Finite Group Databasebeta

← #13158957

Subgroups — #13158957

Order 1728.

1168 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 186 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
1C4×C4⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
2C32×C4×D12C211yesmaximal, characteristic, complemented
3C2×#88790C211yesmaximal, characteristic
4C3×#63630C211yesmaximal, characteristic, complemented
5#11722054C211yesmaximal, characteristic, complemented
6#11722035C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10592713—31nomaximal
9#10590346—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)—21no—
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3—21no—
14C33×C42C2211yescharacteristic
15#88859—21no—
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×(C2×D3)⋊C4—21no—
20C32×C4⋊Dic3C2211yescharacteristic
21#88790C414yescomplemented
22C32×C4×D4—31no—
23C2×#3753—31no—
24C4×C4⋊(C3⋊D3)—31no—
25C4×C4⋊(C3⋊D3)—34no—
26C4×C4⋊(C3⋊D3)—64no—
27C3×C4×D12D314yescomplemented
28C2×#3714—34no—
29#63659—31no—
30#63636—31no—
31#63635—31no—
32#63630D311yescharacteristic, complemented
33C3×#803—34no—
34#63573—34no—
35#63564—34no—
36#63563—34no—
37C22×C3⋊(C3⋊D3)—21no—
38C22×C32×D3—21no—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
45C4×C3⋊(C3⋊D3)—42no—
46C2×C32×Dic3—21no—
47C32×D12—21no—
48C32×D12C2×C412yescomplemented
49C32×C4×D3—42no—
50C3×C3⋊(C3⋊C8)—21no—
51C3×C3⋊(C3⋊C8)C2×C412yescomplemented
52#5014—94no—
53#4711—91no—
54C2×C32×D4—31no—
55C22×C32×C4—61no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4⋊(C3⋊D3)—34no—
58C2×C4⋊(C3⋊D3)—64no—
59C2×C4×C3⋊D3—68no—
60C2×C4×C3⋊D3—61no—
61C2×C4×C3⋊D3—64no—
62C2×C3×D12C2×D314yes—
63C2×C3×C4×D3—24no—
64C32×C4⋊C4—31no—
65C32×C22⋊C4—61no—
66C32×C42—24no—
67C32×C42C2×D311yescharacteristic
68C32×C42C2×D314yes—
69#3753—34no—
70#3752—68no—
71#3752—61no—
72#3752—64no—
73C4⋊(C3⋊Dic3)—31no—
74C4⋊(C3⋊Dic3)—34no—
75C4⋊(C3⋊Dic3)—64no—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C3×(C2×D3)⋊C4—24no—
79C3×C4⋊Dic3C2×D314yes—
80C2×C3×C3⋊C8—34no—
81C2×C3×C3⋊C8—34no—
82#3714—316no—
83C22×C33C2×D411yescharacteristic
84C2×C3⋊(C3⋊D3)—22no—
85C2×C3⋊(C3⋊D3)—41no—
86C2×C3⋊(C3⋊D3)—41no—
87C2×C32×D3—22no—
88C2×C32×D3—41no—
89C2×C32×D3—41no—
90C33×C4—21no—
91C33×C4—21no—
92C33×C4C22×C411yescharacteristic
93C33×C4#4111yescharacteristic
94C3⋊(C3⋊Dic3)—41no—
95C32×Dic3—41no—
96C3×C4×D4—34no—
97C2×#213—94no—
98C2×D24—91no—
99C4×D12—94no—
100C4×D12—91no—
101C4×D12—184no—
102C4×D12C3⋊D311yescharacteristic, complemented
103C3×#97—91no—
104#832—94no—
105#821—91no—
106#821—91no—
107#817—91no—
108#809—94no—
109#808—94no—
110#803—34no—
111C23×C32—61no—
112C22×C3⋊D3—68no—
113C22×C3⋊D3—61no—
114C22×C3⋊D3—64no—
115C22×C3×D3—24no—
116C32×D4—32no—
117C32×D4—61no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4—24no—
121C2×C32×C4—61no—
122C2×C32×C4—122no—
123C2×C32×C4D1211yescharacteristic
124C2×C32×C4D1211yescharacteristic
125C2×C32×C4#6714yes—
