Finite Group Databasebeta

← #13147786

Subgroups — #13147786

Order 1728.

1193 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1193 conjugacy classes of subgroups — too many to draw. Only its 167 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×#88649C212yesmaximal, complemented
2C42×#272C211yesmaximal, characteristic, complemented
3#11721385C211yesmaximal, characteristic, complemented
4#64078⋊C3C211yesmaximal, characteristic, complemented
5#11721159C214yesmaximal, complemented
6#11721136C214yesmaximal, complemented
7#11721113C212yesmaximal, complemented
8#10593970—31nomaximal
9#10593897—31nomaximal
10C2×C4×#272C2212yes—
11C2×C4×#272C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
13C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
14C2×#634⋊C3C2212yes—
15C2×#6200C2214yes—
16C2×((C3⋊D3)⋊C3)⋊C4C2214yes—
17#88649—28no—
18C42×H(3,𝔽3)C2211yescharacteristic
19C4×#1084C2211yescharacteristic
20C4×#1084C2212yes—
21C4⋊((C3⋊Dic3)⋊C3)C2214yes—
22#88450C2218yes—
23C2×#3799—32no—
24C2×#3795—32no—
25#64083D311yescharacteristic, complemented
26#64078D311yescharacteristic, complemented
27C3×#874—31no—
28C3×C42×D3—31no—
29C3×C42×D3—31no—
30C3×C42×D3—61no—
31C3×#869—31no—
32#63909—34no—
33#63908—34no—
34#63894—34no—
35Dic3×D12—34no—
36#63864—32no—
37#63860—32no—
38C22×#272C2311yescharacteristic
39C22×(C3⋊D3)⋊C3C2314yes—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×C4×H(3,𝔽3)C2312yes—
42C2×#1084C2312yes—
43C2×#1084C2311yescharacteristic
44C4×#272—24no—
45C4×#272D414yescomplemented
46C2×(C3⋊Dic3)⋊C3C2314yes—
47C4⋊((C3⋊D3)⋊C3)—24no—
48C4⋊((C3⋊D3)⋊C3)—28no—
49#634⋊C3—24no—
50#6200—28no—
51((C3⋊D3)⋊C3)⋊C4—28no—
52#5725—91no—
53#5716—91no—
54#5477—91no—
55C2×C4⋊(C3⋊D3)C2×D312yes—
56C2×C4⋊(C3⋊D3)C2×D314yes—
57C2×#634C2×D312yes—
58C2×C3×D12—32no—
59C2×C3×D12—34no—
60C2×C3×C4×D3—31no—
61C2×C3×C4×D3—31no—
62C2×C3×C4×D3—32no—
63C2×C3×C4×D3—32no—
64C2×C3×C4×D3—61no—
65C2×C3×C4×D3—62no—
66C2×C3×Dic6—32no—
67C2×#626—34no—
68C2×#625—34no—
69C2×#624—34no—
70C2×D3×Dic3—34no—
71#3799—68no—
72#3795—68no—
73C32×C42—21no—
74C32×C42C2×D311yescharacteristic
75C32×C42C2×D311yescharacteristic
76C4⋊(C3⋊Dic3)C2×D314yes—
77C3×C4⋊Dic3—34no—
78C3×C4×Dic3—31no—
79C3×C4×Dic3—31no—
80C3×C4×Dic3—32no—
81C3×C4×Dic3—32no—
82C3×C4×Dic3—62no—
83C3×C4×Dic3—61no—
84#3722—38no—
85(C2×D3)⋊Dic3—38no—
86C22×H(3,𝔽3)C2411yescharacteristic
87C2×#272—22no—
88C2×#272C2×D412yes—
89C2×(C3⋊D3)⋊C3—24no—
90C2×(C3⋊D3)⋊C3—28no—
91C4×H(3,𝔽3)C2×D412yes—
92C4×H(3,𝔽3)#4114yescomplemented
93#1084—22no—
94#1084C2×D412yes—
95(C3⋊Dic3)⋊C3—24no—
96C2×#239—92no—
97C2×#237—92no—
98C2×#235—92no—
99C3×C4⋊Q8—91no—
100C3×#127—91no—
101C2×C3×C42—91no—
102#947—92no—
103#938—94no—
104#936—92no—
105D4×Dic3—94no—
106#930—94no—
107#895—94no—
108#893—94no—
109#875—92no—
110#874—31no—
111#874—91no—
112C4×D12—94no—
113C42×D3—31no—
114C42×D3—31no—
115C42×D3—61no—
116#869—31no—
117#869—91no—
118C22×C3⋊D3C22×D314yes—
119C22×C3×D3—31no—
