Finite Group Databasebeta

← #13161011

Subgroups — #13161011

Order 1728.

1144 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1144 conjugacy classes of subgroups — too many to draw. Only its 172 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×#88860C211yesmaximal, characteristic, complemented
2C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C32×#213C211yesmaximal, characteristic, complemented
4#11723524C211yesmaximal, characteristic, complemented
5#11722051C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592849—31nomaximal
9#10591481—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C22×C3×C3⋊Dic3—21no—
13C2×C32×D12C2211yescharacteristic
14C2×#6291—21no—
15C2×#6291—21no—
16#88860—22no—
17C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C3×#3757—21no—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C3×C4×C3⋊Dic3C2211yescharacteristic
21C32×#213—22no—
22C2×C32×C3⋊C8C2211yescharacteristic
23C2×C32×D8—31no—
24#64149—31no—
25D4×C3⋊Dic3D311yescharacteristic, complemented
26C2×#3753—31no—
27C2×#3753—34no—
28C2×#3753—64no—
29C3×D4×Dic3—34no—
30C2×C3×#213D314yescomplemented
31#63911—34no—
32#63659—31no—
33#63648—31no—
34#63641—31no—
35Dic3⋊(C3⋊C8)—34no—
36#63564—34no—
37#63562—34no—
38C23×C33—21no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C32×D3—21no—
41C33×D4—21no—
42C33×D4D412yescomplemented
43C2×C33×C4C2311yescharacteristic
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))—21no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3—42no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C2×C3×C3⋊Dic3D411yescharacteristic
50C32×D12—21no—
51C32×D12—21no—
52#6291—42no—
53#6291—42no—
54C3⋊(C3⋊(C3⋊C8))—21no—
55C32×C3⋊C8—21no—
56#5150—94no—
57#5015—91no—
58C2×C32×D4—24no—
59C2×C32×D4—31no—
60C2×C32×D4C2×D311yescharacteristic
61C2×C32×D4C2×D314yes—
62C2×#638—61no—
63C2×#638—61no—
64C22×C3⋊Dic3—21no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4⋊(C3⋊D3)—34no—
67C2×C4⋊(C3⋊D3)—64no—
68C22×C3×Dic3—64no—
69C2×C3×D12C2×D314yes—
70C2×#626—64no—
71C2×#626—64no—
72C32×D8—62no—
73C2×C32×C8—31no—
74#3757—21no—
75#3753—616no—
76#3753—62no—
77#3753—68no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79C4×C3⋊Dic3C2×D311yescharacteristic
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—34no—
82C2×C3⋊(C3⋊C8)—64no—
83C3×#217—64no—
84C3×#213—28no—
85C3×C4⋊Dic3—34no—
86C3×C4×Dic3—34no—
87C2×C3×C3⋊C8C2×D314yes—
88C22×C33—22no—
89C22×C33—41no—
90C22×C33—41no—
91C22×C33C2×D411yescharacteristic
92C2×C3⋊(C3⋊D3)—41no—
93C2×C3⋊(C3⋊D3)—41no—
94C2×C3⋊(C3⋊D3)—41no—
95C2×C32×D3—41no—
96C2×C32×D3—41no—
97C2×C32×D3—41no—
98C33×C4C2×D411yescharacteristic
99C33×C4#4111yescharacteristic
100C3×C3⋊Dic3—21no—
101C3×C3⋊Dic3—41no—
102C3×C3⋊Dic3D812yescomplemented
103C2×C3×D8—34no—
104C3×C4×D4—91no—
105#940—94no—
106D4×Dic3—34no—
107C2×#213—94no—
108C2×#213—91no—
109C2×#213—184no—
110C2×#213C3⋊D311yescharacteristic, complemented
111#874—91no—
112#832—94no—
113#821—94no—
114Dic3⋊C8—94no—
115#809—91no—
116#809—91no—
117C4⋊(C3⋊C8)—91no—
118C23×C32—21no—
119C23×C32—24no—
120C23×C32—44no—
121C23×C32—61no—
122C22×C3⋊D3—61no—
123C22×C3⋊D3—64no—
124C22×C3⋊D3—124no—
125C22×C3×D3—24no—
126C32×D4—21no—
127C32×D4—28no—
128C32×D4—24no—
129C32×D4—44no—
