Finite Group Databasebeta

← #13175193

Subgroups — #13175193

Order 1728.

2292 conjugacy classes of subgroups; only the 316 normal subgroups are shown.

normal subgroups only

This group has 2292 conjugacy classes of subgroups — too many to draw. Only its 316 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
1C2×#89017C212yesmaximal, complemented
2C42×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C3×#64083C211yesmaximal, characteristic, complemented
4C32×#869C211yesmaximal, characteristic, complemented
5#11723532C214yesmaximal, complemented
6#11723501C214yesmaximal, complemented
7#11723436C212yesmaximal, complemented
8#10596196—31nomaximal
9#10593907—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2212yes—
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2214yes—
13C2×C3×C4⋊(C3⋊D3)C2212yes—
14C2×C32×Dic6C2212yes—
15C2×#6292C2214yes—
16C2×Dic3×C3⋊D3C2214yes—
17#89017—28no—
18C33×C42C2211yescharacteristic
19C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
20C4×C3⋊(C3⋊Dic3)C2212yes—
21C32×C4⋊Dic3C2214yes—
22#88803C2218yes—
23C2×#3832—32no—
24C2×#3796—38no—
25C32×C4⋊Q8—31no—
26#64156—32no—
27#64104—34no—
28#64102—34no—
29#64083D311yescharacteristic, complemented
30C42×C3⋊D3—31no—
31C42×C3⋊D3—34no—
32C42×C3⋊D3—64no—
33C3×#874—34no—
34C3×#869D314yescomplemented
35#63916—316no—
36#63898—316no—
37#63863—38no—
38C22×C3⋊(C3⋊D3)C2311yescharacteristic
39C22×C3×C3⋊D3C2314yes—
40C2×C33×C4C2311yescharacteristic
41C2×C33×C4C2312yes—
42C2×C3⋊(C3⋊Dic3)C2312yes—
43C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
44C4×C3⋊(C3⋊D3)—24no—
45C4×C3⋊(C3⋊D3)D414yescomplemented
46C3×C4⋊(C3⋊D3)—24no—
47C3×C4⋊(C3⋊D3)—28no—
48C2×C32×Dic3C2314yes—
49C32×Dic6—24no—
50#6292—28no—
51Dic3×C3⋊D3—28no—
52#5725—94no—
53#5716—91no—
54C2×C32×Q8—32no—
55#3832—68no—
56C2×C4⋊(C3⋊D3)—34no—
57C2×C4⋊(C3⋊D3)C2×D314yes—
58C2×C4⋊(C3⋊D3)C2×D312yes—
59C2×C4×C3⋊D3—31no—
60C2×C4×C3⋊D3—34no—
61C2×C4×C3⋊D3—34no—
62C2×C4×C3⋊D3—32no—
63C2×C4×C3⋊D3—38no—
64C2×C4×C3⋊D3—64no—
65C2×C4×C3⋊D3—68no—
66C2×C3×D12—38no—
67C2×C3×D12—316no—
68C2×C3×Dic6C2×D318yes—
69C2×#626—316no—
70C2×D3×Dic3—316no—
71#3796—632no—
72C32×C4⋊C4—34no—
73C32×C42—24no—
74C32×C42C2×D314yes—
75C32×C42C2×D311yescharacteristic
76#3752—38no—
77C4×C3⋊Dic3—32no—
78C4×C3⋊Dic3—31no—
79C4×C3⋊Dic3—34no—
80C4×C3⋊Dic3—38no—
81C4×C3⋊Dic3—68no—
82C4×C3⋊Dic3—64no—
83C3×C4⋊Dic3C2×D3116yes—
84(C2×D3)⋊Dic3—332no—
85C22×C33C2411yescharacteristic
86C2×C3⋊(C3⋊D3)—22no—
87C2×C3⋊(C3⋊D3)C2×D412yes—
88C2×C3×C3⋊D3—24no—
89C2×C3×C3⋊D3—28no—
90C33×C4C2×D412yes—
91C33×C4#4114yescomplemented
92C3⋊(C3⋊Dic3)—22no—
93C3⋊(C3⋊Dic3)C2×D412yes—
94C32×Dic3—24no—
95C2×#239—98no—
96C2×#237—92no—
97C3×C4⋊Q8—34no—
98C3×#127—91no—
99#947—98no—
100#938—94no—
101#936—92no—
102D4×Dic3—94no—
103#895—916no—
