Finite Group Databasebeta

← #13159395

Subgroups — #13159395

Order 1728.

1096 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1096 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#11723936C211yesmaximal, characteristic
2#11723449C211yesmaximal, characteristic, complemented
3C2×#88797C211yesmaximal, characteristic, complemented
4C2×#88788C211yesmaximal, characteristic, complemented
5C3×#63637C211yesmaximal, characteristic, complemented
6C32×#807C211yesmaximal, characteristic, complemented
7#11722035C211yesmaximal, characteristic, complemented
8#10592585—31nomaximal
9#10590522—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15#88859—21no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20#88804—21no—
21#88797—22no—
22#88788—22no—
23C2×#3744—31no—
24C2×#3743—31no—
25#64106—31no—
26#64104—31no—
27#64104—34no—
28#64104—64no—
29#63871—34no—
30C2×#3717—34no—
31C2×#3712—34no—
32C32×#107—31no—
33#63637D311yescharacteristic, complemented
34#63636—31no—
35C3×#810—34no—
36C3×#807D314yescomplemented
37#63564—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3⋊(C3⋊D3)D411yescharacteristic
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))—42no—
46C4×C3⋊(C3⋊D3)—21no—
47C4×C3⋊(C3⋊D3)—21no—
48C2×C3×C3⋊Dic3—21no—
49C3×#634—21no—
50C3×#634—21no—
51C2×C32×Dic3—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C32×C3⋊C8D412yescomplemented
54#4886—94no—
55#4812—91no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4⋊(C3⋊D3)—34no—
58C2×C4⋊(C3⋊D3)—61no—
59C2×C4⋊(C3⋊D3)—64no—
60C2×C4⋊(C3⋊D3)—64no—
61C2×C4⋊(C3⋊D3)—68no—
62C2×C4×C3⋊D3—31no—
63C2×C4×C3⋊D3—34no—
64C2×C4×C3⋊D3—64no—
65C2×#634C2×D311yescharacteristic
66C2×C3×Dic6—34no—
67C2×C32×C8—31no—
68C32×C4⋊C4—24no—
69C32×C4⋊C4—31no—
70C32×C4⋊C4C2×D314yes—
71C32×C4⋊C4C2×D311yescharacteristic
72#3752—61no—
73#3752—61no—
74#3752—64no—
75#3752—68no—
76C4⋊(C3⋊Dic3)—61no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78#3744—62no—
79#3743—62no—
80C3×C4⋊Dic3C2×D314yes—
81C2×C3×C3⋊C8—34no—
82C2×C3×C3⋊C8C2×D314yes—
83Dic3⋊Dic3—64no—
84#3722—64no—
85#3717—68no—
86#3712—68no—
87C22×C33C2×D411yescharacteristic
88C2×C3⋊(C3⋊D3)—21no—
89C2×C3⋊(C3⋊D3)—21no—
90C2×C3⋊(C3⋊D3)—21no—
91C2×C3⋊(C3⋊D3)—41no—
92C2×C3⋊(C3⋊D3)—41no—
93C2×C3⋊(C3⋊D3)—41no—
94C33×C4—41no—
95C33×C4C2×D411yescharacteristic
96C33×C4#4111yescharacteristic
97C3⋊(C3⋊Dic3)—21no—
98C3×C3⋊Dic3—41no—
99C32×Dic3—41no—
100#946—91no—
101C2×#215—91no—
102C2×#204—94no—
103C2×ΓL(1,𝔽25)—94no—
104C2×ΓL(1,𝔽25)—91no—
105#897—94no—
106#895—94no—
107#895—91no—
108#895—184no—
109C3×#107—34no—
110C3×#103—91no—
111#821—91no—
112#810—34no—
113#809—94no—
114#807C3⋊D311yescharacteristic, complemented
115C22×C3⋊D3—31no—
116C22×C3⋊D3—34no—
117C22×C3⋊D3—64no—
118C22×C3⋊D3—68no—
119C22×C3⋊D3—64no—
120C22×C3⋊D3—61no—
121C2×C32×C4—28no—
122C2×C32×C4—24no—
123C2×C32×C4—21no—
124C2×C32×C4—24no—
125C2×C32×C4—61no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D311yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—31no—
130C2×C3⋊Dic3—34no—
131C2×C3⋊Dic3—64no—
