Finite Group Databasebeta

← #13159327

Subgroups — #13159327

Order 1728.

1004 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 168 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
1C32×#895C211yesmaximal, characteristic, complemented
2#11723516C211yesmaximal, characteristic, complemented
3C2×#88790C211yesmaximal, characteristic, complemented
4C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5#11722179C211yesmaximal, characteristic, complemented
6C3×#63634C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592737—31nomaximal
9#10590576—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C32×D12—21no—
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×#6291—21no—
15C33×C4⋊C4C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×(C2×D3)⋊C4—21no—
20#88804—21no—
21#88790—22no—
22D3×C3⋊(C3⋊C8)—22no—
23C32×#121—31no—
24C2×#3753—31no—
25#64139—31no—
26C22×C3⋊(C3⋊C8)—31no—
27C3×#895D314yescomplemented
28#63909—34no—
29C2×#3714—34no—
30C2×D3×C3⋊C8—34no—
31#63659—31no—
32#63636—31no—
33#63636—34no—
34#63636—64no—
35#63634D311yescharacteristic, complemented
36C3×#807—34no—
37#63562—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C32×D3—21no—
40C22×C32×D3D411yescharacteristic
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C32×Dic3D411yescharacteristic
47C32×D12—21no—
48C32×D12—21no—
49C32×D12—42no—
50C32×C4×D3—21no—
51C32×C4×D3—21no—
52#6291—42no—
53C3⋊(C3⋊(C3⋊C8))—21no—
54C3×C3⋊(C3⋊C8)D412yescomplemented
55#5038—94no—
56#4883—91no—
57C2×C32×D4—31no—
58C2×C32×D4—61no—
59C22×C32×C4—31no—
60C2×#638—61no—
61C2×C4⋊(C3⋊D3)—31no—
62C2×C4⋊(C3⋊D3)—34no—
63C2×C4⋊(C3⋊D3)—64no—
64C2×C3×D12—24no—
65C2×C3×D12C2×D314yes—
66C2×C3×C4×D3C2×D314yes—
67C2×#626—64no—
68C32×C4⋊C4—24no—
69C32×C4⋊C4C2×D314yes—
70C32×C4⋊C4C2×D311yescharacteristic
71C32×C22⋊C4—61no—
72#3753—62no—
73#3752—61no—
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—62no—
78C2×C3⋊(C3⋊C8)—64no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80C3×(C2×D3)⋊C4—24no—
81C3×C4⋊Dic3—34no—
82C2×C3×C3⋊C8—34no—
83#3722—64no—
84#3714—68no—
85D3×C3⋊C8—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—41no—
90C2×C32×D3—21no—
91C2×C32×D3—21no—
92C2×C32×D3—41no—
93C2×C32×D3—41no—
94C2×C32×D3—41no—
95C2×C32×D3D812yescomplemented
96C33×C4—41no—
97C33×C4C2×D411yescharacteristic
98C33×C4#4111yescharacteristic
99C3×C3⋊Dic3—41no—
100C32×Dic3—21no—
101C3×#121—34no—
102C2×#213—94no—
103#930—94no—
104C22×C3⋊C8—94no—
105C2×D24—91no—
106C2×D3×C8—91no—
107#895—91no—
108#895C3⋊D311yescharacteristic, complemented
109C3×#107—91no—
110#832—94no—
111#809—91no—
112#809—94no—
113#809—91no—
114#809—184no—
115#807—34no—
116C23×C32—31no—
117C23×C32—61no—
118C22×C3⋊D3—61no—
119C22×C3⋊D3—64no—
120C22×C3⋊D3—124no—
121C22×C3×D3—24no—
122C22×C3×D3#6714yes—
123C32×D4—61no—
124C32×D4—61no—
125C32×D4—122no—
126C2×C32×C4—21no—
127C2×C32×C4—24no—
128C2×C32×C4—24no—
129C2×C32×C4—31no—
130C2×C32×C4—44no—
131C2×C32×C4—61no—
132C2×C32×C4—61no—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C22×D311yescharacteristic
