Finite Group Databasebeta

← #13161079

Subgroups — #13161079

Order 1728.

1200 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 156 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#11723981C211yesmaximal, characteristic
2C2×C32×#214C211yesmaximal, characteristic, complemented
3#11723397C211yesmaximal, characteristic, complemented
4C2×#88795C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6#11722050C211yesmaximal, characteristic, complemented
7#11722036C211yesmaximal, characteristic, complemented
8#10592862—31nomaximal
9#10591444—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×#6312—21no—
12C2×#6312—21no—
13C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#214—22no—
19C2×C32×C3⋊C8C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3—21no—
22#88795—22no—
23C2×C32×SD16—31no—
24Q8×C3⋊Dic3—31no—
25C2×#3755—31no—
26#64149—31no—
27#64149—34no—
28#64149—64no—
29C2×C3×#214D314yescomplemented
30#63843—34no—
31C2×#3717—34no—
32#63659D311yescharacteristic, complemented
33#63648—31no—
34#63641—31no—
35C3×#832—34no—
36#63572—34no—
37#63564—34no—
38C23×C33—21no—
39C22×C3⋊(C3⋊D3)—21no—
40C33×D4—21no—
41C33×D4—21no—
42C2×C33×C4C2311yescharacteristic
43#6312—42no—
44#6312—42no—
45C2×C3⋊(C3⋊Dic3)D411yescharacteristic
46C2×C3⋊(C3⋊Dic3)D411yescharacteristic
47C4⋊(C3⋊(C3⋊D3))—21no—
48C4⋊(C3⋊(C3⋊D3))—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C32×Dic3—21no—
51C32×Dic6—21no—
52C32×Dic6D412yescomplemented
53C3×C3⋊(C3⋊C8)—21no—
54C32×C3⋊C8—21no—
55#5163—94no—
56#4762—91no—
57C2×C32×Q8—31no—
58C2×C32×D4—24no—
59C2×C32×D4C2×D314yes—
60C2×C32×D4C2×D311yescharacteristic
61C2×#638—61no—
62C2×#638—61no—
63C2×#638—64no—
64C2×#638—64no—
65C2×#638—68no—
66C2×#638—68no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4⋊(C3⋊D3)—34no—
69C2×C4⋊(C3⋊D3)—64no—
70C2×C3×Dic6C2×D314yes—
71C32×SD16—62no—
72C2×C32×C8—31no—
73#3755—62no—
74C4⋊(C3⋊Dic3)—61no—
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76C4×C3⋊Dic3—31no—
77C4×C3⋊Dic3—34no—
78C4×C3⋊Dic3—61no—
79C4×C3⋊Dic3—64no—
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C3×#214—28no—
82C3×C4⋊Dic3—34no—
83C2×C3×C3⋊C8—34no—
84C2×C3×C3⋊C8C2×D314yes—
85Dic3⋊Dic3—64no—
86Dic32—64no—
87#3717—68no—
88C22×C33—41no—
89C22×C33—41no—
90C22×C33—41no—
91C22×C33C2×D411yescharacteristic
92C2×C3⋊(C3⋊D3)—41no—
93C2×C3⋊(C3⋊D3)—41no—
94C2×C3⋊(C3⋊D3)—41no—
95C33×C4C2×D411yescharacteristic
96C33×C4#4111yescharacteristic
97C3⋊(C3⋊Dic3)—21no—
98C3⋊(C3⋊Dic3)SD1612yescomplemented
99C3×C3⋊Dic3—41no—
100C32×Dic3—22no—
101C2×C3×SD16—34no—
102Q8×Dic3—94no—
103C2×#215—94no—
104#940—94no—
105#940—91no—
106#940—184no—
107C2×#214C3⋊D311yescharacteristic, complemented
108C2×#204—91no—
109#887—91no—
110C3×#102—91no—
111#832—34no—
112#821—94no—
113Dic3⋊C8—94no—
114Dic3⋊C8—91no—
115#809—91no—
116C23×C32—28no—
117C23×C32—24no—
118C23×C32—21no—
119C22×C3⋊D3—68no—
120C22×C3⋊D3—64no—
121C22×C3⋊D3—61no—
122C32×Q8—32no—
123C32×Q8—61no—
124C32×D4—21no—
125C32×D4—21no—
126C32×D4—24no—
127C32×D4—24no—
128C32×D4—44no—
129C32×D4—44no—
130C2×C32×C4—24no—
