Finite Group Databasebeta

← #13138989

Subgroups — #13138989

Order 1728.

462 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721132C211yesmaximal, characteristic
2#11721083C211yesmaximal, characteristic, complemented
3C2×#88438C211yesmaximal, characteristic
4#11720549C211yesmaximal, characteristic, complemented
5#63648⋊C3C211yesmaximal, characteristic
6#63642⋊C3C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10590795—31nomaximal
9#10590743—31nomaximal
10C2×#6229C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×((C3⋊D3)⋊C3)⋊C4—21no—
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#88450—21no—
19#88449—21no—
20#88447—21no—
21#88438C414yescomplemented
22#63898—31no—
23#63894—31no—
24#63843—31no—
25Dic3×Dic6—31no—
26C2×#3717—31no—
27C2×#3715—31no—
28#63648D311yescharacteristic
29#63642D311yescharacteristic, complemented
30C3×#821—31no—
31C3×#816—31no—
32C3×#816—31no—
33C3×#816—61no—
34C3×#815—31no—
35#63576—31no—
36#63575—31no—
37C22×(C3⋊D3)⋊C3—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C2×#1084—21no—
41#6229—21no—
42#6229C2×C412yescomplemented
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45C2×(C3⋊Dic3)⋊C3D411yescharacteristic
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)C2×C412yes—
48((C3⋊D3)⋊C3)⋊C4—42no—
49C8×H(3,𝔽3)—21no—
50(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
51#4817—91no—
52#4757—91no—
53#4710—91no—
54C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
55C2×C3×D12—31no—
56C2×C3×Dic6—31no—
57C2×C3×Dic6—31no—
58C2×C3×Dic6—61no—
59C2×#624—61no—
60C2×D3×Dic3—61no—
61C2×C32×C8—21no—
62C2×C32×C8C2×D311yescharacteristic
63C2×C32×C8C2×D311yescharacteristic
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65C4×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—31no—
70C3×C4⋊Dic3—61no—
71C3×C4×Dic3—31no—
72C2×C3×C3⋊C8—31no—
73Dic3⋊Dic3—61no—
74Dic3⋊Dic3—61no—
75#3722—61no—
76(C2×D3)⋊Dic3—61no—
77Dic32—61no—
78Dic32—61no—
79#3717—34no—
80#3715—34no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C2×(C3⋊D3)⋊C3—22no—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C4×H(3,𝔽3)C22×C411yescharacteristic
86C4×H(3,𝔽3)#4111yescharacteristic
87#1084—22no—
88#1084—41no—
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3—21no—
91(C3⋊Dic3)⋊C3—41no—
92Q8×Dic3—91no—
93C2×#215—91no—
94D4×Dic3—91no—
95C2×#214—91no—
96C2×#204—91no—
97#893—91no—
98#887—91no—
99C4×D12—91no—
100C4×Dic6—91no—
101C3×#103—91no—
102C3×#102—91no—
103C3×#97—91no—
104#821—31no—
105#821—91no—
106#818—91no—
107#816—31no—
108#816—31no—
109#816—61no—
110#815—31no—
111#815—91no—
112#807—91no—
113#807—91no—
114C22×C3⋊D3—21no—
115C22×C3×D3—61no—
116C2×C32×C4—21no—
117C2×C32×C4C22×D311yescharacteristic
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3D1211yescharacteristic
121C2×C3⋊Dic3D1211yescharacteristic
122C4⋊(C3⋊D3)—21no—
123C4⋊(C3⋊D3)C4×D312yes—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—121no—
132C2×C3×Dic3—121no—
133C3×D12—32no—
134C3×D12—61no—
135C3×Dic6—32no—
136C3×Dic6—32no—
137C3×Dic6—62no—
138C3×Dic6—61no—
139C3×Dic6—61no—
140C3×Dic6—121no—
141#624—122no—
142D3×Dic3—122no—
143C32×C8—21no—
144C32×C8—21no—
145C32×C8—41no—
146C3⋊(C3⋊C8)C4×D312yescomplemented
147C3×C3⋊C8—32no—
148#440—271noSylow
149C2×H(3,𝔽3)C4×D411yescharacteristic
150C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
151C2×H(3,𝔽3)#13711yescharacteristic
152(C3⋊D3)⋊C3—42no—
153C2×C3×Q8—91no—
154C2×C3×D4—91no—
155C22×Dic3—181no—
156C2×D12—31no—
157C2×D12—91no—
158C2×C4×D3—181no—
159C2×C4×D3—181no—
160C2×Dic6—31no—
161C2×Dic6—31no—
162C2×Dic6—61no—
163C2×C3×C8—31no—
164C2×C3×C8—31no—
165C2×C3×C8—61no—
166C2×C3×C8—91no—
167C2×C3×C8D3211yescharacteristic
168C3×C4⋊C4—91no—
169C3×C4⋊C4—91no—
170C3×C42—91no—
171#217—181no—
172#215—94no—
173#214—94no—
174(C2×D3)⋊C4—181no—
175(C2×D3)⋊C4—181no—
176C4⋊Dic3—31no—
177C4⋊Dic3—31no—
178C4⋊Dic3—31no—
179C4⋊Dic3—61no—
180C4⋊Dic3—91no—
181C4⋊Dic3—181no—
182Dic3⋊C4—181no—
183Dic3⋊C4—181no—
184C4×Dic3—31no—
185C4×Dic3—91no—
186C4×Dic3—181no—
187C4×Dic3—181no—
188C4×Dic3—181no—
189C2×C3⋊C8—31no—
190C2×C3⋊C8—91no—
191#204—94no—
192C22×C32—21no—
193C22×C32C2×D1211yescharacteristic
194C22×C32D3×D411yescharacteristic
195C2×C3⋊D3—22no—
