Finite Group Databasebeta

← #13177969

Subgroups — #13177969

Order 1728.

1736 conjugacy classes of subgroups; only the 332 normal subgroups are shown.

normal subgroups only

This group has 1736 conjugacy classes of subgroups — too many to draw. Only its 332 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×#64367C211yesmaximal, characteristic, complemented
2C32×#1010C211yesmaximal, characteristic, complemented
3#11723989C211yesmaximal, characteristic, complemented
4C2×#88798C213yesmaximal, complemented
5#11723375C213yesmaximal, complemented
6#11723373C213yesmaximal, complemented
7#11723363C213yesmaximal, complemented
8#10596309—31nomaximal
9#10595445—34nomaximal
10C33×#41C2211yescharacteristic
11C3×Q8×C3⋊D3—21no—
12C3×#3830—23no—
13C3×#3828C2213yes—
14C2×C3×#634C2213yes—
15C32×D3×Q8—21no—
16C32×#237—23no—
17C32×#235C2213yes—
18C2×C32×Dic6C2213yes—
19#88855C2213yes—
20C2×C3⋊(C3⋊(C3⋊C8))C2213yes—
21#88798C2213yes—
22#88798C2216yes—
23#88793C2213yes—
24#88792C2213yes—
25#88789C2211yescharacteristic
26C32×2−1+4—31no—
27#64367D311yescharacteristic, complemented
28C3×#1010—34no—
29C3×#1010D314yescomplemented
30#64160—33no—
31#64159—33no—
32#64157—31no—
33#64157—34no—
34#64157—64no—
35C2×#3756—33no—
36#64151—33no—
37C2×#3718—312no—
38#63827—312no—
39#63827—312no—
40#63822—312no—
41C33×Q8D411yescharacteristic
42C33×D4D413yes—
43C2×C33×C4C2313yes—
44C3×#638—23no—
45C2×C3×C3⋊Dic3—23no—
46C3×C4⋊(C3⋊D3)C2311yescharacteristic
47C3×C4×C3⋊D3—23no—
48C3×#634—23no—
49C3×#634C2313yes—
50C32×#67—23no—
51C2×C32×Dic3—23no—
52C32×D12C2311yescharacteristic
53C32×C4×D3—23no—
54C32×Dic6—23no—
55C32×Dic6C2313yes—
56C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
57C3⋊(C3⋊(C3⋊C8))C2313yes—
58#5838—94no—
59#5838—91no—
60C32×#41—24no—
61C32×#41—33no—
62C32×#41—63no—
63C32×#41C2×D314yes—
64C32×#41C2×D311yescharacteristic
65C2×C32×Q8—33no—
66C2×C32×Q8—61no—
67Q8×C3⋊D3—21no—
68#3830—23no—
69#3828C2×D313yes—
70C2×#634C2×D313yes—
71C3×D3×Q8—24no—
72C3×D3×Q8—64no—
73C3×#237—212no—
74C3×#237—612no—
75C3×#235—312no—
76C3×#235C2×D3112yes—
77C2×C3×Dic6—312no—
78C2×C3×Dic6C2×D3112yes—
79#3756—36no—
80#3756—33no—
81#3755—33no—
82#3754—33no—
83#3753—31no—
84#3748—33no—
85#3748—312no—
86#3748—612no—
87C2×C3⋊(C3⋊C8)—33no—
88C2×C3⋊(C3⋊C8)—312no—
89C2×C3⋊(C3⋊C8)—612no—
90#3718—312no—
91#3718—324no—
92#3715—312no—
93#3715—312no—
94#3713—34no—
95C22×C33C2×D413yes—
96C2×C3×C3⋊D3—21no—
97C2×C32×D3—21no—
98C33×C4C2×D413yes—
99C33×C4C2411yescharacteristic
100C3×C3⋊Dic3—23no—
101C32×Dic3—23no—
102C3×2−1+4—34no—
103C3×2−1+4—91no—
104#1010—34no—
105#1010C3⋊D311yescharacteristic, complemented
106#951—912no—
