47C3×#235C2×D312yes—
48C3×#235C2×D312yes—
49C32×C42—31no—
50#3748—62no—
51#3748C2×D312yes—
52C2×C3⋊(C3⋊C8)—32no—
53C2×C3⋊(C3⋊C8)—31no—
54C2×C3⋊(C3⋊C8)—62no—
55C2×C3⋊(C3⋊C8)—61no—
56C2×C3⋊(C3⋊C8)—62no—
57C2×C3⋊(C3⋊C8)—121no—
58C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
59C3×#216—32no—
60C3×#215—32no—
61C3×#214—32no—
62C3×#213—32no—
63C3×C4×Dic3—21no—
64C3×C4×Dic3C2×D312yes—
65C3×#208—32no—
66C3×#208—32no—
67C3×#208—62no—
68C2×C3×C3⋊C8—32no—
69C2×C3×C3⋊C8—61no—
70#3718Dic311yescharacteristic, complemented
71#3715Dic312yescomplemented
72#3713Dic311yescharacteristic, complemented
73#3711—62no—
74#3711—62no—
75#3711—64no—
76D3×C3⋊C8—62no—
77D3×C3⋊C8—62no—
78D3×C3⋊C8—64no—
79C22×C33#4111yescharacteristic
80C2×C32×D3—22no—
81C33×C4C22×C411yescharacteristic
82C33×C4C2×D411yescharacteristic
83C32×Dic3—22no—
84C32×Dic3—22no—
85C3×#135—91no—
86#950—32no—
87#948—92no—
88#948—92no—
89#948—182no—
90#907—92no—
91C3×#104—92no—
92#837—32no—
93#822—92no—
94#819—91no—
95C8×Dic3—91no—
96#805—32no—
97C4×C3⋊C8—92no—
98C4×C3⋊C8—181no—
99C32×Q8—32no—
100C32×D4—32no—
101C32×D4—62no—
102C2×C32×C4—21no—
103C2×C32×C4—22no—
104C2×C32×C4—41no—
105C2×C32×C4—61no—
106C2×C32×C4—62no—
107C2×C32×C4C22×D312yes—
108C2×C32×C4C22×D311yescharacteristic
109C3×#67—24no—
110C3×#67—22no—
111C3×#67—22no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—22no—
114C3×D12—22no—
115C3×D12C4×D312yes—
116C3×D12C2×Dic312yes—
117C3×C4×D3—24no—
118C3×C4×D3—22no—
119C3×C4×D3—22no—
120C3×Dic6—22no—
121C3×Dic6C4×D312yes—
122C3×Dic6C2×Dic312yes—
123C3⋊(C3⋊C8)—22no—
124C3⋊(C3⋊C8)—31no—
125C3⋊(C3⋊C8)—32no—
126C3⋊(C3⋊C8)—31no—
127C3⋊(C3⋊C8)—32no—
128C3⋊(C3⋊C8)—62no—
129C3⋊(C3⋊C8)—62no—
130C3⋊(C3⋊C8)—61no—
131C3⋊(C3⋊C8)—61no—
132C3⋊(C3⋊C8)—121no—
133C3⋊(C3⋊C8)—121no—
134C3⋊(C3⋊C8)C2×Dic311yescharacteristic
135C3⋊(C3⋊C8)C2×Dic311yescharacteristic
136C3×C3⋊C8—32no—
137C3×C3⋊C8—32no—
138C3×C3⋊C8—61no—
139C3×C3⋊C8—61no—
140C3×C3⋊C8—64no—
141C3×C3⋊C8—62no—
142C3×C3⋊C8—62no—
143#443—271noSylow
144C2×C33—21no—
145C2×C33C4×D411yescharacteristic
146C32×D3—42no—
147C3×#41—32no—
148C3×#41—32no—
149C3×#41—62no—
150#235D3212yes—
151C3×Q16—91no—
152C3×SD16—92no—
153C3×D8—91no—
154C3×M16—92no—
155C2×C3×C8—91no—
156C3×C42—32no—
157C3×C42—61no—
158#216—32no—
159#215—32no—
160#214—32no—
161#213—32no—
162C4×Dic3D3211yescharacteristic
163#208—32no—
164#208—32no—
165#208—62no—
166#208—182no—
167#208—182no—
168#208—184no—
169C2×C3⋊C8—32no—
170C2×C3⋊C8—61no—
171C2×C3⋊C8—92no—
172C2×C3⋊C8—91no—
173C2×C3⋊C8—181no—
174C2×C3⋊C8—182no—
175C2×C3⋊C8—182no—
176C2×C3⋊C8—182no—
177C2×C3⋊C8—184no—
178C2×C3⋊C8—361no—
179ΓL(1,𝔽25)—182no—
180D3×C8—182no—
181C22×C32—21no—
182C22×C32—22no—
183C22×C32—41no—
184C22×C32—62no—
185C22×C32#23712yes—
186C22×C32#23711yescharacteristic
187C2×C3×D3—24no—
188C2×C3×D3—22no—
189C2×C3×D3—22no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—22no—
193C32×C4—22no—
194C32×C4—41no—
195C32×C4—41no—
196C32×C4—62no—
197C32×C4—62no—
198C32×C4C2×C4×D312yes—
199C32×C4D3×D412yes—
200C32×C4D3×D411yescharacteristic
201C32×C4C22×Dic311yescharacteristic
202C3×Dic3—24no—
203C3×Dic3—24no—
204C3×Dic3—24no—
205C3×Dic3—22no—
206C3×Dic3—22no—
207#135—91no—
208#131—272no—
209#108—271no—
210#104—92no—
211C4×C8—271no—
