Finite Group Databasebeta

← #13138833

Subgroups — #13138833

Order 1728.

449 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 107 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×#3743⋊C3C211yesmaximal, characteristic
2#11721129C211yesmaximal, characteristic
3#11721081C211yesmaximal, characteristic
4#11720551C211yesmaximal, characteristic, complemented
5#63644⋊C3C211yesmaximal, characteristic, complemented
6#11720307C211yesmaximal, characteristic, complemented
7#11720306C211yesmaximal, characteristic, complemented
8#10590688—31nomaximal
9#10590683—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×((C3⋊D3)⋊C3)⋊C4—21no—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#3743⋊C3C414yescomplemented
19#88450—21no—
20#88448—21no—
21#88447—21no—
22C2×#3743D311yescharacteristic
23C2×C3×ΓL(1,𝔽25)—31no—
24#63900—31no—
25D3×C4⋊Dic3—31no—
26#63849—31no—
27#63841—31no—
28#63644D311yescharacteristic, complemented
29C3×#818—31no—
30C3×#818—31no—
31C3×#818—61no—
32C3×#817—31no—
33#63576—31no—
34#63575—31no—
35#63574—31no—
36#63573—31no—
37C22×(C3⋊D3)⋊C3D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C2×#1084—21no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C4×(C3⋊D3)⋊C3C2×C412yes—
45C4×(C3⋊D3)⋊C3Q812yes—
46((C3⋊D3)⋊C3)⋊C4—42no—
47C8×H(3,𝔽3)C2×C412yescomplemented
48(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
49#5121—91no—
50#4881—91no—
51#4861—91no—
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C2×C3×C4×D3—31no—
54C2×#624—61no—
55C2×D3×Dic3—61no—
56C2×C32×C8—21no—
57C2×C32×C8C2×D311yescharacteristic
58C2×C32×C8C2×D311yescharacteristic
59C4⋊(C3⋊Dic3)C2×D311yescharacteristic
60C4⋊(C3⋊Dic3)C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
62#3743Dic314yescomplemented
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—61no—
70C3×C4⋊Dic3—61no—
71C2×C3×C3⋊C8—31no—
72C3×ΓL(1,𝔽25)—34no—
73#3724—61no—
74Dic3⋊Dic3—61no—
75#3722—61no—
76(C2×D3)⋊Dic3—61no—
77Dic32—61no—
78Dic32—61no—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C2×(C3⋊D3)⋊C3—21no—
81C2×(C3⋊D3)⋊C3—21no—
82C2×(C3⋊D3)⋊C3C4⋊C412yes—
83C4×H(3,𝔽3)C22×C411yescharacteristic
84C4×H(3,𝔽3)C2×Q811yescharacteristic
85#1084—41no—
86#1084—41no—
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89(C3⋊Dic3)⋊C3C4⋊C412yes—
90C2×C3×M16—91no—
91#926—91no—
92C2×C4⋊Dic3—91no—
93C2×ΓL(1,𝔽25)—31no—
94C2×ΓL(1,𝔽25)—91no—
95#892—91no—
96#892—91no—
97D3×C4⋊C4—91no—
98D3×C4⋊C4—91no—
99C3×#107—91no—
100C3×#106—91no—
101#818—31no—
102#818—31no—
103#818—61no—
104#818—91no—
105#817—31no—
106#817—91no—
107#817—91no—
108#808—91no—
109#808—91no—
110#807—91no—
111#807—91no—
112C22×C3⋊D3D1211yescharacteristic
113C22×C3×D3—31no—
114C2×C32×C4—21no—
115C2×C32×C4C22×D311yescharacteristic
116C2×C32×C4C22×D311yescharacteristic
117C2×C3⋊Dic3—21no—
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3D1211yescharacteristic
120C4×C3⋊D3Dic612yes—
121C4×C3⋊D3C2×Dic312yes—
122C2×C3×Dic3—31no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—121no—
130C2×C3×Dic3—121no—
131C3×C4×D3—32no—
132C3×C4×D3—32no—
133#624—122no—
134D3×Dic3—122no—
135C32×C8—22no—
136C32×C8C4×D312yescomplemented
137C32×C8C2×Dic312yescomplemented
138C3⋊(C3⋊C8)C2×Dic312yescomplemented
139C3×C3⋊C8—32no—
140#428—271noSylow
141C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
142C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
143C2×H(3,𝔽3)#13711yescharacteristic
144(C3⋊D3)⋊C3—22no—
145C22×C3×C4—91no—
146C22×Dic3—181no—
147C2×C4×D3—31no—
148C2×C4×D3—91no—
149C2×C4×D3—181no—
150C2×C4×D3—181no—
151C3×M16—94no—
152C2×C3×C8—31no—
153C2×C3×C8—31no—
154C2×C3×C8—61no—
155C2×C3×C8—91no—
156C2×C3×C8D3211yescharacteristic
157C3×C4⋊C4—91no—
158C3×C4⋊C4—91no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—91no—
161#217—181no—
162(C2×D3)⋊C4—181no—
163(C2×D3)⋊C4—181no—
164C4⋊Dic3—31no—
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—61no—
171C4⋊Dic3—61no—
172C4⋊Dic3—91no—
173C4⋊Dic3—91no—
174C4⋊Dic3—181no—
175Dic3⋊C4—181no—
176Dic3⋊C4—181no—
177C4×Dic3—181no—
