Finite Group Databasebeta

← #13161277

Subgroups — #13161277

Order 1728.

1016 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 154 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#11723987C211yesmaximal, characteristic, complemented
2C2×C32×#215C211yesmaximal, characteristic, complemented
3#11723531C211yesmaximal, characteristic
4C2×#88788C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722055C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10592873—31nomaximal
9#10591590—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6291—21no—
14#88859—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16#88857—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×#215—22no—
20C2×C32×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88788—22no—
23C2×C32×SD16—31no—
24#64155—31no—
25#64155—34no—
26#64155—64no—
27#64147—31no—
28C2×#3743—31no—
29C2×C3×#215D314yescomplemented
30#63916—34no—
31C2×#3712—34no—
32#63662D311yescharacteristic, complemented
33#63659—31no—
34#63645—31no—
35C3×#835—34no—
36#63575—34no—
37#63563—34no—
38C22×C3⋊(C3⋊D3)D411yescharacteristic
39C22×C32×D3—21no—
40C33×Q8—21no—
41C33×Q8—21no—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3⋊(C3⋊Dic3)D411yescharacteristic
46C4×C3⋊(C3⋊D3)—21no—
47C4×C3⋊(C3⋊D3)—21no—
48C2×C3×C3⋊Dic3—21no—
49C32×D12—21no—
50C32×D12—21no—
51#6291—42no—
52C3×C3⋊(C3⋊C8)—21no—
53C32×C3⋊C8D412yescomplemented
54#5174—94no—
55#4812—91no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×D4—31no—
60C2×#638—61no—
61C2×C4×C3⋊D3—31no—
62C2×C4×C3⋊D3—34no—
63C2×C4×C3⋊D3—64no—
64C2×C3×D12C2×D314yes—
65C2×#626—64no—
66C32×SD16—62no—
67C2×C32×C8—31no—
68#3757—61no—
69#3752—61no—
70#3752—64no—
71#3752—68no—
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—34no—
74C4⋊(C3⋊Dic3)—64no—
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76#3750—61no—
77#3750—64no—
78#3750—68no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80#3743—62no—
81C3×#215—28no—
82C3×C4⋊Dic3—34no—
83C2×C3×C3⋊C8—34no—
84C2×C3×C3⋊C8C2×D314yes—
85(C2×D3)⋊Dic3—64no—
86#3712—68no—
87C22×C33C2×D411yescharacteristic
88C2×C3⋊(C3⋊D3)—21no—
89C2×C3⋊(C3⋊D3)—21no—
90C2×C3⋊(C3⋊D3)—21no—
91C2×C32×D3—41no—
92C2×C32×D3—41no—
93C2×C32×D3—41no—
94C33×C4—41no—
95C33×C4C2×D411yescharacteristic
96C33×C4#4111yescharacteristic
97C3⋊(C3⋊Dic3)—21no—
98C3⋊(C3⋊Dic3)—41no—
99C3×C3⋊Dic3—41no—
100C2×C3×SD16—34no—
101#946—94no—
102#946—91no—
103#946—184no—
104C2×#215C3⋊D311yescharacteristic, complemented
105#938—94no—
106C2×ΓL(1,𝔽25)—94no—
107C2×ΓL(1,𝔽25)—91no—
108#895—91no—
109C3×#103—91no—
110#835—34no—
111#832—94no—
112#821—91no—
113#818—94no—
114#807—91no—
115C23×C32—61no—
116C22×C3⋊D3—31no—
117C22×C3⋊D3—34no—
118C22×C3⋊D3—64no—
119C22×C3×D3—24no—
120C32×Q8—21no—
121C32×Q8—21no—
122C32×Q8—24no—
123C32×Q8—24no—
124C32×Q8—44no—
125C32×Q8—44no—
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—