130C32×D4—61no—
131C32×D4—61no—
132C32×D4D1212yescomplemented
133C32×D4#6718yescomplemented
134C2×C32×C4—24no—
135C2×C32×C4C22×D314yes—
136C2×C32×C4C22×D311yescharacteristic
137#638—122no—
138#638—122no—
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—42no—
141C2×C3⋊Dic3#6711yescharacteristic
142C2×C3⋊Dic3#6711yescharacteristic
143C4⋊(C3⋊D3)—68no—
144C4⋊(C3⋊D3)—61no—
145C4⋊(C3⋊D3)—61no—
146C4⋊(C3⋊D3)—64no—
147C4⋊(C3⋊D3)—64no—
148C4⋊(C3⋊D3)—124no—
149C2×C3×Dic3—34no—
150C2×C3×Dic3—34no—
151C2×C3×Dic3—64no—
152C2×C3×Dic3—128no—
153C3×D12—24no—
154C3×D12—24no—
155#626—128no—
156#626—128no—
157C32×C8—61no—
158C3⋊(C3⋊C8)—68no—
159C3⋊(C3⋊C8)—61no—
160C3⋊(C3⋊C8)—64no—
161C3×C3⋊C8—24no—
162#459—271noSylow
163C2×C33—42no—
164C2×C33#12111yescharacteristic
165C2×C33C2×D811yescharacteristic
166C2×C33AGL(1,ℤ/8)11yescharacteristic
167C3⋊(C3⋊D3)—81no—
168C32×D3—81no—
169C2×C3×D4—24no—
170C2×C3×D4—34no—
171C2×C3×D4D3214yes—
172C2×C3×D4C2×C3⋊D311yescharacteristic
173C22×C3×C4—181no—
174C2×#67—184no—
175C2×#67—184no—
176C22×Dic3—64no—
177C2×D12—94no—
178C2×D12—91no—
179C2×D12—184no—
180C2×D12—181no—
181C2×D12—181no—
182C2×D12C2×C3⋊D311yescharacteristic
183C3×D8—68no—
184C2×C3×C8—34no—
185C3×C4⋊C4—91no—
186C3×C22⋊C4—181no—
187C3×C42—91no—
188#217—64no—
189#213—22no—
190#213—1816no—
191#213—188no—
192#213—182no—
193C4⋊Dic3—34no—
194C4×Dic3—34no—
195C2×C3⋊C8—94no—
196C2×C3⋊C8—91no—
197C2×C3⋊C8—184no—
198C2×C3⋊C8C2×C3⋊D311yescharacteristic
199C22×C32—22no—
200C22×C32—24no—
201C22×C32—28no—
202C22×C32—48no—
203C22×C32—41no—
204C22×C32—41no—
205C22×C32—44no—
206C22×C32—44no—
207C22×C32—84no—
208C22×C32—84no—
209C22×C32—121no—
210C22×C32—121no—
211C22×C32—121no—
212C22×C32D3×D414yes—
213C22×C32C2×#6711yescharacteristic
214C2×C3⋊D3—128no—
215C2×C3⋊D3—121no—
216C2×C3⋊D3—121no—
217C2×C3⋊D3—124no—
218C2×C3⋊D3—121no—
219C2×C3⋊D3—124no—
220C2×C3⋊D3—124no—
221C2×C3⋊D3—244no—
222C2×C3⋊D3—244no—
223C2×C3×D3—44no—
224C2×C3×D3—44no—
225C2×C3×D3—44no—
226C32×C4—24no—
227C32×C4—24no—
228C32×C4C2×D1211yescharacteristic
229C32×C4#23511yescharacteristic
230C32×C4#23714yes—
231C32×C4C2×#6714yes—
232C3⋊Dic3—21no—
233C3⋊Dic3—41no—
234C3⋊Dic3#21312yescomplemented
235C3×Dic3—38no—
236C3×Dic3—64no—
237C3×Dic3—124no—
238C2×D8—31no—
239C2×D8—271no—
240#127—271no—
241C4×D4—91no—
242C4⋊C8—271no—
243#102—271no—
244#102—271no—
245C33#45911yescharacteristic, complemented, Sylow
246C23×C3—24no—
247C23×C3—21no—
248C23×C3—44no—
249C23×C3—64no—
250C22×D3—21no—
251C22×D3—184no—
252C22×D3—181no—
253C22×D3—364no—
254C3×D4—24no—
255C3×D4—21no—
256C3×D4—28no—
257C3×D4—44no—
258C3×D4—64no—
259C3×D4—64no—
260C3×D4#62618yescomplemented
261C3×D4#63812yescomplemented
262C2×C3×C4—24no—
263C2×C3×C4—91no—
264C2×C3×C4—91no—
265C2×C3×C4—181no—
266C2×C3×C4—362no—
267C2×C3×C4C2×D3214yes—
268C2×C3×C4C22×C3⋊D311yescharacteristic
269#67—368no—
270#67—368no—
271C2×Dic3—34no—
272C2×Dic3—34no—
273C2×Dic3—64no—
274C2×Dic3—128no—
275D12—21no—
276D12—21no—
277D12—188no—
278D12—184no—
279D12—184no—
280D12—181no—
281D12—181no—
282D12—364no—
283D12—362no—
284D12—362no—
285C3×C8—64no—
286C3⋊C8—21no—
287C3⋊C8—188no—
288C3⋊C8—184no—
289C3⋊C8—181no—
290C2×C32—24no—
291C2×C32—24no—
292C2×C32—24no—
293C2×C32—42no—
294C2×C32—48no—
295C2×C32—88no—