126C2×C32×C4#6714yes—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—68no—
130C2×C3⋊Dic3—61no—
131C2×C3⋊Dic3—64no—
132C4⋊(C3⋊D3)—32no—
133C4⋊(C3⋊D3)—38no—
134C4⋊(C3⋊D3)—68no—
135C4⋊(C3⋊D3)—68no—
136C4⋊(C3⋊D3)—64no—
137C4⋊(C3⋊D3)—61no—
138C4×C3⋊D3—1216no—
139C4×C3⋊D3—122no—
140C4×C3⋊D3—128no—
141C2×C3×Dic3—24no—
142C3×D12—24no—
143C3×D12C4×D318yescomplemented
144C3×C4×D3—48no—
145C3⋊(C3⋊C8)—21no—
146C3⋊(C3⋊C8)C4×D312yescomplemented
147C3×C3⋊C8—38no—
148C3×C3⋊C8—64no—
149#442—271noSylow
150C2×C33C4×D411yescharacteristic
151C2×C33AGL(1,ℤ/8)11yescharacteristic
152C2×C33AGL(1,ℤ/8)11yescharacteristic
153C3⋊(C3⋊D3)—42no—
154C32×D3—42no—
155C2×C3×D4—34no—
156C22×C3×C4—64no—
157C2×D12—94no—
158C2×D12—91no—
159C2×D12—184no—
160C2×D12C2×C3⋊D311yescharacteristic
161C2×C4×D3—21no—
162C2×C4×D3—188no—
163C2×C4×D3—184no—
164C2×C4×D3—181no—
165C2×C3×C8—91no—
166C2×C3×C8—91no—
167C3×C4⋊C4—34no—
168C3×C22⋊C4—64no—
169C3×C42—24no—
170C3×C42D3214yes—
171C3×C42C2×C3⋊D311yescharacteristic
172#213—916no—
173(C2×D3)⋊C4—21no—
174(C2×D3)⋊C4—188no—
175(C2×D3)⋊C4—184no—
176(C2×D3)⋊C4—181no—
177C4⋊Dic3—94no—
178C4⋊Dic3—91no—
179C4⋊Dic3—184no—
180C4⋊Dic3C2×C3⋊D311yescharacteristic
181C2×C3⋊C8—34no—
182C2×C3⋊C8—34no—
183D24—94no—
184C22×C32—24no—
185C22×C32—62no—
186C22×C32—121no—
187C22×C32—121no—
188C22×C32C2×D1211yescharacteristic
189C22×C32C2×#6714yes—
190C2×C3⋊D3—616no—
191C2×C3⋊D3—62no—
192C2×C3⋊D3—68no—
193C2×C3⋊D3—128no—
194C2×C3⋊D3—128no—
195C2×C3⋊D3—121no—
196C2×C3⋊D3—121no—
197C2×C3⋊D3—124no—
198C2×C3⋊D3—124no—
199C2×C3×D3—28no—
200C2×C3×D3—44no—
201C2×C3×D3—44no—
202C32×C4—28no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—24no—
206C32×C4—24no—
207C32×C4—24no—
208C32×C4—24no—
209C32×C4—44no—
210C32×C4—121no—
211C32×C4C2×C4×D314yes—
212C32×C4C2×C4×D311yescharacteristic
213C32×C4#23514yes—
214C32×C4#23511yescharacteristic
215C3⋊Dic3—128no—
216C3⋊Dic3—121no—
217C3⋊Dic3—124no—
218C3×Dic3—44no—
219C2×D8—271no—
220C4×D4—31no—
221C4×D4—271no—
222#106—271no—
223#102—271no—
224#102—271no—
225#97—91no—
226C33#44211yescharacteristic, complemented, Sylow
227C23×C3—64no—
228C22×D3—21no—
229C22×D3—188no—
230C22×D3—184no—
231C22×D3—181no—
232C3×D4—38no—
233C3×D4—64no—
234C2×C3×C4—24no—
235C2×C3×C4—24no—
236C2×C3×C4—24no—
237C2×C3×C4—64no—
238C2×C3×C4—128no—
239C2×C3×C4#62614yes—
240C2×C3×C4#62614yes—
241C2×C3×C4#63811yescharacteristic
242C2×C3×C4#63811yescharacteristic
243C2×C3×C4C2×D3214yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245C2×Dic3—21no—
246C2×Dic3—188no—
247C2×Dic3—184no—
248C2×Dic3—181no—
249D12—21no—
250D12—98no—
251D12—92no—
252D12—188no—
253D12—188no—
254D12—184no—
255D12—181no—
256D12C4×C3⋊D312yescomplemented
257C4×D3—42no—
258C4×D3—3616no—
259C4×D3—368no—
260C4×D3—362no—
261C3×C8—92no—
262C3×C8—181no—
263C3⋊C8—38no—
264C3⋊C8—64no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—122no—
269C2×C32C4×D1211yescharacteristic
270C2×C32#90811yescharacteristic
271C2×C32#90811yescharacteristic
272C2×C32C4×#6714yes—
273C2×C32#93214yes—
