Finite Group Databasebeta

← #13140912

Subgroups — #13140912

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#64153⋊C3C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic
3#11721132C211yesmaximal, characteristic
4C2×#88431C211yesmaximal, characteristic
5#11720583C211yesmaximal, characteristic, complemented
6#11720311C211yesmaximal, characteristic, complemented
7#11720309C211yesmaximal, characteristic, complemented
8#10591569—31nomaximal
9#10591565—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×#634⋊C3C2211yescharacteristic
13C2×((C3⋊D3)⋊C3)⋊C4—21no—
14C4⋊#1084C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16#3755⋊C3—22no—
17#3750⋊C3—21no—
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#88450—21no—
22#88431—22no—
23#64153D311yescharacteristic, complemented
24C2×#3755D311yescharacteristic
25C3×#944—31no—
26C2×C3×#215—31no—
27#63898—31no—
28#63894—31no—
29C2×#3717—31no—
30C2×#3716—31no—
31C3×#835—31no—
32C3×#835—31no—
33C3×#835—61no—
34#63572—31no—
35Dic3⋊(C3⋊C8)—31no—
36#63567—31no—
37#63565—31no—
38C22×(C3⋊D3)⋊C3—21no—
39Q8×H(3,𝔽3)—21no—
40Q8×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×#1084—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C2×(C3⋊Dic3)⋊C3D411yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)D412yes—
49#634⋊C3—21no—
50#634⋊C3—21no—
51((C3⋊D3)⋊C3)⋊C4—42no—
52#6180—21no—
53(C3⋊(C3⋊C8))⋊C3—21no—
54#5162—91no—
55#5083—91no—
56#4818—91no—
57C2×C32×Q8—21no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C2×D311yescharacteristic
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×#634C2×D311yescharacteristic
62C2×C3×D12—31no—
63C2×C3×Dic6—31no—
64C2×#624—61no—
65C2×D3×Dic3—61no—
66#3755—22no—
67#3750—21no—
68#3750—21no—
69C4×C3⋊Dic3C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×#215—62no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—61no—
75C3×Dic3⋊C4—61no—
76C3×Dic3⋊C4—61no—
77C3×C4×Dic3—31no—
78C2×C3×C3⋊C8—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81C2×C3×C3⋊C8—61no—
82#3722—61no—
83(C2×D3)⋊Dic3—61no—
84#3717—62no—
85#3716—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
97C2×C3×SD16—91no—
98C3×C4⋊Q8—91no—
99#944—31no—
100#944—91no—
101C2×#215—31no—
102C2×#215—91no—
103C2×#215—91no—
104D4×Dic3—91no—
105C2×#214—91no—
106C2×#204—91no—
107#893—91no—
108C4×D12—91no—
109C3×#103—91no—
110#835—31no—
111#835—31no—
112#835—61no—
113#835—91no—
114#816—91no—
115Dic3⋊C8—91no—
116Dic3⋊C8—91no—
117#810—91no—
118C4⋊(C3⋊C8)—91no—
119C22×C3⋊D3—21no—
120C22×C3×D3—61no—
121C32×Q8—22no—
122C32×Q8—21no—
123C32×Q8—21no—
124C32×Q8—21no—
125C32×Q8—21no—
126C32×Q8—41no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—41no—
131C2×C32×C4C22×D311yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3#6711yescharacteristic
135C2×C3⋊Dic3#6711yescharacteristic
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)#6712yes—
138#634—21no—
139#634—21no—
140C2×C3×Dic3—31no—
141C2×C3×Dic3—31no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—121no—
146C3×D12—32no—
147C3×D12—61no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150#624—122no—
151D3×Dic3—122no—
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—31no—
163C2×C3×Q8—31no—
164C2×C3×Q8—61no—
165C2×C3×Q8—91no—
166C2×C3×Q8D3211yescharacteristic
167C2×C3×D4—91no—
168C22×Dic3—181no—
169C2×D12—31no—
170C2×D12—91no—
171C2×C4×D3—181no—
172C2×C4×D3—181no—
173C2×Dic6—31no—
174C2×Dic6—91no—
175C3×SD16—182no—
176C2×C3×C8—91no—
177C2×C3×C8—91no—
178C3×C4⋊C4—91no—
179C3×C4⋊C4—181no—
180C3×C4⋊C4—181no—
181C3×C42—91no—
182#217—181no—
183#215—62no—
184#215—182no—
185#215—182no—
186#214—182no—
187(C2×D3)⋊C4—181no—
188(C2×D3)⋊C4—181no—
189C4⋊Dic3—31no—
190C4⋊Dic3—31no—
191C4⋊Dic3—61no—
192Dic3⋊C4—61no—
193Dic3⋊C4—61no—
194Dic3⋊C4—181no—