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—31no—
137C2×C3⋊Dic3—34no—
138C2×C3⋊Dic3—64no—
139C2×C3⋊Dic3—68no—
140C2×C3⋊Dic3—64no—
141C2×C3⋊Dic3—61no—
142C4×C3⋊D3—61no—
143C4×C3⋊D3—61no—
144C4×C3⋊D3—64no—
145C4×C3⋊D3—64no—
146C4×C3⋊D3—124no—
147C4×C3⋊D3—124no—
148C2×C3×Dic3—64no—
149C3×D12—24no—
150C3×D12—24no—
151#626—128no—
152C32×C8—32no—
153C3⋊(C3⋊C8)—21no—
154C3×C3⋊C8—64no—
155C3×C3⋊C8#6718yescomplemented
156#468—271noSylow
157C2×C33#12111yescharacteristic
158C2×C33AGL(1,ℤ/8)11yescharacteristic
159C2×C33#13711yescharacteristic
160C3⋊(C3⋊D3)—41no—
161C32×D3—81no—
162C2×C3×Q8—24no—
163C2×C3×Q8D3214yes—
164C2×C3×Q8C2×C3⋊D311yescharacteristic
165C2×C3×D4—34no—
166C2×#67—184no—
167C2×D12—181no—
168C2×D12C2×C3⋊D311yescharacteristic
169C2×C4×D3—94no—
170C2×C4×D3—91no—
171C2×C4×D3—184no—
172C3×SD16—68no—
173C2×C3×C8—34no—
174C2×C3×C8—91no—
175C3×C4⋊C4—91no—
176#217—184no—
177#215—22no—
178(C2×D3)⋊C4—181no—
179(C2×D3)⋊C4—184no—
180(C2×D3)⋊C4—181no—
181(C2×D3)⋊C4—188no—
182C4⋊Dic3—34no—
183C4⋊Dic3—94no—
184C4⋊Dic3—91no—
185C4⋊Dic3—184no—
186Dic3⋊C4—184no—
187Dic3⋊C4—181no—
188Dic3⋊C4—188no—
189C2×C3⋊C8—34no—
190C2×C3⋊C8C2×C3⋊D311yescharacteristic
191ΓL(1,𝔽25)—188no—
192ΓL(1,𝔽25)—182no—
193C22×C32—24no—
194C22×C32—121no—
195C22×C32—121no—
196C22×C32—121no—
197C22×C32D3×D414yes—
198C22×C32D3×D411yescharacteristic
199C2×C3⋊D3—68no—
200C2×C3⋊D3—61no—
201C2×C3⋊D3—61no—
202C2×C3⋊D3—64no—
203C2×C3⋊D3—61no—
204C2×C3⋊D3—64no—
205C2×C3⋊D3—64no—
206C2×C3⋊D3—124no—
207C2×C3⋊D3—124no—
208C2×C3×D3—44no—
209C2×C3×D3—44no—
210C2×C3×D3—44no—
211C32×C4—24no—
212C32×C4—24no—
213C32×C4—48no—
214C32×C4—41no—
215C32×C4—44no—
216C32×C4#23511yescharacteristic
217C32×C4D3×D411yescharacteristic
218C32×C4#23714yes—
219C32×C4C2×#6714yes—
220C3⋊Dic3—41no—
221C3⋊Dic3—61no—
222C3⋊Dic3—64no—
223C3⋊Dic3—124no—
224C3⋊Dic3—128no—
225C3⋊Dic3—124no—
226C3⋊Dic3—121no—
227C3×Dic3—124no—
228C2×SD16—31no—
229C2×M16—271no—
230C22⋊Q8—271no—
231#121—271no—
232#107—271no—
233#103—91no—
234#102—271no—
235C33#46811yescharacteristic, complemented, Sylow
236C23×C3—64no—
237C22×D3—21no—
238C22×D3—94no—
239C22×D3—91no—
240C22×D3—184no—
241C3×Q8—24no—
242C3×Q8—24no—
243C3×Q8—21no—
244C3×Q8—21no—
245C3×Q8—44no—
246C3×Q8—44no—
247C3×D4—64no—
248C3×D4—64no—
249C2×C3×C4—28no—
250C2×C3×C4—24no—
251C2×C3×C4—21no—
252C2×C3×C4—24no—
253C2×C3×C4—181no—
254C2×C3×C4C2×D3214yes—
255C2×C3×C4C22×C3⋊D311yescharacteristic
256#67—368no—
257C2×Dic3—64no—
258C2×Dic3—94no—
259C2×Dic3—91no—
260C2×Dic3—184no—
261C2×Dic3—188no—
262C2×Dic3—184no—
263C2×Dic3—181no—
264D12—21no—
265D12—21no—
266D12—362no—
267C4×D3—184no—
268C4×D3—184no—
269C4×D3—181no—
270C4×D3—181no—
271C4×D3—364no—
272C4×D3—364no—
273C3×C8—38no—
274C3×C8—181no—
275C3⋊C8—64no—
276C3⋊C8#63812yescomplemented
277C2×C32—24no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—241no—
281C2×C32#88411yescharacteristic
282C2×C32#91414yes—
