Finite Group Databasebeta

← #13159999

Subgroups — #13159999

Order 1728.

932 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C2×#88850C211yesmaximal, characteristic
2#11723456C211yesmaximal, characteristic
3#11723449C211yesmaximal, characteristic
4C2×#88798C211yesmaximal, characteristic, complemented
5C3×#63643C211yesmaximal, characteristic, complemented
6C32×#816C211yesmaximal, characteristic, complemented
7#11722053C211yesmaximal, characteristic, complemented
8#10592515—31nomaximal
9#10590856—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15#88850—22no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18#88807—21no—
19Dic3⋊(C3⋊Dic3)—21no—
20#88804—21no—
21#88804—21no—
22#88798—22no—
23#64155—31no—
24C2×#3756—31no—
25C2×#3743—31no—
26C2×#3743—34no—
27C2×#3743—64no—
28#64106—31no—
29#63871—34no—
30#63871—34no—
31C2×#3718—34no—
32C32×#103—31no—
33#63643D311yescharacteristic, complemented
34#63635—31no—
35C3×#816—34no—
36C3×#816D314yescomplemented
37#63574—34no—
38C22×C3⋊(C3⋊D3)D411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C4×C3⋊(C3⋊D3)—21no—
42C4×C3⋊(C3⋊D3)—21no—
43C2×C3×C3⋊Dic3—21no—
44C2×C3×C3⋊Dic3—21no—
45C3×#634—21no—
46C3×#634—21no—
47C2×C32×Dic3—21no—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51C33×C8—21no—
52C3⋊(C3⋊(C3⋊C8))D412yes—
53#4816—94no—
54#4816—91no—
55C2×C32×Q8—31no—
56C2×C4×C3⋊D3—31no—
57C2×C4×C3⋊D3—34no—
58C2×C4×C3⋊D3—64no—
59C2×#634C2×D311yescharacteristic
60C2×C3×Dic6—34no—
61C2×C3×Dic6C2×D314yes—
62C2×C32×C8—24no—
63C2×C32×C8C2×D314yes—
64C2×C32×C8C2×D311yescharacteristic
65C32×C4⋊C4—31no—
66#3756—62no—
67#3752—61no—
68#3752—61no—
69C4⋊(C3⋊Dic3)—61no—
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71#3750—61no—
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—34no—
74C2×C3⋊(C3⋊C8)—64no—
75#3743—62no—
76#3743—68no—
77#3743—616no—
78C3×C4⋊Dic3—34no—
79C3×C4⋊Dic3C2×D314yes—
80Dic3⋊Dic3—64no—
81Dic3⋊Dic3—64no—
82#3722—64no—
83#3722—64no—
84#3718—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3⋊(C3⋊D3)—21no—
87C2×C3⋊(C3⋊D3)—21no—
88C2×C3⋊(C3⋊D3)—21no—
89C33×C4C2×D411yescharacteristic
90C33×C4#4111yescharacteristic
91C3⋊(C3⋊Dic3)—21no—
92C3×C3⋊Dic3—41no—
93C3×C3⋊Dic3—41no—
94C32×Dic3—41no—
95C32×Dic3—41no—
96#946—94no—
97#946—91no—
98C2×#216—94no—
99C2×#216—91no—
100C2×ΓL(1,𝔽25)—94no—
101C2×ΓL(1,𝔽25)—91no—
102C2×ΓL(1,𝔽25)—184no—
103#897—94no—
104#897—91no—
105C3×#103—34no—
106C3×#103—91no—
107#816—34no—
108#816C3⋊D311yescharacteristic, complemented
109#808—94no—
110#808—91no—
111C22×C3⋊D3—31no—
112C22×C3⋊D3—34no—
113C22×C3⋊D3—64no—
114C32×Q8—61no—
115C32×Q8—61no—
116C2×C32×C4—24no—
117C2×C32×C4—61no—
