Finite Group Databasebeta

← #13159354

Subgroups — #13159354

Order 1728.

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

normal subgroups only

This group has 956 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#11723938C211yesmaximal, characteristic
2#11723456C211yesmaximal, characteristic
3C2×#88799C211yesmaximal, characteristic, complemented
4C2×#88788C211yesmaximal, characteristic
5C3×#63635C211yesmaximal, characteristic, complemented
6C32×#810C211yesmaximal, characteristic, complemented
7#11722042C211yesmaximal, characteristic, complemented
8#10592515—31nomaximal
9#10590524—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C33×C4⋊C4C2211yescharacteristic
14#88859—21no—
15C4⋊(C3⋊(C3⋊Dic3))—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20#88804—21no—
21#88799—22no—
22#88788—22no—
23#64155—31no—
24C2×#3756—31no—
25C2×#3743—31no—
26#64106—31no—
27#64106—34no—
28#64106—64no—
29#63871—34no—
30C2×#3719—34no—
31C2×#3712—34no—
32C32×#103—31no—
33#63643—31no—
34#63635D311yescharacteristic, complemented
35C3×#810D314yescomplemented
36C3×#808—34no—
37#63567—34no—
38C22×C3⋊(C3⋊D3)D411yescharacteristic
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C4×C3⋊(C3⋊D3)—21no—
44C4×C3⋊(C3⋊D3)—21no—
45#6308—21no—
46#6308—21no—
47C2×C3×C3⋊Dic3—21no—
48C2×C32×Dic3—21no—
49C32×Dic6—21no—
50C32×Dic6—21no—
51C3×C3⋊(C3⋊C8)D412yes—
52C32×C3⋊C8—21no—
53#4867—91no—
54#4816—94no—
55C2×C32×Q8—31no—
56C2×C4×C3⋊D3—31no—
57C2×C4×C3⋊D3—34no—
58C2×C4×C3⋊D3—64no—
59C2×#634—31no—
60C2×#634—34no—
61C2×#634—64no—
62C2×C3×Dic6C2×D314yes—
63C2×C32×C8—31no—
64C32×C4⋊C4—24no—
65C32×C4⋊C4C2×D314yes—
66C32×C4⋊C4C2×D311yescharacteristic
67#3756—62no—
68#3752—61no—
69#3752—61no—
70#3752—64no—
71#3752—68no—
72C4⋊(C3⋊Dic3)—61no—
73C4⋊(C3⋊Dic3)—64no—
74C4⋊(C3⋊Dic3)—68no—
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76#3750—61no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78#3743—62no—
79C3×C4⋊Dic3—34no—
80C2×C3×C3⋊C8—34no—
81C2×C3×C3⋊C8C2×D314yes—
82Dic3⋊Dic3—64no—
83#3722—64no—
84#3719—68no—
85#3712—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—21no—
88C2×C3⋊(C3⋊D3)—21no—
89C2×C3⋊(C3⋊D3)—21no—
90C33×C4—41no—
91C33×C4C2×D411yescharacteristic
92C33×C4#4111yescharacteristic
93C3⋊(C3⋊Dic3)—21no—
94C3⋊(C3⋊Dic3)—41no—
95C3×C3⋊Dic3—41no—
96C32×Dic3—41no—
97#946—94no—
98C2×#216—94no—
99C2×Dic12—91no—
100C2×ΓL(1,𝔽25)—94no—
101C2×ΓL(1,𝔽25)—91no—
102#897—91no—
103#897—94no—
104#897—91no—
105#897—184no—
106C3×#106—91no—
107C3×#103—34no—
108#816—94no—
109#810—91no—
110#810C3⋊D311yescharacteristic, complemented
111#808—34no—
112C22×C3⋊D3—31no—
113C22×C3⋊D3—34no—
114C22×C3⋊D3—64no—
115C32×Q8—61no—
116C32×Q8—61no—
117C2×C32×C4—28no—
118C2×C32×C4—24no—
119C2×C32×C4—21no—