196C2×C3⋊D3—41no—
197C2×C3⋊D3—41no—
198C2×C3×D3—62no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C32×C4—21no—
202C32×C4—21no—
203C32×C4C2×C4×D311yescharacteristic
204C32×C4C2×C4×D311yescharacteristic
205C32×C4#23511yescharacteristic
206C32×C4#23711yescharacteristic
207C3⋊Dic3—21no—
208C3⋊Dic3—21no—
209C3⋊Dic3—41no—
210C3×Dic3—62no—
211C3×Dic3—62no—
212C3×Dic3—61no—
213C3×Dic3—61no—
214C3×Dic3—122no—
215C3×Dic3—121no—
216C3×Dic3—121no—
217C3×Dic3—121no—
218C3×Dic3—241no—
219C2×SD16—271no—
220C4×Q8—271no—
221C4×D4—271no—
222#107—271no—
223#103—91no—
224#102—91no—
225#97—91no—
226H(3,𝔽3)#44011yescharacteristic, complemented, Sylow
227C23×C3—181no—
228C22×D3—61no—
229C22×D3—181no—
230C3×Q8—92no—
231C3×Q8—181no—
232C3×D4—92no—
233C3×D4—181no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—61no—
237C2×C3×C4—91no—
238C2×C3×C4—91no—
239C2×C3×C4—181no—
240C2×C3×C4—181no—
241C2×C3×C4—181no—
242C2×C3×C4C2×D3211yescharacteristic
243C2×Dic3—31no—
244C2×Dic3—31no—
245C2×Dic3—61no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—91no—
251C2×Dic3—91no—
252C2×Dic3—121no—
253C2×Dic3—121no—
254C2×Dic3—181no—
255C2×Dic3—362no—
256D12—32no—
257D12—61no—
258D12—92no—
259D12—181no—
260C4×D3—362no—
261C4×D3—362no—
262Dic6—32no—
263Dic6—32no—
264Dic6—62no—
265Dic6—61no—
266Dic6—61no—
267Dic6—121no—
268C3×C8—21no—
269C3×C8—61no—
270C3×C8—61no—
271C3×C8—92no—
272C3×C8—121no—
273C3⋊C8—32no—
274C3⋊C8—92no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32C4×D1211yescharacteristic
279C2×C32#67⋊C411yescharacteristic
280C2×C32#90811yescharacteristic
281C2×C32#90911yescharacteristic
282C2×C32#91111yescharacteristic
283C2×C32#91511yescharacteristic
284C3⋊D3—42no—
285C3×D3—122no—
286C2×Q8—91no—
287C2×D4—91no—
288C22×C4—541no—
289SD16—274no—
290C2×C8—91no—
291C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
292C4⋊C4—91no—
293C4⋊C4—91no—
294C4⋊C4—541no—
295C22⋊C4—541no—
296C42—91no—
297C42—541no—
298C22×C3—31no—
299C22×C3—31no—
300C22×C3—61no—
301C22×C3—182no—
302C22×C3—361no—
303C22×C3—361no—
304C22×C3D3×D1211yescharacteristic
305C2×D3—62no—
306C2×D3—121no—
307C2×D3—121no—
308C2×D3—182no—
309C2×D3—361no—
310C2×D3—361no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—61no—
316C3×C4—61no—
317C3×C4—181no—
318C3×C4—181no—
319C3×C4—182no—
320C3×C4—361no—
321C3×C4—361no—
322C3×C4C4×D3211yescharacteristic
323C3×C4#380411yescharacteristic
324Dic3—62no—
325Dic3—62no—
326Dic3—61no—
327Dic3—61no—
328Dic3—122no—
329Dic3—121no—
330Dic3—121no—
331Dic3—121no—
332Dic3—181no—
333Dic3—181no—
334Dic3—241no—
335Dic3—361no—
336C32—21no—
337C32#471011yescharacteristic, complemented
338C32#475711yescharacteristic, complemented
339C23—181no—
340Q8—92no—
341Q8—181no—
342D4—92no—
343D4—181no—
344C2×C4—91no—
345C2×C4—91no—
346C2×C4—181no—
347C2×C4—181no—
348C2×C4—181no—
349C2×C4—1082no—
350C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
351C8—21no—
352C8—92no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—31no—
358C2×C3—31no—
359C2×C3—61no—
360C2×C3—61no—
361C2×C3—61no—
362C2×C3—362no—
363C2×C3#6379511yescharacteristic
364C2×C3#6380011yescharacteristic
365C2×C3#6388011yescharacteristic
366D3—122no—
367D3—362no—
368C22—182no—
369C22—361no—
370C22—361no—
371C22#8865411yescharacteristic
372C4—181no—
373C4—181no—
374C4—182no—
375C4—361no—
376C4—361no—
377C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
378C4#8867111yescharacteristic
379C3—31no—
380C3—31no—
381C3—61no—
382C3#1059074711yescharacteristic
383C2—362no—
384C2#1172084911yescharacteristic
385C2#1172085211yescharacteristic
386C2#1172112311yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#440
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11721083C2semidirect
4. #11720549C2semidirect
6. #63642⋊C3C2semidirect
7. #11720306C2semidirect
21. #88438C4semidirect
29. #63642D3semidirect
42. #6229C2×C4semidirect
50. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
146. C3⋊(C3⋊C8)C4×D3semidirect
226. H(3,𝔽3)#440semidirect
337. C32#4710semidirect
338. C32#4757semidirect

Maximal subgroups

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

subgroupindex
#117211322
#117210832
C2×#884382
#117205492
#63648⋊C32
#63642⋊C32
#117203062
#105907953
#105907433