195Dic3⋊C4—181no—
196C4×Dic3—31no—
197C4×Dic3—91no—
198C2×C3⋊C8—31no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—61no—
202C2×C3⋊C8—91no—
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—42no—
216C32×C4—41no—
217C32×C4—41no—
218C32×C4D3×D411yescharacteristic
219C32×C4#23711yescharacteristic
220C32×C4#23711yescharacteristic
221C32×C4C2×#6711yescharacteristic
222C3⋊Dic3—21no—
223C3⋊Dic3—41no—
224C3⋊Dic3#21512yescomplemented
225C3×Dic3—32no—
226C3×Dic3—61no—
227C3×Dic3—121no—
228C3×Dic3—121no—
229C3×Dic3—121no—
230C3×Dic3—241no—
231C2×SD16—91no—
232C2×SD16—271no—
233C4⋊Q8—91no—
234C4×D4—271no—
235C4⋊C8—271no—
236#103—91no—
237#103—271no—
238H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
239C23×C3—181no—
240C22×D3—61no—
241C22×D3—181no—
242C3×Q8—21no—
243C3×Q8—21no—
244C3×Q8—61no—
245C3×Q8—61no—
246C3×Q8—61no—
247C3×Q8—61no—
248C3×Q8—62no—
249C3×Q8—121no—
250C3×Q8—181no—
251C3×Q8—181no—
252C3×D4—92no—
253C3×D4—181no—
254C2×C3×C4—21no—
255C2×C3×C4—31no—
256C2×C3×C4—31no—
257C2×C3×C4—61no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—91no—
261C2×C3×C4—91no—
262C2×C3×C4—121no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4C2×D3211yescharacteristic
266C2×Dic3—31no—
267C2×Dic3—31no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—91no—
272C2×Dic3—91no—
273C2×Dic3—121no—
274C2×Dic3—181no—
275C2×Dic3—362no—
276D12—32no—
277D12—61no—
278D12—92no—
279D12—181no—
280C4×D3—362no—
281C4×D3—362no—
282Dic6—61no—
283Dic6—61no—
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×C32D3×SD1611yescharacteristic
297C2×C32#91511yescharacteristic
298C2×C32#93811yescharacteristic
299C2×C32#93911yescharacteristic
300C2×C32C2×#21511yescharacteristic
301C2×C32#94211yescharacteristic
302C3⋊D3—42no—
303C3×D3—122no—
304C2×Q8—91no—
305C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
306C2×D4—91no—
307C22×C4—541no—
308SD16—182no—
309SD16—542no—
310C2×C8—91no—
311C2×C8—91no—
312C4⋊C4—91no—
313C4⋊C4—181no—
314C4⋊C4—181no—
315C22⋊C4—541no—
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—122no—
341C3×C4—181no—
342C3×C4—361no—
343C3×C4—361no—
344C3×C4#380511yescharacteristic
345C3×C4D3×#6711yescharacteristic
346Dic3—32no—
347Dic3—61no—
348Dic3—92no—
349Dic3—121no—
350Dic3—121no—
351Dic3—121no—
352Dic3—181no—
353Dic3—241no—
354Dic3—361no—
355C32—21no—
356C32#508311yescharacteristic, complemented
357C32#516211yescharacteristic, complemented
358C23—181no—
359Q8—21no—
360Q8—21no—
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—1082no—
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×C3D3×#21511yescharacteristic
385C2×C3#6394611yescharacteristic
386C2×C3#6397011yescharacteristic
387D3—122no—
388D3—362no—
389C22—182no—
390C22—361no—
391C22—361no—
392C22#8867611yescharacteristic
393C4—41no—
394C4—92no—
395C4—181no—
396C4—361no—
397C4—361no—
398C4#8867211yescharacteristic
399C4#8867611yescharacteristic
400C3—31no—
401C3—31no—
402C3—61no—
403C3#1059156411yescharacteristic
404C2—362no—
405C2#1172123211yescharacteristic
406C2#1172123811yescharacteristic
407C2#1172131311yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3D3×#67
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC2×Q8((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 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #64153⋊C3C2semidirect
5. #11720583C2semidirect
6. #11720311C2semidirect
7. #11720309C2semidirect
23. #64153D3semidirect
96. (C3⋊Dic3)⋊C3SD16semidirect
224. C3⋊Dic3#215semidirect
238. H(3,𝔽3)#461semidirect
356. C32#5083semidirect
357. C32#5162semidirect

Maximal subgroups

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

subgroupindex
#64153⋊C32
C2×#3755⋊C32
#117211322
C2×#884312
#117205832
#117203112
#117203092
#105915693
#105915653