120C2×C32×C4—24no—
121C2×C32×C4—61no—
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—31no—
126C2×C3⋊Dic3—34no—
127C2×C3⋊Dic3—64no—
128C2×C3⋊Dic3—68no—
129C2×C3⋊Dic3—64no—
130C2×C3⋊Dic3—61no—
131C4×C3⋊D3—61no—
132C4×C3⋊D3—61no—
133C4×C3⋊D3—64no—
134C4×C3⋊D3—64no—
135C4×C3⋊D3—124no—
136C4×C3⋊D3—124no—
137#634—61no—
138#634—61no—
139#634—64no—
140#634—64no—
141#634—124no—
142#634—124no—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—64no—
145C3×Dic6—24no—
146C3×Dic6—24no—
147C32×C8—61no—
148C3⋊(C3⋊C8)D1212yes—
149C3×C3⋊C8—24no—
150C3×C3⋊C8—38no—
151#470—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33#13711yescharacteristic
154C2×C33#13711yescharacteristic
155C3⋊(C3⋊D3)—41no—
156C2×C3×Q8—34no—
157C2×C4×D3—94no—
158C2×C4×D3—91no—
159C2×C4×D3—184no—
160C2×Dic6—94no—
161C2×Dic6—91no—
162C2×Dic6—184no—
163C2×Dic6C2×C3⋊D311yescharacteristic
164C2×C3×C8—34no—
165C2×C3×C8—91no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—91no—
168C3×C4⋊C4D3214yes—
169C3×C4⋊C4C2×C3⋊D311yescharacteristic
170#216—188no—
171(C2×D3)⋊C4—181no—
172(C2×D3)⋊C4—184no—
173(C2×D3)⋊C4—184no—
174(C2×D3)⋊C4—181no—
175(C2×D3)⋊C4—188no—
176C4⋊Dic3—34no—
177C4⋊Dic3—181no—
178C4⋊Dic3—184no—
179C4⋊Dic3—181no—
180C4⋊Dic3—188no—
181Dic3⋊C4—184no—
182C2×C3⋊C8—34no—
183C2×C3⋊C8C2×C3⋊D311yescharacteristic
184Dic12—182no—
185ΓL(1,𝔽25)—182no—
186ΓL(1,𝔽25)—188no—
187C22×C32—24no—
188C22×C32D3×D414yes—
189C22×C32D3×D411yescharacteristic
190C2×C3⋊D3—68no—
191C2×C3⋊D3—61no—
192C2×C3⋊D3—61no—
193C2×C3⋊D3—61no—
194C2×C3⋊D3—64no—
195C2×C3⋊D3—64no—
196C2×C3⋊D3—64no—
197C2×C3⋊D3—124no—
198C2×C3⋊D3—124no—
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—48no—
202C32×C4—44no—
203C32×C4—41no—
204C32×C4—121no—
205C32×C4C2×D1211yescharacteristic
206C32×C4#23514yes—
207C32×C4D3×D414yes—
208C32×C4#23911yescharacteristic
209C3⋊Dic3—41no—
210C3⋊Dic3—61no—
211C3⋊Dic3—64no—
212C3⋊Dic3—124no—
213C3⋊Dic3—128no—
214C3⋊Dic3—124no—
215C3⋊Dic3—121no—
216C3×Dic3—44no—
217C3×Dic3—124no—
218C2×Q16—271no—
219C2×M16—271no—
220C22⋊Q8—271no—
221C22⋊Q8—271no—
222#106—91no—
223#103—31no—
224#103—271no—
225C33#47011yescharacteristic, complemented, Sylow
226C22×D3—94no—
227C22×D3—91no—
228C22×D3—184no—
229C3×Q8—64no—
230C3×Q8—64no—
231C2×C3×C4—28no—
232C2×C3×C4—24no—
233C2×C3×C4—21no—
234C2×C3×C4—24no—
235C2×C3×C4—64no—
236C2×C3×C4—181no—
237C2×C3×C4C2×D3214yes—
238C2×C3×C4C22×C3⋊D311yescharacteristic
239C2×Dic3—21no—
240C2×Dic3—64no—
241C2×Dic3—94no—
242C2×Dic3—91no—
243C2×Dic3—184no—
244C2×Dic3—188no—
245C2×Dic3—184no—
246C2×Dic3—181no—
247C4×D3—184no—
248C4×D3—184no—
249C4×D3—181no—
250C4×D3—181no—
251C4×D3—364no—
252C4×D3—364no—
253Dic6—21no—
254Dic6—21no—
255Dic6—184no—
256Dic6—184no—
257Dic6—181no—
