Finite Group Databasebeta

← #13159277

Subgroups — #13159277

Order 1728.

1048 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1048 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#11723928C211yesmaximal, characteristic, complemented
2#11723529C211yesmaximal, characteristic
3C2×#88796C211yesmaximal, characteristic, complemented
4C2×#88794C211yesmaximal, characteristic, complemented
5C3×#63636C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7#11722050C211yesmaximal, characteristic, complemented
8#10592461—31nomaximal
9#10590553—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6290—21no—
14C2×#6290—21no—
15C33×C4⋊C4C2211yescharacteristic
16#88857—21no—
17C4×C3⋊(C3⋊Dic3)—21no—
18C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88796—22no—
22#88794—22no—
23#64149—31no—
24C2×#3754—31no—
25C2×#3744—31no—
26#64096—31no—
27#64096—34no—
28#64096—64no—
29#63914—34no—
30C2×#3716—34no—
31C2×#3716—34no—
32C32×#102—31no—
33#63641—31no—
34#63636D311yescharacteristic, complemented
35C3×#809—34no—
36C3×#809D314yescomplemented
37#63572—34no—
38C22×C3×C3⋊D3—21no—
39C22×C32×D3—21no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C2×C3⋊(C3⋊Dic3)D411yescharacteristic
45#6308—21no—
46#6308D412yescomplemented
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C32×D12—21no—
50C32×D12—21no—
51#6290—42no—
52#6290—42no—
53C3×C3⋊(C3⋊C8)—21no—
54C32×C3⋊C8—21no—
55#4762—94no—
56#4762—91no—
57C2×C32×D4—31no—
58C2×#638—61no—
59C2×#638—61no—
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×#634—31no—
62C2×#634—34no—
63C2×#634—64no—
64C2×C3×D12—34no—
65C2×C3×D12C2×D314yes—
66C2×#625—64no—
67C2×#625—64no—
68C2×C32×C8—31no—
69C32×C4⋊C4—24no—
70C32×C4⋊C4C2×D314yes—
71C32×C4⋊C4C2×D311yescharacteristic
72#3754—62no—
73#3750—61no—
74#3750—64no—
75#3750—68no—
76C4×C3⋊Dic3—31no—
77C4×C3⋊Dic3—34no—
78C4×C3⋊Dic3—61no—
79C4×C3⋊Dic3—64no—
80C4×C3⋊Dic3—64no—
81C4×C3⋊Dic3—68no—
82C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
83#3744—62no—
84C2×C3×C3⋊C8—34no—
85C2×C3×C3⋊C8C2×D314yes—
86#3716—68no—
87#3716—68no—
88C22×C33C2×D411yescharacteristic
89C2×C3×C3⋊D3—41no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C2×C32×D3—41no—
93C2×C32×D3—41no—
94C2×C32×D3—41no—
95C33×C4—41no—
96C33×C4C2×D411yescharacteristic
97C33×C4#4111yescharacteristic
98C3⋊(C3⋊Dic3)—22no—
99C3⋊(C3⋊Dic3)—21no—
100C3⋊(C3⋊Dic3)SD1612yescomplemented
101#940—94no—
102#940—91no—
103C2×#214—94no—
104C2×#214—91no—
105C2×#204—94no—
106C2×#204—91no—
107#887—94no—
108#887—91no—
109#887—184no—
110C3×#102—34no—
111C3×#102—91no—
112Dic3⋊C8—94no—
113Dic3⋊C8—91no—
114#809—34no—
115#809C3⋊D311yescharacteristic, complemented
116C23×C32—61no—
117C22×C3⋊D3—21no—
118C22×C3×D3—24no—
119C22×C3×D3—64no—
120C32×D4—61no—
121C32×D4—61no—
122C2×C32×C4—28no—
123C2×C32×C4—24no—
124C2×C32×C4—21no—
125C2×C32×C4—24no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D311yescharacteristic