212C33#44311yescharacteristic, complemented, Sylow
213C3×Q8—32no—
214C3×Q8—32no—
215C3×Q8—62no—
216C3×D4—32no—
217C3×D4—32no—
218C3×D4—62no—
219C3×D4—64no—
220C3×D4—62no—
221C3×D4—62no—
222C2×C3×C4—21no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—64no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—62no—
229C2×C3×C4—62no—
230C2×C3×C4C2×D3211yescharacteristic
231C2×C3×C4C2×D3212yes—
232#67—22no—
233C2×Dic3—21no—
234D12D3×Dic312yes—
235C4×D3—22no—
236Dic6D3×Dic312yes—
237C3×C8—91no—
238C3×C8—91no—
239C3×C8—182no—
240C3⋊C8—32no—
241C3⋊C8—32no—
242C3⋊C8—61no—
243C3⋊C8—61no—
244C3⋊C8—64no—
245C3⋊C8—62no—
246C3⋊C8—62no—
247C3⋊C8—91no—
248C3⋊C8—91no—
249C3⋊C8—92no—
250C3⋊C8—92no—
251C3⋊C8—181no—
252C3⋊C8—181no—
253C3⋊C8—182no—
254C3⋊C8—182no—
255C3⋊C8—361no—
256C3⋊C8—361no—
257C2×C32—21no—
258C2×C32—22no—
259C2×C32—22no—
260C2×C32—21no—
261C2×C32—22no—
262C2×C32—41no—
263C2×C32—42no—
264C2×C32—42no—
265C2×C32—122no—
266C2×C32#67⋊C412yes—
267C2×C32D4×Dic311yescharacteristic
268C3×D3—44no—
269C3×D3—42no—
270C3×D3—42no—
271#41—32no—
272Q16—91no—
273SD16—92no—
274D8—91no—
275M16—92no—
276M16—542no—
277C2×C8—91no—
278C2×C8—271no—
279C2×C8—542no—
280C42—31no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—41no—
284C22×C3—64no—
285C22×C3—62no—
286C22×C3—62no—
287C22×C3#380511yescharacteristic
288C22×C3#380512yes—
289C2×D3—22no—
290C3×C4—21no—
291C3×C4—22no—
292C3×C4—22no—
293C3×C4—21no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4—64no—
297C3×C4—64no—
298C3×C4—64no—
299C3×C4—62no—
300C3×C4—62no—
301C3×C4C4×D3211yescharacteristic
302C3×C4#380211yescharacteristic
303C3×C4C2×D3×Dic312yes—
304C3×C4#381112yes—
305Dic3—22no—
306Dic3—22no—
307C32—21no—
308C32—22no—
309C32—41no—
310C32#482412yescomplemented
311C32#519611yescharacteristic, complemented
312Q8—32no—
313D4—32no—
314D4—62no—
315C2×C4—61no—
316C2×C4—62no—
317C2×C4D3311yescharacteristic
318C8—91no—
319C8—91no—
320C8—182no—
321C8—271no—
322C8—271no—
323C2×C3—22no—
324C2×C3—21no—
325C2×C3—22no—
326C2×C3—22no—
327C2×C3—21no—
328C2×C3—42no—
329C2×C3—42no—
330C2×C3—41no—
331C2×C3—124no—
332C2×C3—122no—
333C2×C3—122no—
334C2×C3#6390211yescharacteristic
335C2×C3#67⋊Dic312yes—
336D3—42no—
337C22—62no—
338C22#8896311yescharacteristic
339C4—62no—
340C4—62no—
341C4D32×Dic311yescharacteristic
342C4#8895411yescharacteristic
343C3—21no—
344C3—22no—
345C3—41no—
346C3#1059038911yescharacteristic, complemented
347C3#1059167512yescomplemented
348C2—21no—
349C2—122no—
350C2#1172292511yescharacteristic
subgroupG/H
centerC4#88954
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC2×C33C4×D4
prime radicalC3×#63822C2
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
1. #11723350C2semidirect
3. #11722081C2semidirect
4. Dic3×C3⋊(C3⋊C8)C2semidirect
5. C3×#63569C2semidirect
15. C3×#3718C4semidirect
16. C3×#3715C4semidirect
17. C3×#3713C4semidirect
29. #63569D3semidirect
70. #3718Dic3semidirect
71. #3715Dic3semidirect
72. #3713Dic3semidirect
212. C33#443semidirect
310. C32#4824semidirect
311. C32#5196semidirect
346. C3#10590389semidirect
347. C3#10591675semidirect

Maximal subgroups

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

subgroupindex
#117233502
C3×#638222
#117220812
Dic3×C3⋊(C3⋊C8)2
C3×#635692
#105916753
#105903893