Finite Group Databasebeta

← #13137999

Subgroups — #13137999

Order 1728.

442 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 73 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#64096⋊C3C211yesmaximal, characteristic, complemented
2#11721114C211yesmaximal, characteristic
3C2×#88437C211yesmaximal, characteristic
4C2×#88433C211yesmaximal, characteristic, complemented
5#11720531C211yesmaximal, characteristic
6#63637⋊C3C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10590538—31nomaximal
9#10590532—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C2×#6180C2211yescharacteristic
15#3750⋊C3—21no—
16C4×(C3⋊Dic3)⋊C3—21no—
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88451—21no—
20#88451—21no—
21#88437—22no—
22#88433—22no—
23#64096D311yescharacteristic, complemented
24C3×#887—31no—
25#63863—31no—
26#63860—31no—
27C2×#3719—31no—
28C2×#3719—31no—
29C2×#3716—31no—
30C2×#3715—31no—
31#63637D311yescharacteristic, complemented
32C3×#810—31no—
33C3×#809—31no—
34C3×#809—31no—
35C3×#809—61no—
36#63572—31no—
37Dic3⋊(C3⋊C8)—31no—
38C22×#272—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C4⋊#272—21no—
42C4⋊#272—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47#634⋊C3—21no—
48#634⋊C3—21no—
49#634⋊C3—21no—
50#634⋊C3D412yes—
51#6180—21no—
52(C3⋊(C3⋊C8))⋊C3—21no—
53#5028—91no—
54#4798—91no—
55#4767—91no—
56C2×#634C2×D311yescharacteristic
57C2×#634C2×D311yescharacteristic
58C2×C3×D12—31no—
59C2×C3×D12—31no—
60C2×C3×D12—61no—
61C2×C3×Dic6—31no—
62C2×C3×Dic6—31no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4C2×D311yescharacteristic
65C32×C4⋊C4C2×D311yescharacteristic
66#3750—21no—
67C4×C3⋊Dic3—21no—
68C4×C3⋊Dic3C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
70C3×Dic3⋊C4—61no—
71C3×C4×Dic3—31no—
72C3×C4×Dic3—61no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—61no—
77(C2×D3)⋊Dic3—61no—
78(C2×D3)⋊Dic3—61no—
79(C2×D3)⋊Dic3—61no—
80(C2×D3)⋊Dic3—61no—
81#3719—62no—
82#3719—62no—
83#3716—62no—
84#3715—62no—
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C2×#272—41no—
87C2×#272—41no—
88C2×#272—41no—
89C4×H(3,𝔽3)—41no—
90C4×H(3,𝔽3)C2×D411yescharacteristic
91C4×H(3,𝔽3)#4111yescharacteristic
92(C3⋊Dic3)⋊C3—22no—
93(C3⋊Dic3)⋊C3—21no—
94(C3⋊Dic3)⋊C3—21no—
95(C3⋊Dic3)⋊C3—41no—
96C3×C4×Q8—91no—
97#947—91no—
98C2×#216—91no—
99C2×#216—91no—
100C2×#215—91no—
101#936—91no—
102C2×#214—91no—
103C2×Dic12—91no—
104C2×#204—91no—
105#887—31no—
106#887—91no—
107#875—91no—
108C3×#103—91no—
109C3×#102—91no—
110Dic3⋊C8—91no—
111Dic3⋊C8—91no—
112#810—31no—
113#810—91no—
114#809—31no—
115#809—31no—
116#809—61no—
117C4⋊(C3⋊C8)—91no—
118C22×C3×D3—61no—
119C22×C3×D3—61no—
120C22×C3×D3—121no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—41no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3#6711yescharacteristic
130C2×C3⋊Dic3#6711yescharacteristic
131#634—21no—
