Finite Group Databasebeta

← #13159275

Subgroups — #13159275

Order 1728.

1216 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1216 conjugacy classes of subgroups — too many to draw. Only its 146 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#11723934C211yesmaximal, characteristic, complemented
2#11723529C211yesmaximal, characteristic, complemented
3C2×#88791C211yesmaximal, characteristic, complemented
4C2×#88790C211yesmaximal, characteristic, complemented
5C3×#63636C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7#11722050C211yesmaximal, characteristic, complemented
8#10592485—31nomaximal
9#10590551—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6290—21no—
15C2×#6290—21no—
16C33×C4⋊C4C2211yescharacteristic
17#88859—21no—
18C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88791—22no—
22#88790—22no—
23#64149—31no—
24C2×#3753—31no—
25C2×#3745—31no—
26#64102—31no—
27#64102—34no—
28#64102—64no—
29#63914—34no—
30C2×#3714—34no—
31C2×#3714—34no—
32C32×#102—31no—
33#63641—31no—
34#63636D311yescharacteristic, complemented
35C3×#809—34no—
36C3×#809D314yescomplemented
37#63572—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3—21no—
40C22×C32×D3—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45C4⋊(C3⋊(C3⋊D3))—21no—
46C4⋊(C3⋊(C3⋊D3))D412yescomplemented
47C4×C3⋊(C3⋊D3)—42no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C32×D12—21no—
51C32×D12—21no—
52#6290—42no—
53#6290—42no—
54C3×C3⋊(C3⋊C8)—21no—
55C32×C3⋊C8—21no—
56#4786—94no—
57#4786—91no—
58C2×C32×D4—31no—
59C2×#638—61no—
60C2×#638—61no—
61C2×C4⋊(C3⋊D3)—31no—
62C2×C4⋊(C3⋊D3)—34no—
63C2×C4⋊(C3⋊D3)—64no—
64C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
65C2×C4×C3⋊D3—61no—
66C2×C4×C3⋊D3—64no—
67C2×C4×C3⋊D3—68no—
68C2×C3×D12—34no—
69C2×C3×D12C2×D314yes—
70C2×#625—64no—
71C2×#625—64no—
72C2×C32×C8—31no—
73C32×C4⋊C4—24no—
74C32×C4⋊C4C2×D314yes—
75C32×C4⋊C4C2×D311yescharacteristic
76#3753—62no—
77#3752—61no—
78#3752—64no—
79#3752—68no—
80C4×C3⋊Dic3—31no—
81C4×C3⋊Dic3—34no—
82C4×C3⋊Dic3—64no—
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84#3745—62no—
85C2×C3×C3⋊C8—34no—
86C2×C3×C3⋊C8C2×D314yes—
87#3714—68no—
88#3714—68no—
89C22×C33C2×D411yescharacteristic
90C2×C3⋊(C3⋊D3)—22no—
91C2×C3⋊(C3⋊D3)—41no—
92C2×C3⋊(C3⋊D3)—41no—
93C2×C3×C3⋊D3—41no—
94C2×C3×C3⋊D3—41no—
95C2×C3×C3⋊D3—41no—
96C2×C32×D3—41no—
97C2×C32×D3—41no—
98C2×C32×D3—41no—
99C33×C4—41no—
100C33×C4C2×D411yescharacteristic
101C33×C4#4111yescharacteristic
102C3⋊(C3⋊Dic3)—21no—
103C3⋊(C3⋊Dic3)D812yescomplemented
104#940—94no—
105#940—91no—
106C2×#213—94no—
107C2×#213—91no—
108C2×D24—94no—
109C2×D24—91no—
110#893—94no—
111#893—91no—
112#893—184no—
113C3×#102—34no—
114C3×#102—91no—
115Dic3⋊C8—94no—
116Dic3⋊C8—91no—
117#809—34no—
118#809C3⋊D311yescharacteristic, complemented
119C23×C32—61no—
120C22×C3⋊D3—21no—
121C22×C3⋊D3—68no—
122C22×C3⋊D3—64no—
123C22×C3⋊D3—61no—
124C22×C3×D3—24no—
125C22×C3×D3—64no—
126C32×D4—61no—
127C32×D4—61no—
128C2×C32×C4—28no—
129C2×C32×C4—24no—
130C2×C32×C4—21no—
131C2×C32×C4—24no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D311yescharacteristic
