Finite Group Databasebeta

← #13138021

Subgroups — #13138021

Order 1728.

462 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 77 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#64102⋊C3C211yesmaximal, characteristic
2#11721090C211yesmaximal, characteristic, complemented
3C2×#88438C211yesmaximal, characteristic
4C2×#88431C211yesmaximal, characteristic
5#11720532C211yesmaximal, characteristic, complemented
6#63637⋊C3C211yesmaximal, characteristic, complemented
7#11720311C211yesmaximal, characteristic, complemented
8#10590565—31nomaximal
9#10590559—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3—21no—
13C2×#634⋊C3C2211yescharacteristic
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#3752⋊C3—21no—
17C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
19#88448—21no—
20#88448—21no—
21#88438—22no—
22#88431—22no—
23#64102D311yescharacteristic
24C3×#893—31no—
25#63873—31no—
26#63866—31no—
27C2×#3717—31no—
28C2×#3717—31no—
29C2×#3716—31no—
30C2×#3715—31no—
31#63637D311yescharacteristic, complemented
32C3×#810—31no—
33C3×#810—31no—
34C3×#810—31no—
35C3×#810—61no—
36#63572—31no—
37Dic3⋊(C3⋊C8)—31no—
38C22×(C3⋊D3)⋊C3—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×#1084—21no—
42#6229—21no—
43#6229—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)D412yes—
49C4×(C3⋊D3)⋊C3—42no—
50#634⋊C3—21no—
51#634⋊C3—21no—
52#6180—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5019—91no—
55#4818—91no—
56#4818—91no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×C4×C3⋊D3—21no—
59C2×#634C2×D311yescharacteristic
60C2×C3×D12—31no—
61C2×C3×C4×D3—61no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—31no—
65C2×C3×Dic6—61no—
66C32×C4⋊C4—21no—
67C32×C4⋊C4C2×D311yescharacteristic
68C32×C4⋊C4C2×D311yescharacteristic
69#3752—21no—
70C4×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×(C2×D3)⋊C4—61no—
73C3×C4×Dic3—31no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—61no—
78#3724—61no—
79#3724—61no—
80Dic3⋊Dic3—61no—
81Dic3⋊Dic3—61no—
82#3717—62no—
83#3717—62no—
84#3716—62no—
85#3715—62no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C2×(C3⋊D3)⋊C3—22no—
88C2×(C3⋊D3)⋊C3—41no—
89C2×(C3⋊D3)⋊C3—41no—
90C4×H(3,𝔽3)—41no—
91C4×H(3,𝔽3)C2×D411yescharacteristic
92C4×H(3,𝔽3)#4111yescharacteristic
93#1084—41no—
94(C3⋊Dic3)⋊C3—21no—
95(C3⋊Dic3)⋊C3—41no—
96(C3⋊Dic3)⋊C3SD1612yescomplemented
97C3×C4×D4—91no—
98#944—91no—
99#944—91no—
100C2×#215—91no—
101C2×#215—91no—
102C2×#214—91no—
103C2×#214—91no—
104C2×#204—91no—
105C2×#204—91no—
106#893—31no—
107#893—91no—
108#869—91no—
109C3×#103—91no—
110C3×#103—91no—
111Dic3⋊C8—91no—
112Dic3⋊C8—91no—
113#810—31no—
114#810—31no—
115#810—31no—
116#810—61no—
117#810—91no—
118C4⋊(C3⋊C8)—91no—
119C22×C3⋊D3—21no—
120C22×C3×D3—61no—
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#6711yescharacteristic
129C2×C3⋊Dic3#6711yescharacteristic
130C4⋊(C3⋊D3)—21no—
131C4⋊(C3⋊D3)D1212yes—
132C4×C3⋊D3—42no—
133#634—21no—
134#634—21no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—31no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—121no—
141C3×D12—32no—
142C3×D12—61no—
143C3×C4×D3—122no—
144C3×Dic6—62no—
145C3×Dic6—61no—
146C3×Dic6—61no—
147C3×Dic6—61no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C3×Dic6—61no—
151C3×Dic6—121no—
152C3⋊(C3⋊C8)—21no—
153C3×C3⋊C8—61no—
154C3×C3⋊C8—61no—
155C3×C3⋊C8—61no—
156C3×C3⋊C8—121no—
157#461—271noSylow
158C2×H(3,𝔽3)#12111yescharacteristic
159C2×H(3,𝔽3)C2×SD1611yescharacteristic
160C2×H(3,𝔽3)#13711yescharacteristic
161(C3⋊D3)⋊C3—42no—
162C2×C3×Q8—91no—
163C2×C3×Q8—91no—
164C2×C3×D4—91no—
165C22×C3×C4—181no—
166C2×D12—31no—
167C2×D12—91no—
168C2×C4×D3—61no—
169C2×C4×D3—181no—
170C2×Dic6—31no—
171C2×Dic6—31no—
172C2×Dic6—31no—
173C2×Dic6—61no—
174C2×Dic6—91no—
175C2×C3×C8—91no—
176C2×C3×C8—91no—
177C3×C4⋊C4—31no—
178C3×C4⋊C4—31no—
179C3×C4⋊C4—61no—
180C3×C4⋊C4D3211yescharacteristic
181C3×C22⋊C4—181no—
182C3×C42—91no—
183#215—182no—
184#215—182no—
185#214—182no—
186#214—182no—
187(C2×D3)⋊C4—61no—
188(C2×D3)⋊C4—181no—
189C4⋊Dic3—181no—
190C4⋊Dic3—181no—
191Dic3⋊C4—181no—
192Dic3⋊C4—181no—
193Dic3⋊C4—181no—
194Dic3⋊C4—181no—
195C4×Dic3—31no—
196C4×Dic3—91no—
197C2×C3⋊C8—31no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—61no—