135#638—122no—
136C2×C3⋊Dic3—21no—
137C4⋊(C3⋊D3)—68no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—64no—
140C4⋊(C3⋊D3)—61no—
141C4⋊(C3⋊D3)—64no—
142C4⋊(C3⋊D3)—124no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3#6714yes—
145C3×D12—24no—
146C3×D12—24no—
147C3×D12—48no—
148C3×C4×D3—24no—
149C3×C4×D3—24no—
150#626—128no—
151C3⋊(C3⋊C8)—61no—
152C3⋊(C3⋊C8)—64no—
153C3⋊(C3⋊C8)—124no—
154C3⋊(C3⋊C8)D1212yescomplemented
155C3×C3⋊C8—38no—
156#466—271noSylow
157C2×C33#12111yescharacteristic
158C2×C33C2×D811yescharacteristic
159C2×C33#13511yescharacteristic
160C3⋊(C3⋊D3)—81no—
161C32×D3—22no—
162C32×D3—81no—
163C2×C3×D4—34no—
164C2×C3×D4—64no—
165C22×C3×C4—34no—
166C2×#67—184no—
167C2×D12—21no—
168C2×D12—94no—
169C2×D12—91no—
170C2×D12—184no—
171C2×D12—181no—
172C2×D12C2×C3⋊D311yescharacteristic
173C2×C4×D3C2×C3⋊D311yescharacteristic
174C2×C3×C8—91no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—91no—
177C3×C4⋊C4D3214yes—
178C3×C4⋊C4C2×C3⋊D311yescharacteristic
179C3×C22⋊C4—64no—
180#213—188no—
181(C2×D3)⋊C4—21no—
182(C2×D3)⋊C4—184no—
183(C2×D3)⋊C4—181no—
184C4⋊Dic3—34no—
185C2×C3⋊C8—34no—
186C2×C3⋊C8—94no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—184no—
189C2×C3⋊C8—188no—
190D24—182no—
191D3×C8—182no—
192C22×C32—24no—
193C22×C32—32no—
194C22×C32—61no—
195C22×C32—61no—
196C22×C32—121no—
197C22×C32—121no—
198C22×C32—121no—
199C22×C32D3×D411yescharacteristic
200C22×C32C2×#6714yes—
201C2×C3⋊D3—128no—
202C2×C3⋊D3—121no—
203C2×C3⋊D3—121no—
204C2×C3⋊D3—121no—
205C2×C3⋊D3—124no—
206C2×C3⋊D3—124no—
207C2×C3⋊D3—124no—
208C2×C3⋊D3—244no—
209C2×C3⋊D3—244no—
210C2×C3×D3—24no—
211C2×C3×D3—24no—
212C2×C3×D3—44no—
213C2×C3×D3—44no—
214C2×C3×D3—44no—
215C2×C3×D3#21318yescomplemented
216C32×C4—24no—
217C32×C4—24no—
218C32×C4—41no—
219C32×C4—44no—
220C32×C4—61no—
221C32×C4—84no—
222C32×C4C2×D1211yescharacteristic
223C32×C4D3×D414yes—
224C32×C4#23714yes—
225C32×C4#23911yescharacteristic
226C3⋊Dic3—41no—
227C3×Dic3—24no—
228C3×Dic3—124no—
229C2×D8—271no—
230C22×C8—271no—
231#121—31no—
232#121—271no—
233#107—91no—
234#102—271no—
235#102—271no—
236C33#46611yescharacteristic, complemented, Sylow
237C23×C3—34no—
238C23×C3—64no—
239C22×D3—21no—
240C22×D3—184no—
241C22×D3—181no—
242C22×D3—364no—
243C22×D3#63811yescharacteristic
244C3×D4—64no—
245C3×D4—64no—
246C3×D4—128no—
247C2×C3×C4—24no—
248C2×C3×C4—21no—
249C2×C3×C4—24no—
250C2×C3×C4—34no—
251C2×C3×C4—44no—
252C2×C3×C4—64no—
253C2×C3×C4—64no—
254C2×C3×C4—181no—
255C2×C3×C4C2×D3214yes—
256C2×C3×C4C22×C3⋊D311yescharacteristic
257#67—368no—
258C2×Dic3—64no—
259C2×Dic3#63811yescharacteristic
260D12—21no—
261D12—21no—
262D12—42no—
263D12—188no—
264D12—184no—
265D12—184no—
266D12—181no—
267D12—181no—
268D12—364no—
269D12—362no—
270C4×D3—21no—
271C4×D3—21no—
272C3×C8—92no—
273C3⋊C8—38no—
274C3⋊C8—184no—
275C3⋊C8—181no—
276C3⋊C8—364no—
277C2×C32—24no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—62no—
281C2×C32—241no—
282C2×C32#89511yescharacteristic
283C2×C32D3×D811yescharacteristic
284C2×C32#91911yescharacteristic