118C2×C32×C4—61no—
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—31no—
124C2×C3⋊Dic3—34no—
125C2×C3⋊Dic3—64no—
126C4×C3⋊D3—68no—
127C4×C3⋊D3—68no—
128C4×C3⋊D3—64no—
129C4×C3⋊D3—61no—
130C4×C3⋊D3—64no—
131C4×C3⋊D3—61no—
132#634—21no—
133#634—21no—
134C2×C3×Dic3—24no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—64no—
137C2×C3×Dic3—64no—
138C3×Dic6—24no—
139C3×Dic6—24no—
140C3×Dic6—64no—
141C3×Dic6—64no—
142C32×C8—28no—
143C32×C8—24no—
144C32×C8—21no—
145C3⋊(C3⋊C8)—32no—
146C3⋊(C3⋊C8)—38no—
147C3⋊(C3⋊C8)—68no—
148#470—271noSylow
149C2×C33#12111yescharacteristic
150C2×C33#13711yescharacteristic
151C2×C33#13711yescharacteristic
152C3⋊(C3⋊D3)—41no—
153C2×C3×Q8—34no—
154C2×C3×Q8—91no—
155C2×C4×D3—94no—
156C2×C4×D3—91no—
157C2×C4×D3—184no—
158C2×Dic6—34no—
159C2×Dic6C2×C3⋊D311yescharacteristic
160C2×C3×C8—24no—
161C2×C3×C8D3214yes—
162C2×C3×C8C2×C3⋊D311yescharacteristic
163C3×C4⋊C4—34no—
164C3×C4⋊C4—91no—
165#216—182no—
166#216—188no—
167(C2×D3)⋊C4—181no—
168(C2×D3)⋊C4—181no—
169(C2×D3)⋊C4—184no—
170(C2×D3)⋊C4—184no—
171C4⋊Dic3—34no—
172C4⋊Dic3—181no—
173C4⋊Dic3—184no—
174C4⋊Dic3C2×C3⋊D311yescharacteristic
175Dic3⋊C4—181no—
176Dic3⋊C4—184no—
177C2×C3⋊C8—94no—
178C2×C3⋊C8—91no—
179C2×C3⋊C8—184no—
180ΓL(1,𝔽25)—188no—
181ΓL(1,𝔽25)—182no—
182ΓL(1,𝔽25)—1816no—
183C22×C32—24no—
184C22×C32D3×D414yes—
185C22×C32D3×D411yescharacteristic
186C2×C3⋊D3—68no—
187C2×C3⋊D3—64no—
188C2×C3⋊D3—61no—
189C2×C3⋊D3—61no—
190C2×C3⋊D3—61no—
191C2×C3⋊D3—64no—
192C2×C3⋊D3—64no—
193C2×C3⋊D3—124no—
194C2×C3⋊D3—124no—
195C32×C4—24no—
196C32×C4—24no—
197C32×C4—121no—
198C32×C4—121no—
199C32×C4#23514yes—
200C32×C4#23511yescharacteristic
201C32×C4D3×D414yes—
202C32×C4D3×D411yescharacteristic
203C3⋊Dic3—41no—
204C3⋊Dic3—41no—
205C3⋊Dic3—68no—
206C3⋊Dic3—64no—
207C3⋊Dic3—61no—
208C3×Dic3—44no—
209C3×Dic3—44no—
210C3×Dic3—124no—
211C3×Dic3—124no—
212C2×Q16—271no—
213C2×M16—271no—
214C22⋊Q8—271no—
215C22⋊Q8—271no—
216#106—271no—
217#103—31no—
218#103—91no—
219C33#47011yescharacteristic, complemented, Sylow
220C22×D3—94no—
221C22×D3—91no—
222C22×D3—184no—
223C3×Q8—64no—
224C3×Q8—64no—
225C3×Q8—181no—
226C3×Q8—181no—
227C2×C3×C4—24no—
228C2×C3×C4—64no—
229C2×C3×C4—64no—
230C2×C3×C4—181no—
231C2×C3×C4—181no—
232C2×C3×C4C2×D3214yes—
233C2×C3×C4C22×C3⋊D311yescharacteristic
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—64no—
237C2×Dic3—64no—
238C2×Dic3—94no—
239C2×Dic3—91no—
240C2×Dic3—184no—
241C4×D3—188no—
242C4×D3—188no—
243C4×D3—184no—
244C4×D3—184no—
245C4×D3—181no—
246C4×D3—181no—
247Dic6—21no—
248Dic6—21no—
249Dic6—64no—
250Dic6—64no—
251C3×C8—28no—
252C3×C8—24no—
253C3×C8—21no—