120C22×C3×D3—31no—
121C22×C3×D3—34no—
122C22×C3×D3—61no—
123C2×C32×C4—22no—
124C2×C32×C4—21no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C22×D312yes—
128C2×C32×C4C22×D312yes—
129C2×C3⋊Dic3C22×D314yes—
130C4⋊(C3⋊D3)—24no—
131C4⋊(C3⋊D3)—28no—
132#634—24no—
133C2×C3×Dic3—31no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—34no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—32no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—62no—
140C3×D12—64no—
141C3×D12—68no—
142C3×C4×D3—34no—
143C3×C4×D3—34no—
144C3×C4×D3—64no—
145C3×C4×D3—68no—
146C3×C4×D3—64no—
147C3×C4×D3—64no—
148C3×Dic6—64no—
149#626—68no—
150#625—68no—
151#624—68no—
152D3×Dic3—68no—
153#532—271noSylow
154C2×H(3,𝔽3)C22×D411yescharacteristic
155C2×H(3,𝔽3)C2×#4112yes—
156#272—22no—
157(C3⋊D3)⋊C3—44no—
158C2×C3×Q8—92no—
159C2×C3×D4—94no—
160C2×C3×D4—92no—
161C22×C3×C4—91no—
162C22×C3×C4—92no—
163C2×#67—94no—
164C2×#67—94no—
165C22×Dic3—94no—
166#239—188no—
167#237—188no—
168#235—188no—
169C2×D12—34no—
170C2×D12—32no—
171C2×D12—94no—
172C2×D12—92no—
173C2×D12—94no—
174C2×C4×D3—31no—
175C2×C4×D3—31no—
176C2×C4×D3—32no—
177C2×C4×D3—32no—
178C2×C4×D3—61no—
179C2×C4×D3—62no—
180C2×C4×D3—94no—
181C2×C4×D3—94no—
182C2×Dic6—32no—
183C2×Dic6—92no—
184C3×C4⋊C4—94no—
185C3×C42—31no—
186C3×C42—31no—
187C3×C42—61no—
188C3×C42—92no—
189C3×C42—91no—
190C3×C42D3211yescharacteristic
191#217—98no—
192(C2×D3)⋊C4—98no—
193(C2×D3)⋊C4—98no—
194C4⋊Dic3—34no—
195C4⋊Dic3—94no—
196C4×Dic3—32no—
197C4×Dic3—32no—
198C4×Dic3—31no—
199C4×Dic3—31no—
200C4×Dic3—62no—
201C4×Dic3—61no—
202C22×C32—21no—
203C22×C32C23×D311yescharacteristic
204C22×C32C23×D311yescharacteristic
205C2×C3⋊D3—24no—
206C2×C3⋊D3—28no—
207C2×C3×D3—32no—
208C2×C3×D3—32no—
209C2×C3×D3—62no—
210C2×C3×D3—64no—
211C2×C3×D3—62no—
212C2×C3×D3—62no—
213C2×C3×D3—64no—
214C2×C3×D3—68no—
215C32×C4—24no—
216C32×C4—22no—
217C32×C4D3×D412yes—
218C32×C4D3×D412yes—
219C32×C4#23714yescomplemented
220C32×C4#23914yescomplemented
221C3⋊Dic3—24no—
222C3×Dic3—32no—
223C3×Dic3—32no—
224C3×Dic3—62no—
225C3×Dic3—64no—
226C3×Dic3—64no—
227C3×Dic3—62no—
228C3×Dic3—62no—
229C2×#41—272no—
230C4⋊Q8—91no—
231#127—91no—
232#124—272no—
233#121—274no—
234C4×D4—274no—
235C2×C42—91no—
236H(3,𝔽3)#53211yescharacteristic, complemented, Sylow
237C23×C3—94no—
238C23×C3—91no—
239C22×D3—31no—
240C22×D3—31no—
241C22×D3—34no—
242C22×D3—61no—
243C22×D3—94no—
244C3×Q8—184no—
245C3×D4—184no—
246C3×D4—188no—
247C2×C3×C4—32no—
248C2×C3×C4—32no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—62no—
252C2×C3×C4—61no—
253C2×C3×C4—91no—
254C2×C3×C4—94no—
255C2×C3×C4—94no—
256C2×C3×C4—92no—
257C2×C3×C4—184no—
258C2×C3×C4C2×D3212yes—
259C2×C3×C4C2×D3211yescharacteristic
260#67—188no—
261#67—188no—
262C2×Dic3—31no—
263C2×Dic3—31no—
264C2×Dic3—32no—
265C2×Dic3—32no—
266C2×Dic3—34no—
267C2×Dic3—61no—
268C2×Dic3—62no—
269C2×Dic3—94no—
270C2×Dic3—188no—
271D12—68no—
272D12—64no—
273D12—188no—
274D12—184no—