132#634—21no—
133#634—21no—
134#634D1212yes—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—31no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C3×D12—62no—
140C3×D12—61no—
141C3×D12—61no—
142C3×D12—61no—
143C3×D12—61no—
144C3×D12—121no—
145C3×Dic6—32no—
146C3×Dic6—61no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C3⋊(C3⋊C8)—21no—
150C3×C3⋊C8—61no—
151C3×C3⋊C8—61no—
152C3×C3⋊C8—61no—
153C3×C3⋊C8—121no—
154#464—271noSylow
155C2×H(3,𝔽3)#12111yescharacteristic
156C2×H(3,𝔽3)#13511yescharacteristic
157C2×H(3,𝔽3)#13711yescharacteristic
158#272—81no—
159C2×C3×Q8—91no—
160C2×C3×Q8—91no—
161C2×C3×D4—91no—
162C2×D12—31no—
163C2×D12—31no—
164C2×D12—61no—
165C2×Dic6—31no—
166C2×Dic6—31no—
167C2×Dic6—91no—
168C2×Dic6—91no—
169C2×C3×C8—91no—
170C2×C3×C8—91no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—31no—
173C3×C4⋊C4—61no—
174C3×C4⋊C4—181no—
175C3×C4⋊C4D3211yescharacteristic
176C3×C42—91no—
177C3×C42—181no—
178#217—181no—
179#217—181no—
180#216—182no—
181#216—182no—
182#215—182no—
183#214—182no—
184(C2×D3)⋊C4—181no—
185(C2×D3)⋊C4—181no—
186(C2×D3)⋊C4—181no—
187(C2×D3)⋊C4—181no—
188Dic3⋊C4—61no—
189Dic3⋊C4—181no—
190C4×Dic3—31no—
191C4×Dic3—61no—
192C4×Dic3—91no—
193C4×Dic3—181no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—31no—
196C2×C3⋊C8—31no—
197C2×C3⋊C8—61no—
198C2×C3⋊C8—91no—
199Dic12—182no—
200#204—182no—
201C22×C32—21no—
202C22×C32D3×D411yescharacteristic
203C22×C32C2×#6711yescharacteristic
204C2×C3×D3—122no—
205C2×C3×D3—121no—
206C2×C3×D3—121no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C2×C3×D3—121no—
211C2×C3×D3—241no—
212C2×C3×D3—241no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4—41no—
216C32×C4—41no—
217C32×C4—81no—
218C32×C4C2×D1211yescharacteristic
219C32×C4#23511yescharacteristic
220C32×C4#23511yescharacteristic
221C32×C4D3×D411yescharacteristic
222C3⋊Dic3—22no—
223C3⋊Dic3—21no—
224C3⋊Dic3—21no—
225C3⋊Dic3—41no—
226C3×Dic3—62no—
227C3×Dic3—61no—
228C3×Dic3—61no—
229C3×Dic3—121no—
230C2×Q16—271no—
231C2×SD16—271no—
232#124—271no—
233C4×Q8—91no—
234C4⋊C8—271no—
235#103—91no—
236#102—91no—
237H(3,𝔽3)#46411yescharacteristic, complemented, Sylow
238C23×C3—181no—
239C22×D3—61no—
240C22×D3—61no—
241C22×D3—121no—
242C3×Q8—92no—
243C3×Q8—181no—
244C3×Q8—181no—
245C3×Q8—181no—
246C3×D4—181no—
247C3×D4—181no—
248C2×C3×C4—21no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—91no—
255C2×C3×C4—91no—
256C2×C3×C4—121no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259C2×C3×C4C2×D3211yescharacteristic
260C2×Dic3—31no—
261C2×Dic3—31no—
262C2×Dic3—61no—
263C2×Dic3—61no—
264C2×Dic3—91no—
265C2×Dic3—91no—
266C2×Dic3—181no—
267C2×Dic3—181no—
268D12—61no—
269D12—61no—
270D12—61no—
271D12—61no—
272D12—62no—
273D12—121no—
274Dic6—32no—
275Dic6—61no—
276Dic6—61no—
277Dic6—61no—
278Dic6—92no—
279Dic6—181no—
280Dic6—181no—
281Dic6—181no—
282C3×C8—181no—