128#638—122no—
129#638—122no—
130C2×C3⋊Dic3—31no—
131C2×C3⋊Dic3—31no—
132C2×C3⋊Dic3—34no—
133C2×C3⋊Dic3—34no—
134C2×C3⋊Dic3—64no—
135C2×C3⋊Dic3—64no—
136C2×C3⋊Dic3—68no—
137C2×C3⋊Dic3—64no—
138C2×C3⋊Dic3—61no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)—21no—
141#634—32no—
142#634—38no—
143#634—68no—
144#634—68no—
145#634—64no—
146#634—61no—
147C3×D12—24no—
148C3×D12—24no—
149C3×D12—64no—
150C3×D12—64no—
151#625—128no—
152#625—128no—
153C32×C8—61no—
154C3⋊(C3⋊C8)—21no—
155C3×C3⋊C8—24no—
156C3×C3⋊C8—64no—
157#460—271noSylow
158C2×C33#12111yescharacteristic
159C2×C33C2×SD1611yescharacteristic
160C2×C33AGL(1,ℤ/8)11yescharacteristic
161C3×C3⋊D3—81no—
162C32×D3—81no—
163C2×C3×D4—34no—
164C2×C3×D4—91no—
165C2×#67—181no—
166C2×#67—181no—
167C2×#67—184no—
168C2×#67—184no—
169C2×D12—34no—
170C2×D12C2×C3⋊D311yescharacteristic
171C2×Dic6—94no—
172C2×Dic6—91no—
173C2×Dic6—184no—
174C2×C3×C8—34no—
175C2×C3×C8—91no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4D3214yes—
178C3×C4⋊C4C2×C3⋊D311yescharacteristic
179#214—182no—
180#214—188no—
181Dic3⋊C4—184no—
182Dic3⋊C4—181no—
183Dic3⋊C4—188no—
184C4×Dic3—94no—
185C4×Dic3—91no—
186C4×Dic3—184no—
187C4×Dic3—184no—
188C4×Dic3—181no—
189C4×Dic3—188no—
190C2×C3⋊C8—34no—
191C2×C3⋊C8C2×C3⋊D311yescharacteristic
192#204—188no—
193#204—182no—
194C22×C32—24no—
195C22×C32—121no—
196C22×C32—121no—
197C22×C32—121no—
198C22×C32D3×D414yes—
199C22×C32D3×D411yescharacteristic
200C2×C3⋊D3—41no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3×D3—44no—
204C2×C3×D3—44no—
205C2×C3×D3—44no—
206C2×C3×D3—124no—
207C2×C3×D3—124no—
208C2×C3×D3—124no—
209C32×C4—24no—
210C32×C4—24no—
211C32×C4—48no—
212C32×C4—44no—
213C32×C4—41no—
214C32×C4#23514yes—
215C32×C4#23511yescharacteristic
216C32×C4D3×D414yes—
217C32×C4D3×D411yescharacteristic
218C3⋊Dic3—32no—
219C3⋊Dic3—38no—
220C3⋊Dic3—68no—
221C3⋊Dic3—616no—
222C3⋊Dic3—68no—
223C3⋊Dic3—62no—
224C3⋊Dic3—61no—
225C3⋊Dic3—64no—
226C3⋊Dic3—124no—
227C2×SD16—271no—
228C2×SD16—271no—
229#127—271no—
230C4×Q8—271no—
231C4⋊C8—271no—
232#102—31no—
233#102—91no—
234C33#46011yescharacteristic, complemented, Sylow
235C23×C3—64no—
236C23×C3—181no—
237C22×D3—21no—
238C22×D3—64no—
239C3×D4—64no—
240C3×D4—64no—
241C3×D4—181no—
242C3×D4—181no—
243C2×C3×C4—28no—
244C2×C3×C4—24no—
245C2×C3×C4—21no—
246C2×C3×C4—24no—
247C2×C3×C4C2×D3214yes—
248C2×C3×C4C22×C3⋊D311yescharacteristic
249#67—362no—
250#67—368no—
251#67—362no—
252#67—368no—
253C2×Dic3—94no—
254C2×Dic3—91no—
255C2×Dic3—94no—
256C2×Dic3—91no—
257C2×Dic3—184no—
258C2×Dic3—184no—
259C2×Dic3—188no—
260C2×Dic3—184no—
261C2×Dic3—181no—
262D12—21no—
263D12—21no—
264D12—64no—
265D12—64no—
266Dic6—98no—
267Dic6—92no—
268Dic6—188no—
269Dic6—188no—
270Dic6—184no—
271Dic6—181no—
272C3×C8—64no—
273C3×C8—181no—
274C3⋊C8—21no—
275C3⋊C8—64no—
276C2×C32—24no—
277C2×C32—24no—
278C2×C32—24no—