132C4⋊(C3⋊D3)—61no—
133C4⋊(C3⋊D3)—61no—
134C4⋊(C3⋊D3)—64no—
135C4⋊(C3⋊D3)—64no—
136C4⋊(C3⋊D3)—124no—
137C4⋊(C3⋊D3)—124no—
138C4⋊(C3⋊D3)—1216no—
139C4⋊(C3⋊D3)—128no—
140C4⋊(C3⋊D3)—122no—
141C4×C3⋊D3—61no—
142C4×C3⋊D3—61no—
143C4×C3⋊D3—64no—
144C4×C3⋊D3—64no—
145C4×C3⋊D3—124no—
146C4×C3⋊D3—124no—
147#634—21no—
148#634—21no—
149C2×C3×Dic3—24no—
150C2×C3×Dic3—64no—
151C3×Dic6—64no—
152C3×Dic6—64no—
153C32×C8—32no—
154C3⋊(C3⋊C8)—21no—
155C3×C3⋊C8—64no—
156C3×C3⋊C8D1218yescomplemented
157#468—271noSylow
158C2×C33#12111yescharacteristic
159C2×C33AGL(1,ℤ/8)11yescharacteristic
160C2×C33#13711yescharacteristic
161C3⋊(C3⋊D3)—41no—
162C3⋊(C3⋊D3)—81no—
163C2×C3×Q8—91no—
164C2×D12—94no—
165C2×D12—91no—
166C2×D12—184no—
167C2×D12—184no—
168C2×D12—181no—
169C2×D12—188no—
170C2×C4×D3—94no—
171C2×C4×D3—91no—
172C2×C4×D3—184no—
173C2×Dic6—34no—
174C2×C3×C8—34no—
175C2×C3×C8—91no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—34no—
178C3×C4⋊C4D3214yes—
179C3×C4⋊C4C2×C3⋊D311yescharacteristic
180#215—182no—
181(C2×D3)⋊C4—181no—
182(C2×D3)⋊C4—184no—
183(C2×D3)⋊C4—184no—
184(C2×D3)⋊C4—181no—
185(C2×D3)⋊C4—188no—
186C4⋊Dic3—184no—
187C4⋊Dic3C2×C3⋊D311yescharacteristic
188Dic3⋊C4—181no—
189C2×C3⋊C8—34no—
190C2×C3⋊C8C2×C3⋊D311yescharacteristic
191#204—188no—
192ΓL(1,𝔽25)—188no—
193ΓL(1,𝔽25)—182no—
194C22×C32—24no—
195C22×C32D3×D414yes—
196C22×C32D3×D411yescharacteristic
197C2×C3⋊D3—68no—
198C2×C3⋊D3—61no—
199C2×C3⋊D3—64no—
200C2×C3⋊D3—61no—
201C2×C3⋊D3—61no—
202C2×C3⋊D3—64no—
203C2×C3⋊D3—64no—
204C2×C3⋊D3—124no—
205C2×C3⋊D3—124no—
206C2×C3⋊D3—128no—
207C2×C3⋊D3—128no—
208C2×C3⋊D3—128no—
209C2×C3⋊D3—121no—
210C2×C3⋊D3—124no—
211C2×C3⋊D3—124no—
212C2×C3⋊D3—121no—
213C2×C3⋊D3—124no—
214C2×C3⋊D3—121no—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4—48no—
218C32×C4—44no—
219C32×C4—41no—
220C32×C4—121no—
221C32×C4C2×D1214yes—
222C32×C4#23511yescharacteristic
223C32×C4D3×D411yescharacteristic
224C32×C4#23914yes—
225C3⋊Dic3—41no—
226C3⋊Dic3—61no—
227C3⋊Dic3—64no—
228C3⋊Dic3—124no—
229C3×Dic3—44no—
230C3×Dic3—124no—
231C2×SD16—271no—
232C2×M16—271no—
233C22⋊Q8—271no—
234#121—271no—
235#107—31no—
236#103—91no—
237#102—271no—
238C33#46811yescharacteristic, complemented, Sylow
239C22×D3—94no—
240C22×D3—91no—
241C22×D3—184no—
242C22×D3—188no—
243C22×D3—184no—
244C22×D3—181no—
245C3×Q8—181no—
246C3×Q8—181no—
247C2×C3×C4—28no—
248C2×C3×C4—24no—
249C2×C3×C4—21no—
250C2×C3×C4—24no—
251C2×C3×C4—64no—
252C2×C3×C4—181no—
253C2×C3×C4C2×D3214yes—
254C2×C3×C4C22×C3⋊D311yescharacteristic
255C2×Dic3—21no—
256C2×Dic3—64no—
257C2×Dic3—94no—
258C2×Dic3—91no—
259C2×Dic3—184no—
260D12—184no—
261D12—184no—
262D12—181no—
263D12—181no—
264D12—364no—
265D12—364no—
266D12—3616no—
267D12—362no—
268D12—368no—
269C4×D3—184no—
270C4×D3—184no—
271C4×D3—181no—
272C4×D3—181no—
273C4×D3—364no—
274C4×D3—364no—
275Dic6—64no—
276Dic6—64no—
277C3×C8—38no—
278C3×C8—181no—
279C3⋊C8—64no—
280C3⋊C8C4⋊(C3⋊D3)12yescomplemented
281C2×C32—24no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32#88411yescharacteristic