283C3×C8—181no—
284C3⋊C8—61no—
285C3⋊C8—61no—
286C3⋊C8—61no—
287C3⋊C8—121no—
288C3⋊C8—181no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32#88411yescharacteristic
293C2×C32#91611yescharacteristic
294C2×C32#91811yescharacteristic
295C2×C32#93011yescharacteristic
296C2×C32#94211yescharacteristic
297C2×C32#95011yescharacteristic
298C3×D3—242no—
299C3×D3—241no—
300C3×D3—241no—
301C2×Q8—91no—
302C2×Q8—91no—
303C2×D4—91no—
304Q16—542no—
305SD16—542no—
306C2×C8—91no—
307C2×C8—91no—
308C4⋊C4—181no—
309C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
310C22⋊C4—541no—
311C22⋊C4—541no—
312C42—91no—
313C42—181no—
314C22×C3—31no—
315C22×C3—31no—
316C22×C3—61no—
317C22×C3—361no—
318C22×C3—361no—
319C22×C3—361no—
320C22×C3D3×#6711yescharacteristic
321C2×D3—121no—
322C2×D3—121no—
323C2×D3—121no—
324C2×D3—121no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—122no—
328C2×D3—241no—
329C2×D3—241no—
330C3×C4—31no—
331C3×C4—31no—
332C3×C4—31no—
333C3×C4—31no—
334C3×C4—41no—
335C3×C4—61no—
336C3×C4—61no—
337C3×C4—121no—
338C3×C4—121no—
339C3×C4—181no—
340C3×C4—181no—
341C3×C4—182no—
342C3×C4—241no—
343C3×C4—361no—
344C3×C4#379811yescharacteristic
345C3×C4D3×D1211yescharacteristic
346Dic3—62no—
347Dic3—61no—
348Dic3—61no—
349Dic3—121no—
350Dic3—182no—
351Dic3—181no—
352Dic3—181no—
353Dic3—361no—
354C32—21no—
355C32#479811yescharacteristic, complemented
356C32#502811yescharacteristic, complemented
357C23—181no—
358Q8—92no—
359Q8—181no—
360Q8—181no—
361Q8—181no—
362D4—181no—
363D4—181no—
364C2×C4—21no—
365C2×C4—91no—
366C2×C4—91no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
370C8—181no—
371C8—181no—
372C2×C3—31no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—61no—
379C2×C3—61no—
380C2×C3—61no—
381C2×C3—721no—
382C2×C3#6390611yescharacteristic
383C2×C3#6393511yescharacteristic
384C2×C3#6394811yescharacteristic
385D3—241no—
386D3—241no—
387D3—242no—
388C22—361no—
389C22—361no—
390C22—361no—
391C22#8867611yescharacteristic
392C4—41no—
393C4—181no—
394C4—182no—
395C4—181no—
396C4—361no—
397C4#8865011yescharacteristic
398C4#8865411yescharacteristic
399C3—31no—
400C3—31no—
401C3—61no—
402C3#1059053611yescharacteristic
403C2—721no—
404C2#1172114011yescharacteristic
405C2#1172119711yescharacteristic
406C2#1172123411yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×C4⋊#272C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#464
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64096⋊C3C2semidirect
4. C2×#88433C2semidirect
6. #63637⋊C3C2semidirect
7. #11720311C2semidirect
23. #64096D3semidirect
31. #63637D3semidirect
237. H(3,𝔽3)#464semidirect
355. C32#4798semidirect
356. C32#5028semidirect

Maximal subgroups

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

subgroupindex
#64096⋊C32
#117211142
C2×#884372
C2×#884332
#117205312
#63637⋊C32
#117203112
#105905383
#105905323