107#951—93no—
108#950—912no—
109#950—93no—
110#948—94no—
111#948—91no—
112#948—184no—
113C2×#216—912no—
114C2×#216—93no—
115#942—912no—
116#942—93no—
117C32×Q8—24no—
118C32×Q8—33no—
119C32×Q8—63no—
120C32×Q8#6714yes—
121C32×Q8#6711yescharacteristic
122C32×D4—212no—
123C32×D4—31no—
124C32×D4—63no—
125C32×D4#67112yes—
126C32×D4#6713yes—
127C2×C32×C4—212no—
128C2×C32×C4—63no—
129C2×C32×C4—63no—
130C2×C32×C4C22×D3112yes—
131C2×C32×C4C22×D313yes—
132#638—23no—
133C2×C3⋊Dic3—23no—
134C4⋊(C3⋊D3)C22×D311yescharacteristic
135C4×C3⋊D3—23no—
136#634—23no—
137#634C22×D313yes—
138C3×#67—212no—
139C3×#67—612no—
140C2×C3×Dic3—212no—
141C2×C3×Dic3—612no—
142C3×D12—34no—
143C3×D12C22×D314yes—
144C3×C4×D3—212no—
145C3×C4×D3—612no—
146C3×Dic6—212no—
147C3×Dic6—312no—
148C3×Dic6—612no—
149C3×Dic6C22×D3112yes—
150C3⋊(C3⋊C8)—31no—
151C3⋊(C3⋊C8)—33no—
152C3⋊(C3⋊C8)—34no—
153C3⋊(C3⋊C8)—312no—
154C3⋊(C3⋊C8)—64no—
155C3⋊(C3⋊C8)—612no—
156#578—271noSylow
157C2×C33—23no—
158C2×C33C22×D411yescharacteristic
159C3×C3⋊D3—41no—
160C32×D3—41no—
161C3×#41—24no—
162C3×#41—312no—
163C3×#41—612no—
164C3×#41—93no—
165C3×#41—183no—
166C3×#41D3214yes—
167C3×#41C2×C3⋊D311yescharacteristic
168C2×C3×Q8—312no—
169C2×C3×Q8—64no—
170C2×C3×Q8—93no—
171C2×C3×Q8—181no—
172D3×Q8—21no—
173D3×Q8—64no—
174#237—23no—
175#237—612no—
176#235—312no—
177#235C2×C3⋊D313yes—
178C2×Dic6—312no—
179C2×Dic6C2×C3⋊D313yes—
180#216—924no—
181#216—912no—
182#216—96no—
183#216—93no—
184#215—912no—
185#215—93no—
186#214—912no—
187#214—93no—
188#213—94no—
189#213—91no—
190#208—912no—
191#208—93no—
192#208—1812no—
193C2×C3⋊C8—912no—
194C2×C3⋊C8—93no—
195C2×C3⋊C8—1812no—
196C22×C32—212no—
197C22×C32—61no—
198C22×C32C2×#67112yes—
199C22×C32C2×#6713yes—
200C2×C3⋊D3—21no—
201C2×C3×D3—24no—
202C2×C3×D3—64no—
203C32×C4—24no—
204C32×C4—212no—
205C32×C4—63no—
206C32×C4C2×#67112yes—
207C32×C4C2×#6713yes—
208C32×C4C23×D314yes—
209C32×C4C23×D311yescharacteristic
210C3⋊Dic3—23no—
211C3×Dic3—212no—
212C3×Dic3—612no—
2132−1+4—31no—
2142−1+4—91no—
215#137—273no—
216#137—273no—
217#135—273no—
218C2×Q16—273no—
219#131—271no—
220C33#57811yescharacteristic, complemented, Sylow
221C3×Q8—24no—
222C3×Q8—312no—
223C3×Q8—612no—
224C3×Q8—93no—
225C3×Q8—183no—
226C3×Q8#62514yes—
227C3×Q8#63811yescharacteristic
228C3×D4—212no—
229C3×D4—34no—
230C3×D4—612no—
231C3×D4—91no—
232C3×D4—183no—
233C3×D4#625112yes—
234C3×D4#63813yes—
235C2×C3×C4—212no—
236C2×C3×C4—612no—
237C2×C3×C4—612no—
238C2×C3×C4—183no—
239C2×C3×C4—183no—
240C2×C3×C4C2×D32112yes—
241C2×C3×C4C22×C3⋊D313yes—
242#67—23no—
243#67—612no—
244C2×Dic3—23no—
245C2×Dic3—612no—
246D12—34no—
247D12C22×C3⋊D311yescharacteristic