285C2×C32#89514yes—
286C2×C32#91114yes—
287C2×C32#91411yescharacteristic
288C2×C32#91814yes—
289C2×C32#91811yescharacteristic
290C3⋊D3—128no—
291C3⋊D3—124no—
292C3⋊D3—121no—
293C3⋊D3—248no—
294C3⋊D3—244no—
295C3⋊D3—241no—
296C2×Q8—91no—
297C2×D4—271no—
298C2×D4—541no—
299C22×C4—271no—
300SD16—542no—
301M16—542no—
302C2×C8—31no—
303C2×C8—91no—
304C4⋊C4—31no—
305C4⋊C4—541no—
306C4⋊C4D3×C3⋊D311yescharacteristic
307C22⋊C4—541no—
308C22⋊C4—541no—
309C22×C3—24no—
310C22×C3#381114yes—
311C22×C3D4×C3⋊D311yescharacteristic
312C2×D3—188no—
313C2×D3—181no—
314C2×D3—181no—
315C2×D3—181no—
316C2×D3—184no—
317C2×D3—184no—
318C2×D3—184no—
319C2×D3—364no—
320C2×D3—364no—
321C2×D3—368no—
322C2×D3—368no—
323C2×D3—368no—
324C2×D3—361no—
325C2×D3—361no—
326C2×D3—361no—
327C2×D3—364no—
328C2×D3—364no—
329C2×D3—364no—
330C3×C4—24no—
331C3×C4—24no—
332C3×C4—48no—
333C3×C4—41no—
334C3×C4—44no—
335C3×C4—124no—
336C3×C4—361no—
337C3×C4#379914yes—
338C3×C4D3×D1214yes—
339C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
340C3×C4#383211yescharacteristic
341Dic3—41no—
342Dic3—124no—
343Dic3—181no—
344Dic3—184no—
345Dic3—364no—
346C32—24no—
347C32#481211yescharacteristic, complemented
348C32#488614yescomplemented
349C23—271no—
350C23—541no—
351Q8—181no—
352Q8—181no—
353D4—541no—
354D4—541no—
355D4—1082no—
356C2×C4—21no—
357C2×C4—61no—
358C2×C4—181no—
359C2×C4—271no—
360C2×C4—541no—
361C2×C4—541no—
362C2×C4C2×D3×C3⋊D311yescharacteristic
363C8—32no—
364C8—181no—
365C2×C3—24no—
366C2×C3—24no—
367C2×C3—24no—
368C2×C3#6391014yes—
369C2×C3#6393714yes—
370C2×C3#6394514yes—
371C2×C3#6410411yescharacteristic
372C2×C3#6412011yescharacteristic
373C2×C3#6412711yescharacteristic
374D3—368no—
375D3—364no—
376D3—361no—
377D3—728no—
378D3—724no—
379D3—721no—
380C22—541no—
381C22—541no—
382C22—541no—
383C22—1081no—
384C22—1081no—
385C22—1081no—
386C22#8903911yescharacteristic
387C4—41no—
388C4—121no—
389C4—361no—
390C4—541no—
391C4#8901811yescharacteristic
392C4D3×C4⋊(C3⋊D3)11yescharacteristic
393C3—24no—
394C3#1059052214yescomplemented
395C3#1059258511yescharacteristic, complemented
396C2—1081no—
397C2—2161no—
398C2#1172348611yescharacteristic
399C2#1172356411yescharacteristic
400C2#1172357811yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radical#11723936C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#468
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
2. #11723449C2semidirect
3. C2×#88797C2semidirect
4. C2×#88788C2semidirect
5. C3×#63637C2semidirect
6. C32×#807C2semidirect
7. #11722035C2semidirect
33. #63637D3semidirect
36. C3×#807D3semidirect
53. C32×C3⋊C8D4semidirect
114. #807C3⋊D3semidirect
156. C3×C3⋊C8D12semidirect
238. C33#468semidirect
280. C3⋊C8C4⋊(C3⋊D3)semidirect
347. C32#4812semidirect
348. C32#4886semidirect
394. C3#10590522semidirect
395. C3#10592585semidirect

Maximal subgroups

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

subgroupindex
#117239362
#117234492
C2×#887972
C2×#887882
C3×#636372
C32×#8072
#117220352
#105925853
#105905223