104#893—916no—
105#874—34no—
106C42×D3—94no—
107C42×D3—91no—
108C42×D3—184no—
109#869C3⋊D311yescharacteristic, complemented
110C22×C3⋊D3—31no—
111C22×C3⋊D3—34no—
112C22×C3⋊D3—64no—
113C22×C3⋊D3C22×D314yes—
114C22×C3×D3—316no—
115C32×Q8—64no—
116C2×C32×C4—28no—
117C2×C32×C4—24no—
118C2×C32×C4—34no—
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D318yes—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D312yes—
123C2×C3⋊Dic3—31no—
124C2×C3⋊Dic3—32no—
125C2×C3⋊Dic3—34no—
126C2×C3⋊Dic3—38no—
127C2×C3⋊Dic3—64no—
128C2×C3⋊Dic3—68no—
129C4⋊(C3⋊D3)—24no—
130C4⋊(C3⋊D3)—28no—
131C4⋊(C3⋊D3)—68no—
132C4×C3⋊D3—34no—
133C4×C3⋊D3—316no—
134C4×C3⋊D3—616no—
135C4×C3⋊D3—68no—
136C4×C3⋊D3—632no—
137C4×C3⋊D3—64no—
138C4×C3⋊D3—616no—
139C2×C3×Dic3C22×D3116yes—
140C3×D12—616no—
141C3×D12—632no—
142C3×Dic6—216no—
143#626—632no—
144D3×Dic3—632no—
145#532—271noSylow
146C2×C33C22×D411yescharacteristic
147C2×C33C2×#4112yes—
148C3⋊(C3⋊D3)—22no—
149C3×C3⋊D3—44no—
150C2×C3×Q8—38no—
151C2×C3×D4—94no—
152C2×C3×D4—92no—
153C2×#67—94no—
154C22×Dic3—94no—
155#239—1832no—
156#237—188no—
157C2×D12—316no—
158C2×D12—38no—
159C2×D12—916no—
160C2×C4×D3—94no—
161C2×C4×D3—91no—
162C2×C4×D3—916no—
163C2×C4×D3—98no—
164C2×C4×D3—92no—
165C2×C4×D3—184no—
166C2×C4×D3—188no—
167C2×Dic6C2×C3⋊D312yes—
168C3×C4⋊C4—316no—
169C3×C42—24no—
170C3×C42D3214yes—
171C3×C42C2×C3⋊D311yescharacteristic
172#217—98no—
173(C2×D3)⋊C4—932no—
174C4⋊Dic3C2×C3⋊D314yes—
175C4×Dic3—98no—
176C4×Dic3—94no—
177C4×Dic3—92no—
178C4×Dic3—91no—
179C4×Dic3—188no—
180C4×Dic3—184no—
181C22×C32—24no—
182C22×C32C23×D314yes—
183C22×C32C23×D311yescharacteristic
184C2×C3⋊D3—24no—
185C2×C3⋊D3—28no—
186C2×C3⋊D3—32no—
187C2×C3⋊D3—38no—
188C2×C3⋊D3—68no—
189C2×C3⋊D3—616no—
190C2×C3⋊D3—62no—
191C2×C3⋊D3—68no—
192C2×C3×D3—616no—
193C2×C3×D3—632no—
194C32×C4—216no—
195C32×C4—28no—
196C32×C4—64no—
197C32×C4D3×D418yes—
198C32×C4D3×D412yes—
199C32×C4#23714yescomplemented
200C32×C4#239116yescomplemented
201C3⋊Dic3—32no—
202C3⋊Dic3—38no—
203C3⋊Dic3—68no—
204C3⋊Dic3—616no—
205C3⋊Dic3—62no—
206C3⋊Dic3—68no—
207C3×Dic3—216no—
208C2×#41—272no—
209C4⋊Q8—31no—
210#127—91no—
211#124—272no—
212#121—274no—
213C4×D4—274no—
214C2×C42—271no—
215C33#53211yescharacteristic, complemented, Sylow
216C23×C3—94no—
217C22×D3—316no—
218C22×D3—94no—
219C22×D3—91no—
220C22×D3—184no—
221C3×Q8—616no—
222C3×D4—184no—
223C3×D4—188no—
224C2×C3×C4—28no—
225C2×C3×C4—24no—
226C2×C3×C4—316no—
227C2×C3×C4C2×D3218yes—
228C2×C3×C4C2×D3214yes—
229C2×C3×C4C22×C3⋊D311yescharacteristic
230C2×C3×C4C22×C3⋊D312yes—
231#67—188no—
232C2×Dic3—94no—
233C2×Dic3—91no—
234C2×Dic3—98no—
235C2×Dic3—92no—
236C2×Dic3—184no—
237C2×Dic3—188no—
238C2×Dic3—188no—
239C2×Dic3C22×C3⋊D314yes—
240D12—632no—
241D12—616no—
242D12—1832no—
243C4×D3—916no—
244C4×D3—94no—