134#638—122no—
135#638—122no—
136C2×C3⋊Dic3—31no—
137C2×C3⋊Dic3—31no—
138C2×C3⋊Dic3—34no—
139C2×C3⋊Dic3—34no—
140C2×C3⋊Dic3—64no—
141C2×C3⋊Dic3—64no—
142C4⋊(C3⋊D3)—21no—
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)—32no—
145C4⋊(C3⋊D3)—38no—
146C4⋊(C3⋊D3)—68no—
147C4⋊(C3⋊D3)—68no—
148C4⋊(C3⋊D3)—64no—
149C4⋊(C3⋊D3)—61no—
150C4×C3⋊D3—122no—
151C4×C3⋊D3—128no—
152C4×C3⋊D3—1216no—
153C3×D12—24no—
154C3×D12—24no—
155C3×D12—64no—
156C3×D12—64no—
157#625—128no—
158#625—128no—
159C32×C8—61no—
160C3⋊(C3⋊C8)—21no—
161C3×C3⋊C8—24no—
162C3×C3⋊C8—64no—
163#459—271noSylow
164C2×C33#12111yescharacteristic
165C2×C33C2×D811yescharacteristic
166C2×C33AGL(1,ℤ/8)11yescharacteristic
167C3⋊(C3⋊D3)—42no—
168C3×C3⋊D3—81no—
169C32×D3—81no—
170C2×C3×D4—34no—
171C2×C3×D4—91no—
172C2×#67—181no—
173C2×#67—181no—
174C2×#67—184no—
175C2×#67—184no—
176C2×D12—34no—
177C2×D12—94no—
178C2×D12—91no—
179C2×D12—184no—
180C2×D12C2×C3⋊D311yescharacteristic
181C2×C4×D3—184no—
182C2×C4×D3—181no—
183C2×C4×D3—188no—
184C2×C3×C8—34no—
185C2×C3×C8—91no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4D3214yes—
188C3×C4⋊C4C2×C3⋊D311yescharacteristic
189#213—182no—
190#213—188no—
191(C2×D3)⋊C4—184no—
192(C2×D3)⋊C4—181no—
193(C2×D3)⋊C4—188no—
194C4×Dic3—94no—
195C4×Dic3—91no—
196C4×Dic3—184no—
197C2×C3⋊C8—34no—
198C2×C3⋊C8C2×C3⋊D311yescharacteristic
199D24—188no—
200D24—182no—
201C22×C32—24no—
202C22×C32—121no—
203C22×C32—121no—
204C22×C32—121no—
205C22×C32D3×D414yes—
206C22×C32D3×D411yescharacteristic
207C2×C3⋊D3—41no—
208C2×C3⋊D3—41no—
209C2×C3⋊D3—41no—
210C2×C3⋊D3—616no—
211C2×C3⋊D3—68no—
212C2×C3⋊D3—62no—
213C2×C3⋊D3—128no—
214C2×C3⋊D3—128no—
215C2×C3⋊D3—124no—
216C2×C3⋊D3—121no—
217C2×C3⋊D3—124no—
218C2×C3⋊D3—121no—
219C2×C3×D3—44no—
220C2×C3×D3—44no—
221C2×C3×D3—44no—
222C2×C3×D3—124no—
223C2×C3×D3—124no—
224C2×C3×D3—124no—
225C32×C4—24no—
226C32×C4—24no—
227C32×C4—48no—
228C32×C4—44no—
229C32×C4—41no—
230C32×C4#23514yes—
231C32×C4#23511yescharacteristic
232C32×C4D3×D414yes—
233C32×C4D3×D411yescharacteristic
234C3⋊Dic3—32no—
235C3⋊Dic3—38no—
236C3⋊Dic3—68no—
237C3⋊Dic3—61no—
238C3⋊Dic3—64no—
239C3⋊Dic3—124no—
240C2×D8—271no—
241C2×D8—271no—
242#127—271no—
243C4×D4—271no—
244C4⋊C8—271no—
245#102—31no—
246#102—91no—
247C33#45911yescharacteristic, complemented, Sylow
248C23×C3—64no—
249C23×C3—181no—
250C22×D3—21no—
251C22×D3—64no—
252C22×D3—188no—
253C22×D3—184no—
254C22×D3—181no—
255C3×D4—64no—
256C3×D4—64no—
257C3×D4—181no—
258C3×D4—181no—
259C2×C3×C4—28no—
260C2×C3×C4—24no—
261C2×C3×C4—21no—
262C2×C3×C4—24no—
263C2×C3×C4C2×D3214yes—
264C2×C3×C4C22×C3⋊D311yescharacteristic
265#67—362no—
266#67—368no—
267#67—362no—
268#67—368no—
269C2×Dic3—94no—
270C2×Dic3—91no—
271C2×Dic3—94no—
272C2×Dic3—91no—
273C2×Dic3—184no—
274C2×Dic3—184no—
275D12—21no—
276D12—21no—
277D12—64no—
278D12—64no—
279D12—98no—
280D12—92no—
281D12—188no—
282D12—188no—
283D12—184no—
284D12—181no—
285C4×D3—362no—
286C4×D3—368no—
287C4×D3—3616no—
288C3×C8—64no—
289C3×C8—181no—
290C3⋊C8—21no—
291C3⋊C8—64no—
292C2×C32—24no—
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—241no—