258Dic6—181no—
259Dic6—364no—
260Dic6—364no—
261C3×C8—64no—
262C3×C8—92no—
263C3⋊C8—21no—
264C3⋊C8—38no—
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32#88414yes—
269C2×C32#89511yescharacteristic
270C2×C32#91514yes—
271C2×C32#91511yescharacteristic
272C2×C32#91511yescharacteristic
273C2×C32#91814yes—
274C3⋊D3—128no—
275C3⋊D3—121no—
276C3⋊D3—124no—
277C2×Q8—31no—
278C2×Q8—271no—
279C22×C4—271no—
280Q16—542no—
281M16—542no—
282C2×C8—31no—
283C2×C8—91no—
284C4⋊C4—91no—
285C4⋊C4—541no—
286C4⋊C4—541no—
287C4⋊C4D3×C3⋊D311yescharacteristic
288C22⋊C4—541no—
289C22⋊C4—541no—
290C22×C3—24no—
291C22×C3#381114yes—
292C22×C3D4×C3⋊D311yescharacteristic
293C2×D3—188no—
294C2×D3—181no—
295C2×D3—181no—
296C2×D3—181no—
297C2×D3—184no—
298C2×D3—184no—
299C2×D3—184no—
300C2×D3—364no—
301C2×D3—364no—
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—48no—
305C3×C4—41no—
306C3×C4—44no—
307C3×C4—124no—
308C3×C4—361no—
309C3×C4#379914yes—
310C3×C4D3×D1214yes—
311C3×C4#382811yescharacteristic
312C3×C4D4×C3⋊D311yescharacteristic
313Dic3—41no—
314Dic3—124no—
315Dic3—181no—
316Dic3—184no—
317Dic3—364no—
318Dic3—368no—
319Dic3—361no—
320Dic3—364no—
321C32—24no—
322C32#481614yescomplemented
323C32#486711yescharacteristic, complemented
324C23—271no—
325Q8—61no—
326Q8—61no—
327Q8—541no—
328Q8—541no—
329C2×C4—21no—
330C2×C4—61no—
331C2×C4—181no—
332C2×C4—271no—
333C2×C4—541no—
334C2×C4—541no—
335C2×C4—541no—
336C2×C4C2×D3×C3⋊D311yescharacteristic
337C8—61no—
338C8—92no—
339C2×C3—24no—
340C2×C3—24no—
341C2×C3—24no—
342C2×C3#6391014yes—
343C2×C3#6393114yes—
344C2×C3#6395114yes—
345C2×C3#6409311yescharacteristic
346C2×C3#6412411yescharacteristic
347C2×C3#6412711yescharacteristic
348D3—368no—
349D3—361no—
350D3—364no—
351C22—541no—
352C22—541no—
353C22—541no—
354C22#8903911yescharacteristic
355C4—41no—
356C4—121no—
357C4—361no—
358C4—541no—
359C4—1081no—
360C4#8902211yescharacteristic
361C4C3⋊D3×D1211yescharacteristic
362C3—24no—
363C3#1059052414yescomplemented
364C3#1059251511yescharacteristic, complemented
365C2—1081no—
366C2#1172352511yescharacteristic
367C2#1172355411yescharacteristic
368C2#1172359011yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#470
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
3. C2×#88799C2semidirect
5. C3×#63635C2semidirect
6. C32×#810C2semidirect
7. #11722042C2semidirect
34. #63635D3semidirect
35. C3×#810D3semidirect
110. #810C3⋊D3semidirect
225. C33#470semidirect
322. C32#4816semidirect
323. C32#4867semidirect
363. C3#10590524semidirect
364. C3#10592515semidirect

Maximal subgroups

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

subgroupindex
#117239382
#117234562
C2×#887992
C2×#887882
C3×#636352
C32×#8102
#117220422
#105925153
#105905243