245C4×D3—1816no—
246C4×D3—1832no—
247C4×D3—1832no—
248C4×D3—1816no—
249C4×D3—184no—
250Dic6—24no—
251C2×C32—28no—
252C2×C32—24no—
253C2×C32C2×D3×D414yes—
254C2×C32C2×D3×D411yescharacteristic
255C2×C32C2×#23712yes—
256C2×C32C2×#23918yes—
257C3⋊D3—44no—
258C3⋊D3—616no—
259C3⋊D3—62no—
260C3⋊D3—68no—
261C3×D3—1216no—
262#41—548no—
263C2×Q8—32no—
264C2×D4—94no—
265C2×D4—92no—
266C2×D4—274no—
267C22×C4—274no—
268C22×C4—271no—
269C22×C4—272no—
270C4⋊C4—34no—
271C22⋊C4—278no—
272C42—272no—
273C42—271no—
274C42D3×C3⋊D311yescharacteristic
275C22×C3—24no—
276C22×C3—184no—
277C22×C3—188no—
278C22×C3C22×D3214yes—
279C22×C3C23×C3⋊D311yescharacteristic
280C2×D3—632no—
281C2×D3—616no—
282C2×D3—98no—
283C2×D3—92no—
284C2×D3—188no—
285C2×D3—1816no—
286C2×D3—188no—
287C2×D3—182no—
288C3×C4—216no—
289C3×C4—28no—
290C3×C4—616no—
291C3×C4#3796116yescomplemented
292C3×C4#380218yes—
293C3×C4D4×C3⋊D312yes—
294C3×C4#383214yescomplemented
295Dic3—24no—
296Dic3—98no—
297Dic3—92no—
298Dic3—188no—
299Dic3—1816no—
300Dic3—188no—
301Dic3—182no—
302C32—24no—
303C32#571611yescharacteristic, complemented
304C32#572514yescomplemented
305C23—94no—
306C23—271no—
307Q8—64no—
308D4—188no—
309D4—184no—
310D4—548no—
311C2×C4—34no—
312C2×C4—271no—
313C2×C4—274no—
314C2×C4—272no—
315C2×C4—548no—
316C2×C4—544no—
317C2×C4C2×D3×C3⋊D312yes—
318C2×C4C2×D3×C3⋊D311yescharacteristic
319C2×C3—28no—
320C2×C3—24no—
321C2×C3—364no—
322C2×C3C2×#379618yes—
323C2×C3C2×#380214yes—
324C2×C3C2×D4×C3⋊D311yescharacteristic
325C2×C3C2×#383212yes—
326D3—1216no—
327D3—1816no—
328D3—182no—
329D3—188no—
330C22—188no—
331C22—184no—
332C22—272no—
333C22—542no—
334C22C22×D3×C3⋊D311yescharacteristic
335C4—64no—
336C4—272no—
337C4—542no—
338C4#8901714yescomplemented
339C4#8902612yes—
340C3—24no—
341C3#1059390714yescomplemented
342C3#1059619611yescharacteristic, complemented
343C2—364no—
344C2—542no—
345C2C2×#8901712yes—
346C2C2×#8902611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#532
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#89017C2semidirect
2. C42×C3⋊(C3⋊D3)C2semidirect
3. C3×#64083C2semidirect
4. C32×#869C2semidirect
5. #11723532C2semidirect
6. #11723501C2semidirect
7. #11723436C2semidirect
29. #64083D3semidirect
34. C3×#869D3semidirect
45. C4×C3⋊(C3⋊D3)D4semidirect
91. C33×C4#41semidirect
109. #869C3⋊D3semidirect
199. C32×C4#237semidirect
200. C32×C4#239semidirect
215. C33#532semidirect
291. C3×C4#3796semidirect
294. C3×C4#3832semidirect
303. C32#5716semidirect
304. C32#5725semidirect
338. C4#89017semidirect
341. C3#10593907semidirect
342. C3#10596196semidirect

Maximal subgroups

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

subgroupindex
C2×#890172
C42×C3⋊(C3⋊D3)2
C3×#640832
C32×#8692
#117235322
#117235012
#117234362
#105961963
#105939073