275D12—188no—
276C4×D3—34no—
277C4×D3—34no—
278C4×D3—64no—
279C4×D3—64no—
280C4×D3—64no—
281C4×D3—68no—
282C4×D3—188no—
283C4×D3—188no—
284Dic6—64no—
285Dic6—184no—
286C2×C32—22no—
287C2×C32—21no—
288C2×C32C2×D3×D411yescharacteristic
289C2×C32C2×D3×D411yescharacteristic
290C2×C32C2×#23712yes—
291C2×C32C2×#23912yes—
292C3⋊D3—44no—
293C3×D3—64no—
294C3×D3—62no—
295C3×D3—62no—
296C3×D3—124no—
297#41—548no—
298C2×Q8—92no—
299C2×D4—94no—
300C2×D4—92no—
301C2×D4—274no—
302C22×C4—91no—
303C22×C4—92no—
304C22×C4—274no—
305C4⋊C4—94no—
306C22⋊C4—278no—
307C42—92no—
308C42—91no—
309C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
310C22×C3—31no—
311C22×C3—31no—
312C22×C3—61no—
313C22×C3—92no—
314C22×C3—184no—
315C22×C3—188no—
316C22×C3—182no—
317C22×C3C22×D3211yescharacteristic
318C2×D3—32no—
319C2×D3—32no—
320C2×D3—62no—
321C2×D3—62no—
322C2×D3—62no—
323C2×D3—64no—
324C2×D3—68no—
325C2×D3—64no—
326C2×D3—184no—
327C2×D3—188no—
328C3×C4—34no—
329C3×C4—34no—
330C3×C4—32no—
331C3×C4—32no—
332C3×C4—64no—
333C3×C4—62no—
334C3×C4—92no—
335C3×C4—184no—
336C3×C4—182no—
337C3×C4#379614yes—
338C3×C4#380212yes—
339Dic3—32no—
340Dic3—32no—
341Dic3—62no—
342Dic3—62no—
343Dic3—62no—
344Dic3—64no—
345Dic3—64no—
346Dic3—184no—
347C32—21no—
348C32#571611yescharacteristic, complemented
349C32#572511yescharacteristic, complemented
350C23—91no—
351C23—94no—
352Q8—184no—
353D4—188no—
354D4—184no—
355D4—548no—
356C2×C4—91no—
357C2×C4—94no—
358C2×C4—92no—
359C2×C4—94no—
360C2×C4—184no—
361C2×C4—548no—
362C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
363C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
364C2×C3—32no—
365C2×C3—32no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—62no—
369C2×C3—61no—
370C2×C3—182no—
371C2×C3—364no—
372C2×C3C2×#379612yes—
373C2×C3C2×#380211yescharacteristic
374D3—62no—
375D3—62no—
376D3—64no—
377D3—124no—
378D3—364no—
379C22—92no—
380C22—182no—
381C22—188no—
382C22—184no—
383C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
384C4—92no—
385C4—182no—
386C4—184no—
387C4#8864914yescomplemented
388C4#8865512yes—
389C3—31no—
390C3—31no—
391C3—61no—
392C3#1059390711yescharacteristic
393C2—182no—
394C2—364no—
395C2C2×#8864912yes—
396C2C2×#8865511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#532
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88649C2semidirect
2. C42×#272C2semidirect
3. #11721385C2semidirect
4. #64078⋊C3C2semidirect
5. #11721159C2semidirect
6. #11721136C2semidirect
7. #11721113C2semidirect
25. #64083D3semidirect
26. #64078D3semidirect
45. C4×#272D4semidirect
92. C4×H(3,𝔽3)#41semidirect
219. C32×C4#237semidirect
220. C32×C4#239semidirect
236. H(3,𝔽3)#532semidirect
348. C32#5716semidirect
349. C32#5725semidirect
387. C4#88649semidirect

Maximal subgroups

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

subgroupindex
C2×#886492
C42×#2722
#117213852
#64078⋊C32
#117211592
#117211362
#117211132
#105939703
#105938973