279C2×C32—241no—
280C2×C32#88414yes—
281C2×C32#88411yescharacteristic
282C2×C32#91114yes—
283C2×C32#91111yescharacteristic
284C2×C32D3×SD1614yes—
285C2×C32D3×SD1611yescharacteristic
286C3⋊D3—81no—
287C3×D3—84no—
288C3×D3—244no—
289C2×Q8—271no—
290C2×D4—31no—
291C2×D4—91no—
292C2×D4—541no—
293C2×D4—541no—
294SD16—542no—
295SD16—542no—
296C2×C8—31no—
297C2×C8—91no—
298C4⋊C4—541no—
299C4⋊C4D3×C3⋊D311yescharacteristic
300C42—271no—
301C42—541no—
302C22×C3—24no—
303C22×C3—124no—
304C22×C3—124no—
305C22×C3—124no—
306C22×C3—361no—
307C22×C3—361no—
308C22×C3—361no—
309C22×C3#381114yes—
310C22×C3D4×C3⋊D311yescharacteristic
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—124no—
315C2×D3—124no—
316C2×D3—124no—
317C3×C4—24no—
318C3×C4—24no—
319C3×C4—48no—
320C3×C4—41no—
321C3×C4—44no—
322C3×C4#379814yes—
323C3×C4#380214yes—
324C3×C4#382811yescharacteristic
325C3×C4D4×C3⋊D311yescharacteristic
326Dic3—92no—
327Dic3—98no—
328Dic3—188no—
329Dic3—1816no—
330Dic3—182no—
331Dic3—181no—
332Dic3—188no—
333Dic3—184no—
334Dic3—364no—
335C32—24no—
336C32#476214yescomplemented
337C32#476211yescharacteristic, complemented
338C23—61no—
339C23—181no—
340Q8—272no—
341Q8—541no—
342D4—61no—
343D4—61no—
344D4—181no—
345D4—181no—
346D4—1082no—
347D4—1082no—
348C2×C4—21no—
349C2×C4—271no—
350C2×C4—271no—
351C2×C4—541no—
352C2×C4C2×D3×C3⋊D311yescharacteristic
353C8—61no—
354C8—181no—
355C2×C3—24no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—244no—
359C2×C3—721no—
360C2×C3#6391214yes—
361C2×C3#6392714yes—
362C2×C3#6394014yes—
363C2×C3#6409311yescharacteristic
364C2×C3#6412011yescharacteristic
365C2×C3SD16×C3⋊D311yescharacteristic
366D3—81no—
367D3—244no—
368C22—121no—
369C22—121no—
370C22—121no—
371C22—361no—
372C22—361no—
373C22—361no—
374C22#8903911yescharacteristic
375C4—41no—
376C4—272no—
377C4—542no—
378C4—541no—
379C4#8902011yescharacteristic
380C4#8902611yescharacteristic
381C3—24no—
382C3#1059055314yescomplemented
383C3#1059246111yescharacteristic, complemented
384C2—241no—
385C2—721no—
386C2#1172352711yescharacteristic
387C2#1172354811yescharacteristic
388C2#1172357111yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723529C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#460
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. #11723928C2semidirect
3. C2×#88796C2semidirect
4. C2×#88794C2semidirect
5. C3×#63636C2semidirect
6. C32×#809C2semidirect
7. #11722050C2semidirect
34. #63636D3semidirect
36. C3×#809D3semidirect
46. #6308D4semidirect
100. C3⋊(C3⋊Dic3)SD16semidirect
115. #809C3⋊D3semidirect
234. C33#460semidirect
336. C32#4762semidirect
337. C32#4762semidirect
382. C3#10590553semidirect
383. C3#10592461semidirect

Maximal subgroups

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

subgroupindex
#117239282
#117235292
C2×#887962
C2×#887942
C3×#636362
C32×#8092
#117220502
#105924613
#105905533