283C2×C32#91411yescharacteristic
284C2×C32#91514yes—
285C2×C32#91811yescharacteristic
286C2×C32#93814yes—
287C3⋊D3—128no—
288C3⋊D3—124no—
289C3⋊D3—121no—
290C3×D3—84no—
291C2×Q8D3×C3⋊D311yescharacteristic
292C2×D4—31no—
293C2×D4—541no—
294C22×C4—271no—
295SD16—62no—
296M16—542no—
297C2×C8—31no—
298C2×C8—91no—
299C4⋊C4—91no—
300C4⋊C4—271no—
301C4⋊C4—541no—
302C22⋊C4—541no—
303C22⋊C4—541no—
304C22×C3—24no—
305C22×C3—124no—
306C22×C3—124no—
307C22×C3—124no—
308C22×C3#381114yes—
309C22×C3D4×C3⋊D311yescharacteristic
310C2×D3—41no—
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—188no—
314C2×D3—181no—
315C2×D3—181no—
316C2×D3—181no—
317C2×D3—184no—
318C2×D3—184no—
319C2×D3—184no—
320C2×D3—364no—
321C2×D3—364no—
322C3×C4—24no—
323C3×C4—24no—
324C3×C4—48no—
325C3×C4—41no—
326C3×C4—44no—
327C3×C4—361no—
328C3×C4#380414yes—
329C3×C4D3×#6714yes—
330C3×C4#383011yescharacteristic
331C3×C4C2×#63811yescharacteristic
332Dic3—124no—
333Dic3—181no—
334Dic3—184no—
335Dic3—364no—
336Dic3—368no—
337Dic3—361no—
338Dic3—364no—
339C32—24no—
340C32#481211yescharacteristic, complemented
341C32#517414yescomplemented
342C23—61no—
343C23—271no—
344Q8—21no—
345Q8—21no—
346D4—61no—
347D4—61no—
348D4—1082no—
349C2×C4—21no—
350C2×C4—181no—
351C2×C4—271no—
352C2×C4—541no—
353C2×C4—541no—
354C2×C4—541no—
355C2×C4C2×D3×C3⋊D311yescharacteristic
356C8—32no—
357C8—181no—
358C2×C3—24no—
359C2×C3—24no—
360C2×C3—24no—
361C2×C3—244no—
362C2×C3#6394114yes—
363C2×C3#6395114yes—
364C2×C3#6395814yes—
365C2×C3#6412311yescharacteristic
366C2×C3#6412411yescharacteristic
367C2×C3#6414711yescharacteristic
368D3—81no—
369D3—368no—
370D3—364no—
371D3—361no—
372C22—121no—
373C22—121no—
374C22—121no—
375C22—541no—
376C22—541no—
377C22—541no—
378C22#8903911yescharacteristic
379C4—41no—
380C4—361no—
381C4—541no—
382C4—1081no—
383C4#8902811yescharacteristic
384C4D3×#63811yescharacteristic
385C3—24no—
386C3#1059159014yescomplemented
387C3#1059287311yescharacteristic, complemented
388C2—241no—
389C2—1081no—
390C2#1172357211yescharacteristic
391C2#1172358811yescharacteristic
392C2#1172359911yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radical#11723531C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#468
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #11723987C2semidirect
2. C2×C32×#215C2semidirect
4. C2×#88788C2semidirect
5. C3×#63662C2semidirect
6. #11722055C2semidirect
7. #11722032C2semidirect
29. C2×C3×#215D3semidirect
32. #63662D3semidirect
53. C32×C3⋊C8D4semidirect
104. C2×#215C3⋊D3semidirect
155. C3×C3⋊C8#67semidirect
235. C33#468semidirect
276. C3⋊C8#638semidirect
340. C32#4812semidirect
341. C32#5174semidirect
386. C3#10591590semidirect
387. C3#10592873semidirect

Maximal subgroups

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

subgroupindex
#117239872
C2×C32×#2152
#117235312
C2×#887882
C3×#636622
#117220552
#117220322
#105928733
#105915903