131C2×C32×C4—61no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D311yescharacteristic
134#638—122no—
135#638—128no—
136#638—1216no—
137#638—1216no—
138#638—122no—
139#638—128no—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—31no—
142C2×C3⋊Dic3—31no—
143C2×C3⋊Dic3—34no—
144C2×C3⋊Dic3—34no—
145C2×C3⋊Dic3—64no—
146C2×C3⋊Dic3—64no—
147C4⋊(C3⋊D3)—61no—
148C4⋊(C3⋊D3)—61no—
149C4⋊(C3⋊D3)—64no—
150C4⋊(C3⋊D3)—64no—
151C4⋊(C3⋊D3)—124no—
152C4⋊(C3⋊D3)—124no—
153C2×C3×Dic3—24no—
154C2×C3×Dic3—64no—
155C3×Dic6—24no—
156C3×Dic6#6718yescomplemented
157C32×C8—61no—
158C3⋊(C3⋊C8)—21no—
159C3×C3⋊C8—24no—
160C3×C3⋊C8—64no—
161#460—271noSylow
162C2×C33—81no—
163C2×C33#12111yescharacteristic
164C2×C33C2×SD1611yescharacteristic
165C2×C33AGL(1,ℤ/8)11yescharacteristic
166C3⋊(C3⋊D3)—81no—
167C2×C3×Q8—34no—
168C2×C3×D4—24no—
169C2×C3×D4D3214yes—
170C2×C3×D4C2×C3⋊D311yescharacteristic
171C2×#67—184no—
172C2×#67—181no—
173C2×#67—184no—
174C2×#67—181no—
175C2×#67—188no—
176C2×#67—188no—
177C2×D12—94no—
178C2×D12—91no—
179C2×D12—184no—
180C2×Dic6C2×C3⋊D311yescharacteristic
181C3×SD16—68no—
182C2×C3×C8—34no—
183C2×C3×C8—91no—
184C3×C4⋊C4—91no—
185#215—188no—
186#214—22no—
187C4⋊Dic3—34no—
188C4⋊Dic3—184no—
189Dic3⋊C4—181no—
190C4×Dic3—94no—
191C4×Dic3—91no—
192C4×Dic3—184no—
193C4×Dic3—181no—
194C4×Dic3—184no—
195C2×C3⋊C8—34no—
196C2×C3⋊C8C2×C3⋊D311yescharacteristic
197#204—182no—
198C22×C32—24no—
199C22×C32—48no—
200C22×C32—48no—
201C22×C32—48no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32—44no—
205C22×C32—41no—
206C22×C32—44no—
207C22×C32—44no—
208C22×C32D3×D414yes—
209C22×C32D3×D411yescharacteristic
210C2×C3⋊D3—128no—
211C2×C3⋊D3—128no—
212C2×C3⋊D3—128no—
213C2×C3⋊D3—121no—
214C2×C3⋊D3—121no—
215C2×C3⋊D3—124no—
216C2×C3⋊D3—121no—
217C2×C3⋊D3—124no—
218C2×C3⋊D3—124no—
219C32×C4—24no—
220C32×C4—24no—
221C32×C4—62no—
222C32×C4#23511yescharacteristic
223C32×C4D3×D411yescharacteristic
224C32×C4#23714yes—
225C32×C4C2×#6714yes—
226C3⋊Dic3—32no—
227C3⋊Dic3—38no—
228C3⋊Dic3—41no—
229C3⋊Dic3—68no—
230C3⋊Dic3—68no—
231C3⋊Dic3—61no—
232C3⋊Dic3—64no—
233C3×Dic3—28no—
234C3×Dic3—124no—
235C2×SD16—31no—
236C2×SD16—271no—
237#127—271no—
238C4×Q8—271no—
239C4⋊C8—271no—
240#102—91no—
241#102—271no—
242C33#46011yescharacteristic, complemented, Sylow
243C23×C3—28no—
244C23×C3—24no—
245C23×C3—21no—
246C22×D3—188no—
247C22×D3—184no—
248C22×D3—181no—
249C3×Q8—38no—
250C3×Q8—64no—
251C3×D4—24no—
252C3×D4—24no—
253C3×D4—21no—
254C3×D4—21no—
255C3×D4—44no—
256C3×D4—44no—
257C2×C3×C4—24no—
258C2×C3×C4—64no—
259C2×C3×C4—181no—
260C2×C3×C4C2×D3214yes—
261C2×C3×C4C22×C3⋊D311yescharacteristic
262#67—362no—
263#67—368no—
264#67—3616no—
265#67—3616no—
266#67—362no—
267#67—368no—
268C2×Dic3—21no—
269C2×Dic3—64no—
270C2×Dic3—94no—
271C2×Dic3—91no—
272C2×Dic3—94no—
273C2×Dic3—91no—
274C2×Dic3—184no—
275C2×Dic3—184no—
276D12—184no—
277D12—184no—
278D12—181no—
279D12—181no—
280D12—364no—
281D12—364no—
282Dic6—21no—
283Dic6#63812yescomplemented
284C3×C8—64no—
285C3×C8—181no—
286C3⋊C8—21no—
287C3⋊C8—64no—
288C2×C32—24no—
289C2×C32—24no—
290C2×C32—24no—