248C4×D3—23no—
249C4×D3—612no—
250Dic6—23no—
251Dic6—312no—
252Dic6—612no—
253Dic6C22×C3⋊D313yes—
254C3⋊C8—94no—
255C3⋊C8—912no—
256C3⋊C8—91no—
257C3⋊C8—93no—
258C3⋊C8—184no—
259C3⋊C8—1812no—
260C2×C32—23no—
261C2×C32—24no—
262C2×C32—212no—
263C2×C32—412no—
264C2×C32—121no—
265C2×C32C22×#6714yes—
266C2×C32C22×#6711yescharacteristic
267C3⋊D3—41no—
268C3×D3—44no—
269C3×D3—124no—
270#41—33no—
271#41—63no—
272#41—93no—
273#41—183no—
274#41D3×C3⋊D311yescharacteristic
275C2×Q8—33no—
276C2×Q8—61no—
277C2×Q8—93no—
278C2×Q8—181no—
279Q16—273no—
280Q16—276no—
281SD16—273no—
282SD16—273no—
283D8—271no—
284M16—273no—
285C2×C8—273no—
286C22×C3—212no—
287C22×C3—64no—
288C22×C3—181no—
289C22×C3C2×#625112yes—
290C22×C3C2×#63813yes—
291C2×D3—21no—
292C2×D3—64no—
293C3×C4—24no—
294C3×C4—212no—
295C3×C4—612no—
296C3×C4—183no—
297C3×C4C2×#625112yes—
298C3×C4C2×#63813yes—
299C3×C4C22×D3214yes—
300C3×C4C23×C3⋊D311yescharacteristic
301Dic3—23no—
302Dic3—612no—
303C32—24no—
304C32#583814yescomplemented
305C32#583811yescharacteristic, complemented
306Q8—33no—
307Q8—63no—
308Q8—93no—
309Q8—183no—
310Q8#629011yescharacteristic
311D4—31no—
312D4—63no—
313D4—91no—
314D4—183no—
315D4#629013yes—
316C2×C4—63no—
317C2×C4—63no—
318C2×C4—183no—
319C2×C4—183no—
320C2×C4C2×D3×C3⋊D313yes—
321C8—271no—
322C8—273no—
323C2×C3—212no—
324C2×C3—23no—
325C2×C3—24no—
326C2×C3—412no—
327C2×C3—124no—
328C2×C3—361no—
329C2×C3C22×#62514yes—
330C2×C3C22×#63811yescharacteristic
331D3—41no—
332D3—124no—
333C22—61no—
334C22—181no—
335C22C2×#629013yes—
336C4—63no—
337C4—183no—
338C4C2×#629013yes—
339C4C22×D3×C3⋊D311yescharacteristic
340C3—24no—
341C3#1059544514yescomplemented
342C3#1059630911yescharacteristic, complemented
343C2—23no—
344C2—121no—
345C2—361no—
346C2C22×#629011yescharacteristic
subgroupG/H
centerC2C22×#6290
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C22×D4
hypercenter#41D3×C3⋊D3
hypocenterC33#578
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ #41

Complements

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

NH ≅ G/Ntype
1. C3×#64367C2semidirect
2. C32×#1010C2semidirect
3. #11723989C2semidirect
4. C2×#88798C2semidirect
5. #11723375C2semidirect
6. #11723373C2semidirect
7. #11723363C2semidirect
27. #64367D3semidirect
29. C3×#1010D3semidirect
105. #1010C3⋊D3semidirect
220. C33#578semidirect
304. C32#5838semidirect
305. C32#5838semidirect
341. C3#10595445semidirect
342. C3#10596309semidirect

Maximal subgroups

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

subgroupindex
C3×#643672
C32×#10102
#117239892
C2×#887982
#117233752
#117233732
#117233632
#105963093
#105954453