254C3⋊C8—98no—
255C3⋊C8—92no—
256C3⋊C8—188no—
257C2×C32—24no—
258C2×C32—24no—
259C2×C32—24no—
260C2×C32#88414yes—
261C2×C32#88411yescharacteristic
262C2×C32#91514yes—
263C2×C32#91511yescharacteristic
264C2×C32#91814yes—
265C2×C32#91811yescharacteristic
266C3⋊D3—128no—
267C3⋊D3—124no—
268C3⋊D3—121no—
269C2×Q8—31no—
270C2×Q8—91no—
271C22×C4—271no—
272Q16—542no—
273M16—542no—
274C2×C8—271no—
275C2×C8D3×C3⋊D311yescharacteristic
276C4⋊C4—31no—
277C4⋊C4—91no—
278C4⋊C4—541no—
279C4⋊C4—541no—
280C22⋊C4—541no—
281C22⋊C4—541no—
282C22×C3—24no—
283C22×C3#380214yes—
284C22×C3D4×C3⋊D311yescharacteristic
285C2×D3—188no—
286C2×D3—184no—
287C2×D3—184no—
288C2×D3—181no—
289C2×D3—181no—
290C2×D3—181no—
291C2×D3—184no—
292C2×D3—364no—
293C2×D3—364no—
294C3×C4—24no—
295C3×C4—24no—
296C3×C4—124no—
297C3×C4—124no—
298C3×C4—361no—
299C3×C4—361no—
300C3×C4#379814yes—
301C3×C4#381114yes—
302C3×C4#382811yescharacteristic
303C3×C4D4×C3⋊D311yescharacteristic
304Dic3—41no—
305Dic3—41no—
306Dic3—124no—
307Dic3—124no—
308Dic3—188no—
309Dic3—181no—
310Dic3—184no—
311C32—24no—
312C32#481614yescomplemented
313C32#481611yescharacteristic, complemented
314C23—271no—
315Q8—61no—
316Q8—61no—
317Q8—181no—
318Q8—181no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—181no—
322C2×C4—181no—
323C2×C4—271no—
324C2×C4—541no—
325C2×C4—541no—
326C2×C4C2×D3×C3⋊D311yescharacteristic
327C8—21no—
328C8—272no—
329C2×C3—24no—
330C2×C3—24no—
331C2×C3—24no—
332C2×C3#6381114yes—
333C2×C3#6381114yes—
334C2×C3#6391214yes—
335C2×C3#6409311yescharacteristic
336C2×C3#6412411yescharacteristic
337C2×C3#6412711yescharacteristic
338D3—368no—
339D3—361no—
340D3—364no—
341C22—541no—
342C22—541no—
343C22—541no—
344C22#8902611yescharacteristic
345C4—121no—
346C4—121no—
347C4—361no—
348C4—361no—
349C4—541no—
350C4#8902011yescharacteristic
351C4#8903911yescharacteristic
352C3—24no—
353C3#1059085614yescomplemented
354C3#1059251511yescharacteristic, complemented
355C2—1081no—
356C2#1172334411yescharacteristic
357C2#1172334611yescharacteristic
358C2#1172352711yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#470
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. C2×#88798C2semidirect
5. C3×#63643C2semidirect
6. C32×#816C2semidirect
7. #11722053C2semidirect
33. #63643D3semidirect
36. C3×#816D3semidirect
108. #816C3⋊D3semidirect
219. C33#470semidirect
312. C32#4816semidirect
313. C32#4816semidirect
353. C3#10590856semidirect
354. C3#10592515semidirect

Maximal subgroups

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

subgroupindex
C2×#888502
#117234562
#117234492
C2×#887982
C3×#636432
C32×#8162
#117220532
#105925153
#105908563