274C2×C32#94914yes—
275C3⋊D3—1216no—
276C3⋊D3—122no—
277C3⋊D3—128no—
278C3×D3—48no—
279C2×D4—31no—
280C2×D4—271no—
281C22×C4—61no—
282C22×C4—541no—
283D8—274no—
284C2×C8—91no—
285C2×C8—91no—
286C4⋊C4—31no—
287C4⋊C4—271no—
288C22⋊C4—61no—
289C22⋊C4—541no—
290C42D3×C3⋊D311yescharacteristic
291C22×C3—24no—
292C22×C3—68no—
293C22×C3—124no—
294C22×C3—124no—
295C22×C3C2×#62614yes—
296C22×C3C2×#63811yescharacteristic
297C2×D3—22no—
298C2×D3—41no—
299C2×D3—41no—
300C2×D3—1816no—
301C2×D3—182no—
302C2×D3—188no—
303C2×D3—368no—
304C2×D3—368no—
305C2×D3—364no—
306C2×D3—364no—
307C2×D3—361no—
308C2×D3—361no—
309C3×C4—28no—
310C3×C4—24no—
311C3×C4—24no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—24no—
315C3×C4—24no—
316C3×C4—44no—
317C3×C4—124no—
318C3×C4#379814yes—
319C3×C4C4×D3214yes—
320C3×C4C2×C4×C3⋊D311yescharacteristic
321C3×C4#382811yescharacteristic
322Dic3—41no—
323Dic3—368no—
324Dic3—364no—
325Dic3—361no—
326C32—24no—
327C32#471111yescharacteristic, complemented
328C32#501414yescomplemented
329C23—61no—
330C23—541no—
331D4—32no—
332D4—61no—
333D4—272no—
334D4—541no—
335C2×C4—61no—
336C2×C4—122no—
337C2×C4—541no—
338C2×C4—1082no—
339C2×C4#629111yescharacteristic
340C2×C4#629111yescharacteristic
341C2×C4C2×D3×C3⋊D311yescharacteristic
342C8—92no—
343C8—181no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—24no—
347C2×C3—128no—
348C2×C3#6382514yes—
349C2×C3#6383014yes—
350C2×C3C4×#62614yes—
351C2×C3C4×#63811yescharacteristic
352C2×C3#6414111yescharacteristic
353C2×C3#6415811yescharacteristic
354D3—42no—
355D3—3616no—
356D3—368no—
357D3—362no—
358C22—62no—
359C22—121no—
360C22—121no—
361C22—542no—
362C22—1081no—
363C22—1081no—
364C22C2×#629111yescharacteristic
365C4—21no—
366C4—21no—
367C4—121no—
368C4—1081no—
369C4#8902011yescharacteristic
370C4C4×D3×C3⋊D311yescharacteristic
371C3—24no—
372C3#1059034614yescomplemented
373C3#1059271311yescharacteristic, complemented
374C2—122no—
375C2—1082no—
376C2#1172336711yescharacteristic
377C2#1172337811yescharacteristic
378C2C4×#629111yescharacteristic
subgroupG/H
centerC22C2×#6291
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×#88790C2
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#442
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊(C3⋊(C3⋊D3))C2semidirect
2. C32×C4×D12C2semidirect
4. C3×#63630C2semidirect
5. #11722054C2semidirect
6. #11722035C2semidirect
7. #11722032C2semidirect
21. #88790C4semidirect
27. C3×C4×D12D3semidirect
32. #63630D3semidirect
44. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
48. C32×D12C2×C4semidirect
51. C3×C3⋊(C3⋊C8)C2×C4semidirect
102. C4×D12C3⋊D3semidirect
143. C3×D12C4×D3semidirect
146. C3⋊(C3⋊C8)C4×D3semidirect
226. C33#442semidirect
256. D12C4×C3⋊D3semidirect
327. C32#4711semidirect
328. C32#5014semidirect
372. C3#10590346semidirect
373. C3#10592713semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C3⋊(C3⋊D3))2
C32×C4×D122
C2×#887902
C3×#636302
#117220542
#117220352
#117220322
#105927133
#105903463