Finite Group Databasebeta

← #13138906

Subgroups — #13138906

Order 1728.

481 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 481 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C2×#88527C211yesmaximal, characteristic
2#11721152C211yesmaximal, characteristic
3#11721089C211yesmaximal, characteristic
4#63645⋊C3C211yesmaximal, characteristic, complemented
5#63644⋊C3C211yesmaximal, characteristic, complemented
6#11720315C211yesmaximal, characteristic, complemented
7#11720314C211yesmaximal, characteristic, complemented
8#10590683—31nomaximal
9#10590681—31nomaximal
10C2×C4×#272C2211yescharacteristic
11C2×#6199—21no—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14#88527C414yescomplemented
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#88451—21no—
20#88449—21no—
21#88447—21no—
22C2×C3×ΓL(1,𝔽25)—31no—
23C2×C3×ΓL(1,𝔽25)—31no—
24C2×C3×ΓL(1,𝔽25)—61no—
25D3×C4⋊Dic3—31no—
26D3×C4⋊Dic3—31no—
27#63841—31no—
28#63841—31no—
29#63645D311yescharacteristic, complemented
30#63644D311yescharacteristic, complemented
31C3×#818—31no—
32C3×#817—31no—
33#63575—31no—
34#63575—31no—
35#63573—31no—
36#63573—31no—
37C22×#272D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084D411yescharacteristic
40C4×#272C2×C412yes—
41C4×#272Q812yes—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46#6199—42no—
47C8×H(3,𝔽3)C2×C412yescomplemented
48#6180C2×C412yescomplemented
49#5121—91no—
50#4881—91no—
51#4861—91no—
52C2×C3×C4×D3—31no—
53C2×C3×C4×D3—31no—
54C2×C3×C4×D3—61no—
55C2×D3×Dic3—61no—
56C2×D3×Dic3—61no—
57C2×C32×C8—21no—
58C2×C32×C8C2×D311yescharacteristic
59C2×C32×C8C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62C4⋊(C3⋊Dic3)C2×D311yescharacteristic
63C4⋊(C3⋊Dic3)C2×D311yescharacteristic
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C2×C3×C3⋊C8—31no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8—61no—
71C3×ΓL(1,𝔽25)—34no—
72C3×ΓL(1,𝔽25)—34no—
73C3×ΓL(1,𝔽25)—64no—
74Dic3⋊Dic3—61no—
75Dic3⋊Dic3—61no—
76(C2×D3)⋊Dic3—61no—
77(C2×D3)⋊Dic3—61no—
78Dic32—61no—
79Dic32—61no—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C2×#272—21no—
82C2×#272—21no—
83C2×#272C4⋊C412yes—
84C4×H(3,𝔽3)C22×C411yescharacteristic
85C4×H(3,𝔽3)C2×Q811yescharacteristic
86#1084C4⋊C412yes—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89(C3⋊Dic3)⋊C3—41no—
90(C3⋊Dic3)⋊C3—41no—
91C2×C3×M16—91no—
92#926—91no—
93C2×C4⋊Dic3—91no—
94C2×ΓL(1,𝔽25)—31no—
95C2×ΓL(1,𝔽25)—31no—
96C2×ΓL(1,𝔽25)—61no—
97#892—91no—
98#892—91no—
99D3×C4⋊C4—91no—
100D3×C4⋊C4—91no—
101C3×#107—91no—
102C3×#106—91no—
103#818—31no—
104#818—91no—
105#818—91no—
106#817—31no—
107#817—91no—
108#817—91no—
109#808—91no—
110#808—91no—
111#807—91no—
112#807—91no—
113C22×C3×D3—31no—
114C22×C3×D3—31no—
115C22×C3×D3—61no—
116C2×C32×C4—21no—
117C2×C32×C4C22×D311yescharacteristic
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—21no—