296C2×C32—241no—
297C2×C32D3×D814yes—
298C2×C32#91114yes—
299C2×C32#93011yescharacteristic
300C2×C32C2×#21311yescharacteristic
301C2×C32#93914yes—
302C2×C32#94911yescharacteristic
303C3⋊D3—248no—
304C3⋊D3—244no—
305C3⋊D3—241no—
306C3×D3—84no—
307C2×D4—31no—
308C2×D4—271no—
309C2×D4—541no—
310C2×D4—541no—
311C2×D4D3×C3⋊D311yescharacteristic
312C22×C4—181no—
313D8—62no—
314D8—542no—
315C2×C8—31no—
316C2×C8—271no—
317C4⋊C4—91no—
318C22⋊C4—181no—
319C42—91no—
320C22×C3—22no—
321C22×C3—28no—
322C22×C3—24no—
323C22×C3—48no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—44no—
327C22×C3—44no—
328C22×C3—84no—
329C22×C3—84no—
330C22×C3—124no—
331C22×C3—124no—
332C22×C3—124no—
333C22×C3D3×#6714yes—
334C22×C3D4×C3⋊D311yescharacteristic
335C2×D3—41no—
336C2×D3—41no—
337C2×D3—41no—
338C2×D3—368no—
339C2×D3—361no—
340C2×D3—361no—
341C2×D3—361no—
342C2×D3—364no—
343C2×D3—364no—
344C2×D3—364no—
345C2×D3—724no—
346C2×D3—724no—
347C3×C4—24no—
348C3×C4—24no—
349C3×C4—92no—
350C3×C4—181no—
351C3×C4—361no—
352C3×C4#380414yes—
353C3×C4C2×#62614yes—
354C3×C4#383011yescharacteristic
355C3×C4C2×#63811yescharacteristic
356Dic3—38no—
357Dic3—64no—
358Dic3—124no—
359C32—24no—
360C32#501511yescharacteristic, complemented
361C32#515014yescomplemented
362C23—21no—
363C23—61no—
364C23—541no—
365D4—21no—
366D4—61no—
367D4—61no—
368D4—541no—
369D4—541no—
370D4—1082no—
371D4—1082no—
372D4#629112yescomplemented
373C2×C4—91no—
374C2×C4—91no—
375C2×C4—181no—
376C2×C4—362no—
377C2×C4C2×D3×C3⋊D311yescharacteristic
378C8—61no—
379C8—541no—
380C2×C3—24no—
381C2×C3—24no—
382C2×C3—24no—
383C2×C3—42no—
384C2×C3—48no—
385C2×C3—88no—
386C2×C3—244no—
387C2×C3D3×#21314yes—
388C2×C3#6392514yes—
389C2×C3#6397814yes—
390C2×C3D8×C3⋊D311yescharacteristic
391C2×C3#6412011yescharacteristic
392C2×C3#6414811yescharacteristic
393D3—81no—
394D3—728no—
395D3—721no—
396D3—724no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—121no—
401C22—121no—
402C22—121no—
403C22—1081no—
404C22—1081no—
405C22—1081no—
406C22C3⋊D3×#6711yescharacteristic
407C4—92no—
408C4—181no—
409C4—361no—
410C4#8902811yescharacteristic
411C4C2×#629111yescharacteristic
412C3—24no—
413C3#1059148114yescomplemented
414C3#1059284911yescharacteristic, complemented
415C2—42no—
416C2—241no—
417C2—2161no—
418C2C3⋊D3×#21311yescharacteristic
419C2#1172354611yescharacteristic
420C2#1172363511yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#459
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88860C2semidirect
2. C3×D4×C3⋊Dic3C2semidirect
3. C2×C32×#213C2semidirect
4. #11723524C2semidirect
5. #11722051C2semidirect
6. #11722036C2semidirect
7. #11722031C2semidirect
25. D4×C3⋊Dic3D3semidirect
30. C2×C3×#213D3semidirect
42. C33×D4D4semidirect
102. C3×C3⋊Dic3D8semidirect
110. C2×#213C3⋊D3semidirect
132. C32×D4D12semidirect
133. C32×D4#67semidirect
234. C3⋊Dic3#213semidirect
245. C33#459semidirect
260. C3×D4#626semidirect
261. C3×D4#638semidirect
360. C32#5015semidirect
361. C32#5150semidirect
372. D4#6291semidirect
413. C3#10591481semidirect
414. C3#10592849semidirect

Maximal subgroups

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

subgroupindex
C2×#888602
C3×D4×C3⋊Dic32
C2×C32×#2132
#117235242
#117220512
#117220362
#117220312
#105928493
#105914813