291C2×C32—88no—
292C2×C32—84no—
293C2×C32—81no—
294C2×C32#88411yescharacteristic
295C2×C32#91111yescharacteristic
296C2×C32D3×SD1614yes—
297C2×C32D3×SD1611yescharacteristic
298C2×C32#91414yes—
299C2×C32#93914yes—
300C3⋊D3—248no—
301C3⋊D3—244no—
302C3⋊D3—241no—
303C2×Q8—31no—
304C2×D4—271no—
305C2×D4—541no—
306C2×D4—541no—
307C2×D4D3×C3⋊D311yescharacteristic
308SD16—62no—
309SD16—542no—
310C2×C8—31no—
311C2×C8—91no—
312C4⋊C4—91no—
313C4⋊C4—541no—
314C42—271no—
315C42—541no—
316C22×C3—24no—
317C22×C3—48no—
318C22×C3—48no—
319C22×C3—48no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—44no—
324C22×C3—44no—
325C22×C3—44no—
326C22×C3#381114yes—
327C22×C3D4×C3⋊D311yescharacteristic
328C2×D3—368no—
329C2×D3—368no—
330C2×D3—368no—
331C2×D3—361no—
332C2×D3—361no—
333C2×D3—361no—
334C2×D3—364no—
335C2×D3—364no—
336C2×D3—364no—
337C3×C4—24no—
338C3×C4—24no—
339C3×C4—68no—
340C3×C4—361no—
341C3×C4#380414yes—
342C3×C4D3×#6714yes—
343C3×C4#383011yescharacteristic
344C3×C4C2×#63811yescharacteristic
345Dic3—22no—
346Dic3—92no—
347Dic3—98no—
348Dic3—124no—
349Dic3—188no—
350Dic3—188no—
351Dic3—181no—
352Dic3—184no—
353C32—24no—
354C32#476211yescharacteristic, complemented
355C32#516314yescomplemented
356C23—21no—
357C23—541no—
358Q8—32no—
359Q8—61no—
360D4—21no—
361D4—21no—
362D4—541no—
363D4—541no—
364D4—1082no—
365D4—1082no—
366C2×C4—61no—
367C2×C4—181no—
368C2×C4—271no—
369C2×C4—271no—
370C2×C4C2×D3×C3⋊D311yescharacteristic
371C8—61no—
372C8—181no—
373C2×C3—24no—
374C2×C3—24no—
375C2×C3—24no—
376C2×C3—88no—
377C2×C3—81no—
378C2×C3—84no—
379C2×C3#6393014yes—
380C2×C3#6393714yes—
381C2×C3#6397114yes—
382C2×C3SD16×C3⋊D311yescharacteristic
383C2×C3#6412311yescharacteristic
384C2×C3#6414811yescharacteristic
385D3—728no—
386D3—721no—
387D3—724no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—1081no—
392C22—1081no—
393C22—1081no—
394C22#8903911yescharacteristic
395C4—62no—
396C4—272no—
397C4—361no—
398C4—541no—
399C4#8902811yescharacteristic
400C4D3×#63811yescharacteristic
401C3—24no—
402C3#1059144414yescomplemented
403C3#1059286211yescharacteristic, complemented
404C2—81no—
405C2—2161no—
406C2#1172355311yescharacteristic
407C2#1172356611yescharacteristic
408C2#1172361611yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radical#11723981C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#460
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
2. C2×C32×#214C2semidirect
3. #11723397C2semidirect
4. C2×#88795C2semidirect
5. C3×#63659C2semidirect
6. #11722050C2semidirect
7. #11722036C2semidirect
29. C2×C3×#214D3semidirect
32. #63659D3semidirect
52. C32×Dic6D4semidirect
98. C3⋊(C3⋊Dic3)SD16semidirect
107. C2×#214C3⋊D3semidirect
156. C3×Dic6#67semidirect
242. C33#460semidirect
283. Dic6#638semidirect
354. C32#4762semidirect
355. C32#5163semidirect
402. C3#10591444semidirect
403. C3#10592862semidirect

Maximal subgroups

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

subgroupindex
#117239812
C2×C32×#2142
#117233972
C2×#887952
C3×#636592
#117220502
#117220362
#105928623
#105914443