285C2×C32C2×#21314yes—
286C2×C32#93914yes—
287C2×C32#95014yes—
288C3⋊D3—248no—
289C3⋊D3—241no—
290C3⋊D3—244no—
291C3×D3—28no—
292C3×D3—84no—
293C2×D4—31no—
294C2×D4—61no—
295C2×D4—271no—
296C2×D4—541no—
297C22×C4—31no—
298D8—542no—
299C2×C8—91no—
300C2×C8—271no—
301C2×C8—542no—
302C4⋊C4—91no—
303C4⋊C4D3×C3⋊D311yescharacteristic
304C22⋊C4—61no—
305C22⋊C4—541no—
306C22×C3—24no—
307C22×C3—38no—
308C22×C3—64no—
309C22×C3—64no—
310C22×C3—124no—
311C22×C3—124no—
312C22×C3—124no—
313C22×C3D3×#6714yes—
314C22×C3C2×#63811yescharacteristic
315C2×D3—21no—
316C2×D3—21no—
317C2×D3—41no—
318C2×D3—41no—
319C2×D3—41no—
320C2×D3—368no—
321C2×D3—361no—
322C2×D3—364no—
323C2×D3—361no—
324C2×D3—361no—
325C2×D3—364no—
326C2×D3—364no—
327C2×D3—724no—
328C2×D3—724no—
329C2×D3#375312yescomplemented
330C3×C4—24no—
331C3×C4—24no—
332C3×C4—41no—
333C3×C4—44no—
334C3×C4—64no—
335C3×C4—84no—
336C3×C4—361no—
337C3×C4#379614yes—
338C3×C4D3×D1214yes—
339C3×C4D4×C3⋊D311yescharacteristic
340C3×C4#383011yescharacteristic
341Dic3—21no—
342Dic3—124no—
343C32—24no—
344C32#488311yescharacteristic, complemented
345C32#503814yescomplemented
346C23—31no—
347C23—61no—
348C23—541no—
349D4—61no—
350D4—61no—
351D4—122no—
352D4—541no—
353D4—541no—
354D4—1082no—
355C2×C4—21no—
356C2×C4—31no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—181no—
360C2×C4C2×D3×C3⋊D311yescharacteristic
361C8—92no—
362C8—541no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—24no—
366C2×C3—68no—
367C2×C3—244no—
368C2×C3#6388614yes—
369C2×C3D3×#21314yes—
370C2×C3#6394914yes—
371C2×C3C2×#375311yescharacteristic
372C2×C3#6414811yescharacteristic
373C2×C3#6415911yescharacteristic
374D3—22no—
375D3—81no—
376D3—728no—
377D3—724no—
378D3—721no—
379C22—32no—
380C22—61no—
381C22—61no—
382C22—121no—
383C22—121no—
384C22—121no—
385C22—1081no—
386C22—1081no—
387C22—1081no—
388C22D3×#63811yescharacteristic
389C4—41no—
390C4—61no—
391C4—361no—
392C4#8901511yescharacteristic
393C4C3⋊D3×D1211yescharacteristic
394C3—24no—
395C3#1059057614yescomplemented
396C3#1059273711yescharacteristic, complemented
397C2—62no—
398C2—241no—
399C2—2161no—
400C2#1172347511yescharacteristic
401C2D3×#375311yescharacteristic
402C2#1172358211yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#466
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C32×#895C2semidirect
2. #11723516C2semidirect
3. C2×#88790C2semidirect
4. C2×D3×C3⋊(C3⋊C8)C2semidirect
5. #11722179C2semidirect
6. C3×#63634C2semidirect
7. #11722031C2semidirect
27. C3×#895D3semidirect
35. #63634D3semidirect
54. C3×C3⋊(C3⋊C8)D4semidirect
95. C2×C32×D3D8semidirect
108. #895C3⋊D3semidirect
154. C3⋊(C3⋊C8)D12semidirect
215. C2×C3×D3#213semidirect
236. C33#466semidirect
329. C2×D3#3753semidirect
344. C32#4883semidirect
345. C32#5038semidirect
395. C3#10590576semidirect
396. C3#10592737semidirect

Maximal subgroups

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

subgroupindex
C32×#8952
#117235162
C2×#887902
C2×D3×C3⋊(C3⋊C8)2
#117221792
C3×#636342
#117220312
#105927373
#105905763