296C2×C32#88414yes—
297C2×C32#88411yescharacteristic
298C2×C32D3×D814yes—
299C2×C32D3×D811yescharacteristic
300C2×C32#91414yes—
301C2×C32#91411yescharacteristic
302C3⋊D3—81no—
303C3⋊D3—1216no—
304C3⋊D3—128no—
305C3⋊D3—122no—
306C3×D3—84no—
307C3×D3—244no—
308C2×D4—31no—
309C2×D4—91no—
310C2×D4—271no—
311C2×D4—541no—
312C2×D4—541no—
313C22×C4—541no—
314D8—542no—
315D8—542no—
316C2×C8—31no—
317C2×C8—91no—
318C4⋊C4D3×C3⋊D311yescharacteristic
319C22⋊C4—541no—
320C42—271no—
321C22×C3—24no—
322C22×C3—124no—
323C22×C3—124no—
324C22×C3—124no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3—361no—
328C22×C3#381114yes—
329C22×C3D4×C3⋊D311yescharacteristic
330C2×D3—41no—
331C2×D3—41no—
332C2×D3—41no—
333C2×D3—124no—
334C2×D3—124no—
335C2×D3—124no—
336C2×D3—1816no—
337C2×D3—182no—
338C2×D3—188no—
339C2×D3—368no—
340C2×D3—368no—
341C2×D3—361no—
342C2×D3—361no—
343C2×D3—364no—
344C2×D3—364no—
345C3×C4—24no—
346C3×C4—24no—
347C3×C4—48no—
348C3×C4—41no—
349C3×C4—44no—
350C3×C4#379814yes—
351C3×C4#380214yes—
352C3×C4#382811yescharacteristic
353C3×C4D4×C3⋊D311yescharacteristic
354Dic3—92no—
355Dic3—98no—
356Dic3—188no—
357Dic3—181no—
358Dic3—184no—
359Dic3—364no—
360C32—24no—
361C32#478614yescomplemented
362C32#478611yescharacteristic, complemented
363C23—61no—
364C23—181no—
365C23—541no—
366D4—61no—
367D4—61no—
368D4—181no—
369D4—181no—
370D4—272no—
371D4—541no—
372D4—1082no—
373D4—1082no—
374C2×C4—21no—
375C2×C4—271no—
376C2×C4—271no—
377C2×C4—1082no—
378C2×C4C2×D3×C3⋊D311yescharacteristic
379C8—61no—
380C8—181no—
381C2×C3—24no—
382C2×C3—24no—
383C2×C3—24no—
384C2×C3—244no—
385C2×C3—721no—
386C2×C3#6391214yes—
387C2×C3#6392614yes—
388C2×C3#6394114yes—
389C2×C3#6409311yescharacteristic
390C2×C3D8×C3⋊D311yescharacteristic
391C2×C3#6412311yescharacteristic
392D3—81no—
393D3—244no—
394D3—3616no—
395D3—368no—
396D3—362no—
397C22—121no—
398C22—121no—
399C22—121no—
400C22—361no—
401C22—361no—
402C22—361no—
403C22—542no—
404C22—1081no—
405C22—1081no—
406C22#8903911yescharacteristic
407C4—41no—
408C4—272no—
409C4—541no—
410C4#8902011yescharacteristic
411C4#8902611yescharacteristic
412C3—24no—
413C3#1059055114yescomplemented
414C3#1059248511yescharacteristic, complemented
415C2—241no—
416C2—721no—
417C2—1082no—
418C2#1172352711yescharacteristic
419C2#1172354711yescharacteristic
420C2#1172357211yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#459
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. #11723934C2semidirect
2. #11723529C2semidirect
3. C2×#88791C2semidirect
4. C2×#88790C2semidirect
5. C3×#63636C2semidirect
6. C32×#809C2semidirect
7. #11722050C2semidirect
34. #63636D3semidirect
36. C3×#809D3semidirect
46. C4⋊(C3⋊(C3⋊D3))D4semidirect
103. C3⋊(C3⋊Dic3)D8semidirect
118. #809C3⋊D3semidirect
247. C33#459semidirect
361. C32#4786semidirect
362. C32#4786semidirect
413. C3#10590551semidirect
414. C3#10592485semidirect

Maximal subgroups

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

subgroupindex
#117239342
#117235292
C2×#887912
C2×#887902
C3×#636362
C32×#8092
#117220502
#105924853
#105905513