201C2×C3⋊C8—91no—
202#204—182no—
203#204—182no—
204C22×C32—21no—
205C22×C32D3×D411yescharacteristic
206C22×C32C2×#6711yescharacteristic
207C2×C3⋊D3—22no—
208C2×C3⋊D3—41no—
209C2×C3⋊D3—41no—
210C2×C3×D3—62no—
211C2×C3×D3—121no—
212C2×C3×D3—121no—
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—21no—
223C3⋊Dic3—41no—
224C3⋊Dic3#21412yescomplemented
225C3×Dic3—32no—
226C3×Dic3—61no—
227C3×Dic3—122no—
228C3×Dic3—121no—
229C3×Dic3—121no—
230C3×Dic3—121no—
231C2×SD16—271no—
232C2×SD16—271no—
233C4⋊Q8—271no—
234C4×D4—91no—
235C4⋊C8—271no—
236#103—91no—
237#103—91no—
238H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
239C23×C3—181no—
240C22×D3—61no—
241C22×D3—181no—
242C3×Q8—181no—
243C3×Q8—181no—
244C3×Q8—181no—
245C3×Q8—181no—
246C3×D4—92no—
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×C4—362no—
260C2×C3×C4C2×D3211yescharacteristic
261C2×Dic3—31no—
262C2×Dic3—31no—
263C2×Dic3—61no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—91no—
267C2×Dic3—91no—
268C2×Dic3—121no—
269C2×Dic3—181no—
270D12—32no—
271D12—61no—
272D12—92no—
273D12—181no—
274C4×D3—122no—
275C4×D3—362no—
276Dic6—61no—
277Dic6—61no—
278Dic6—61no—
279Dic6—61no—
280Dic6—62no—
281Dic6—61no—
282Dic6—61no—
283Dic6—121no—
284Dic6—181no—
285Dic6—181no—
286C3×C8—181no—
287C3×C8—181no—
288C3⋊C8—61no—
289C3⋊C8—61no—
290C3⋊C8—61no—
291C3⋊C8—121no—
292C3⋊C8—181no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32#88411yescharacteristic
297C2×C32D3×SD1611yescharacteristic
298C2×C32#91811yescharacteristic
299C2×C32#93011yescharacteristic
300C2×C32C2×#21411yescharacteristic
301C2×C32#95111yescharacteristic
302C3⋊D3—42no—
303C3×D3—122no—
304C2×Q8—91no—
305C2×Q8—91no—
306C2×D4—91no—
307C22×C4—181no—
308SD16—542no—
309SD16—542no—
310C2×C8—91no—
311C2×C8—91no—
312C4⋊C4—541no—
313C4⋊C4—541no—
314C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
315C22⋊C4—181no—
316C42—91no—
317C22×C3—31no—
318C22×C3—31no—
319C22×C3—61no—
320C22×C3—182no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3D3×#6711yescharacteristic
324C2×D3—62no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—182no—
328C2×D3—361no—
329C2×D3—361no—
330C3×C4—31no—
331C3×C4—31no—
332C3×C4—31no—
333C3×C4—31no—
334C3×C4—41no—
335C3×C4—61no—
336C3×C4—61no—
337C3×C4—92no—
338C3×C4—121no—
339C3×C4—121no—
340C3×C4—181no—
341C3×C4—241no—
342C3×C4—361no—
343C3×C4—361no—
344C3×C4#379811yescharacteristic
345C3×C4D3×D1211yescharacteristic
346Dic3—32no—
347Dic3—61no—
348Dic3—92no—
349Dic3—121no—
350Dic3—121no—
351Dic3—122no—
352Dic3—121no—
353Dic3—181no—
354Dic3—361no—
355C32—21no—
356C32#481811yescharacteristic, complemented
357C32#501911yescharacteristic, complemented
358C23—181no—
359Q8—181no—
360Q8—181no—
361Q8—181no—
362Q8—181no—
363D4—92no—
364D4—181no—
365C2×C4—21no—
366C2×C4—91no—
367C2×C4—91no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4—362no—
371C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
372C8—181no—
373C8—181no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—61no—
381C2×C3—61no—
382C2×C3—61no—
383C2×C3—362no—
384C2×C3#6390611yescharacteristic
385C2×C3D3×#21411yescharacteristic
386C2×C3#6394311yescharacteristic
387D3—122no—
388D3—362no—
389C22—182no—
390C22—361no—
391C22—361no—
392C22#8867611yescharacteristic
393C4—41no—
394C4—92no—
395C4—181no—
396C4—361no—
397C4—361no—
398C4#8865011yescharacteristic
399C4#8865411yescharacteristic
400C3—31no—
401C3—31no—
402C3—61no—
403C3#1059055911yescharacteristic
404C2—362no—
405C2#1172114011yescharacteristic
406C2#1172119811yescharacteristic
407C2#1172123711yescharacteristic
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⋊((C3⋊D3)⋊C3)C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#461
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
2. #11721090C2semidirect
5. #11720532C2semidirect
6. #63637⋊C3C2semidirect
7. #11720311C2semidirect
31. #63637D3semidirect
96. (C3⋊Dic3)⋊C3SD16semidirect
224. C3⋊Dic3#214semidirect
238. H(3,𝔽3)#461semidirect
356. C32#4818semidirect
357. C32#5019semidirect

Maximal subgroups

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

subgroupindex
#64102⋊C32
#117210902
C2×#884382
C2×#884312
#117205322
#63637⋊C32
#117203112
#105905653
#105905593