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—21no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C3×C4×D3—32no—
131C3×C4×D3—32no—
132C3×C4×D3—32no—
133C3×C4×D3—32no—
134C3×C4×D3—62no—
135C3×C4×D3—62no—
136D3×Dic3—122no—
137D3×Dic3—122no—
138C32×C8—22no—
139C32×C8C4×D312yescomplemented
140C32×C8C4×D312yescomplemented
141C3×C3⋊C8—32no—
142C3×C3⋊C8—32no—
143C3×C3⋊C8—62no—
144#428—271noSylow
145C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
146C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
147C2×H(3,𝔽3)#13711yescharacteristic
148#272—22no—
149C22×C3×C4—91no—
150C22×Dic3—181no—
151C2×C4×D3—31no—
152C2×C4×D3—31no—
153C2×C4×D3—61no—
154C2×C4×D3—181no—
155C2×C4×D3—181no—
156C3×M16—94no—
157C2×C3×C8—31no—
158C2×C3×C8—31no—
159C2×C3×C8—61no—
160C2×C3×C8—91no—
161C2×C3×C8D3211yescharacteristic
162C3×C4⋊C4—91no—
163C3×C4⋊C4—91no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4—91no—
166#217—181no—
167(C2×D3)⋊C4—181no—
168(C2×D3)⋊C4—181no—
169C4⋊Dic3—31no—
170C4⋊Dic3—31no—
171C4⋊Dic3—31no—
172C4⋊Dic3—31no—
173C4⋊Dic3—91no—
174C4⋊Dic3—91no—
175C4⋊Dic3—91no—
176C4⋊Dic3—91no—
177C4⋊Dic3—181no—
178Dic3⋊C4—181no—
179Dic3⋊C4—181no—
180C4×Dic3—181no—
181C4×Dic3—181no—
182C4×Dic3—181no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8—31no—
185C2×C3⋊C8—61no—
186ΓL(1,𝔽25)—34no—
187ΓL(1,𝔽25)—34no—
188ΓL(1,𝔽25)—64no—
189C22×C32—21no—
190C22×C32D3×D411yescharacteristic
191C22×C32D3×D411yescharacteristic
192C2×C3×D3—32no—
193C2×C3×D3—32no—
194C2×C3×D3—62no—
195C2×C3×D3—61no—
196C2×C3×D3—61no—
197C2×C3×D3—61no—
198C2×C3×D3—61no—
199C2×C3×D3—121no—
200C2×C3×D3—121no—
201C32×C4—21no—
202C32×C4—21no—
203C32×C4C2×C4×D311yescharacteristic
204C32×C4C2×C4×D311yescharacteristic
205C32×C4D3×Q811yescharacteristic
206C32×C4D3×Q811yescharacteristic
207C3⋊Dic3—41no—
208C3⋊Dic3—41no—
209C3⋊Dic3—41no—
210C3⋊Dic3—41no—
211C3×Dic3—32no—
212C3×Dic3—32no—
213C3×Dic3—62no—
214C3×Dic3—121no—
215C3×Dic3—121no—
216C3×Dic3—121no—
217C3×Dic3—121no—
218C2×M16—91no—
219#117—271no—
220C2×C4⋊C4—271no—
221#107—91no—
222#107—271no—
223#106—91no—
224#106—271no—
225H(3,𝔽3)#42811yescharacteristic, complemented, Sylow
226C23×C3—91no—
227C22×D3—31no—
228C22×D3—31no—
229C22×D3—61no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—61no—
233C2×C3×C4—91no—
234C2×C3×C4—92no—
235C2×C3×C4—92no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4C2×D3211yescharacteristic
241C2×Dic3—31no—
242C2×Dic3—31no—
243C2×Dic3—61no—
244C2×Dic3—61no—
245C2×Dic3—61no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C2×Dic3—181no—
249C2×Dic3—181no—
250C2×Dic3—181no—
251C2×Dic3—181no—
252C2×Dic3—362no—
253C4×D3—32no—
254C4×D3—32no—
255C4×D3—32no—
256C4×D3—32no—
257C4×D3—62no—
258C4×D3—62no—
259C4×D3—362no—
260C4×D3—362no—
261C3×C8—32no—
262C3×C8—32no—
263C3×C8—62no—
264C3×C8—92no—
265C3×C8#62412yes—
266C3⋊C8—32no—
267C3⋊C8—32no—
268C3⋊C8—62no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32D3×C4⋊C411yescharacteristic
273C2×C32D3×C4⋊C411yescharacteristic
274C2×C32#91111yescharacteristic
275C2×C32#91411yescharacteristic
276C2×C32#91511yescharacteristic
277C2×C32#91811yescharacteristic
278C3×D3—62no—
279C3×D3—62no—
280C3×D3—122no—
281C22×C4—91no—
282C22×C4—541no—
283M16—94no—
284C2×C8—91no—
285C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
286C4⋊C4—91no—
287C4⋊C4—91no—
288C4⋊C4—91no—
289C4⋊C4—91no—
290C4⋊C4—541no—
291C22⋊C4—541no—
292C42—541no—
293C22×C3—31no—
294C22×C3—31no—
295C22×C3—61no—
296C22×C3—92no—
297C22×C3—181no—
298C22×C3—181no—
299C22×C3#380211yescharacteristic
300C2×D3—32no—
301C2×D3—32no—
302C2×D3—62no—
303C2×D3—61no—
304C2×D3—61no—
305C2×D3—61no—
306C2×D3—61no—
307C2×D3—121no—
308C2×D3—121no—
309C3×C4—31no—
310C3×C4—31no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—61no—
314C3×C4—61no—
315C3×C4—92no—
316C3×C4—361no—
317C3×C4—361no—
318C3×C4—361no—
319C3×C4—361no—
320C3×C4#379711yescharacteristic
321C3×C4C2×#62411yescharacteristic
322Dic3—32no—
323Dic3—32no—
324Dic3—62no—
325Dic3—121no—
326Dic3—121no—
327Dic3—121no—
328Dic3—121no—
329Dic3—361no—
330Dic3—361no—
331Dic3—361no—
332Dic3—361no—
333C32—21no—
334C32#486111yescharacteristic, complemented
335C32#488111yescharacteristic, complemented
336C23—91no—
337C2×C4—91no—
338C2×C4—92no—
339C2×C4—92no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4—181no—
343C2×C4—181no—
344C2×C4—1082no—
345C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
346C8—92no—
347C8#619912yescomplemented
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—61no—
355C2×C3—61no—
356C2×C3—61no—
357C2×C3—182no—
358C2×C3#6379811yescharacteristic
359C2×C3#6381111yescharacteristic
360C2×C3#6390011yescharacteristic
361D3—62no—
362D3—62no—
363D3—122no—
364C22—92no—
365C22—181no—
366C22—181no—
367C22#8865511yescharacteristic
368C4—92no—
369C4—361no—
370C4—361no—
371C4—361no—
372C4—361no—
373C4#8865111yescharacteristic
374C4C2×#619911yescharacteristic
375C3—31no—
376C3—31no—
377C3—61no—
378C3#1059068811yescharacteristic
379C2—182no—
380C2#1172083911yescharacteristic
381C2#1172084511yescharacteristic
382C2#1172115211yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×#272D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#428
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
4. #63645⋊C3C2semidirect
5. #63644⋊C3C2semidirect
6. #11720315C2semidirect
7. #11720314C2semidirect
14. #88527C4semidirect
29. #63645D3semidirect
30. #63644D3semidirect
47. C8×H(3,𝔽3)C2×C4semidirect
48. #6180C2×C4semidirect
139. C32×C8C4×D3semidirect
140. C32×C8C4×D3semidirect
225. H(3,𝔽3)#428semidirect
334. C32#4861semidirect
335. C32#4881semidirect
347. C8#6199semidirect

Maximal subgroups

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

subgroupindex
C2×#885272
#117211522
#117210892
#63645⋊C32
#63644⋊C32
#117203152
#117203142
#105906833
#105906813