178C4×Dic3—181no—
179C4×Dic3—181no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—91no—
182ΓL(1,𝔽25)—34no—
183ΓL(1,𝔽25)—94no—
184C22×C32—21no—
185C22×C32C2×D1211yescharacteristic
186C22×C32D3×D411yescharacteristic
187C2×C3⋊D3—21no—
188C2×C3⋊D3—21no—
189C2×C3⋊D3C4⋊Dic312yes—
190C2×C3×D3—32no—
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4C2×Dic611yescharacteristic
196C32×C4C2×C4×D311yescharacteristic
197C32×C4D3×Q811yescharacteristic
198C32×C4C22×Dic311yescharacteristic
199C3⋊Dic3—41no—
200C3⋊Dic3—41no—
201C3⋊Dic3C4⋊Dic312yes—
202C3×Dic3—32no—
203C3×Dic3—121no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C3×Dic3—121no—
209C3×Dic3—241no—
210C3×Dic3—241no—
211C2×M16—91no—
212#117—271no—
213C2×C4⋊C4—271no—
214#107—91no—
215#107—271no—
216#106—91no—
217#106—271no—
218H(3,𝔽3)#42811yescharacteristic, complemented, Sylow
219C23×C3—91no—
220C22×D3—31no—
221C22×D3—91no—
222C2×C3×C4—31no—
223C2×C3×C4—31no—
224C2×C3×C4—61no—
225C2×C3×C4—92no—
226C2×C3×C4—92no—
227C2×C3×C4—91no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4—181no—
232C2×C3×C4C2×D3211yescharacteristic
233C2×Dic3—31no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3—91no—
241C2×Dic3—121no—
242C2×Dic3—121no—
243C2×Dic3—181no—
244C2×Dic3—181no—
245C2×Dic3—362no—
246C4×D3—32no—
247C4×D3—32no—
248C4×D3—92no—
249C4×D3—92no—
250C4×D3—362no—
251C4×D3—362no—
252C3×C8—32no—
253C3×C8—32no—
254C3×C8—62no—
255C3×C8—92no—
256C3×C8D3×Dic312yes—
257C3⋊C8—32no—
258C3⋊C8—92no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32D3×C4⋊C411yescharacteristic
263C2×C32#90811yescharacteristic
264C2×C32#90911yescharacteristic
265C2×C32#91411yescharacteristic
266C2×C32#91511yescharacteristic
267C2×C32C2×C4⋊Dic311yescharacteristic
268C3⋊D3—22no—
269C3×D3—62no—
270C22×C4—91no—
271C22×C4—541no—
272M16—94no—
273C2×C8—91no—
274C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
275C4⋊C4—91no—
276C4⋊C4—91no—
277C4⋊C4—91no—
278C4⋊C4—91no—
279C4⋊C4—541no—
280C22⋊C4—541no—
281C42—541no—
282C22×C3—31no—
283C22×C3—31no—
284C22×C3—61no—
285C22×C3—92no—
286C22×C3—181no—
287C22×C3—181no—
288C22×C3D3×D1211yescharacteristic
289C2×D3—32no—
290C2×D3—61no—
291C2×D3—61no—
292C2×D3—92no—
293C2×D3—181no—
294C2×D3—181no—
295C3×C4—31no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—31no—
299C3×C4—61no—
300C3×C4—61no—
301C3×C4—92no—
302C3×C4—361no—
303C3×C4—361no—
304C3×C4—361no—
305C3×C4—361no—
306C3×C4D3×Dic611yescharacteristic
307C3×C4C2×D3×Dic311yescharacteristic
308Dic3—32no—
309Dic3—92no—
310Dic3—121no—
311Dic3—121no—
312Dic3—121no—
313Dic3—121no—
314Dic3—121no—
315Dic3—121no—
316Dic3—241no—
317Dic3—241no—
318Dic3—361no—
319Dic3—361no—
320C32—21no—
321C32#486111yescharacteristic, complemented
322C32#512111yescharacteristic, complemented
323C23—91no—
324C2×C4—91no—
325C2×C4—92no—
326C2×C4—92no—
327C2×C4—181no—
328C2×C4—181no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—1082no—
332C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
333C8—92no—
334C8((C3⋊D3)⋊C3)⋊C412yescomplemented
335C2×C3—31no—
336C2×C3—31no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—61no—
342C2×C3—61no—
343C2×C3—61no—
344C2×C3—182no—
345C2×C3#6379411yescharacteristic
346C2×C3#6380011yescharacteristic
347C2×C3D3×C4⋊Dic311yescharacteristic
348D3—62no—
349D3—182no—
350C22—92no—
351C22—181no—
352C22—181no—
353C22#8865411yescharacteristic
354C4—92no—
355C4—361no—
356C4—361no—
357C4—361no—
358C4—361no—
359C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
360C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
361C3—31no—
362C3—31no—
363C3—61no—
364C3#1059068111yescharacteristic
365C2—182no—
366C2#1172084811yescharacteristic
367C2#1172085211yescharacteristic
368C2#1172112911yescharacteristic
subgroupG/H
centerC22#88654
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC22×(C3⋊D3)⋊C3D4
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. #11720551C2semidirect
5. #63644⋊C3C2semidirect
6. #11720307C2semidirect
7. #11720306C2semidirect
18. #3743⋊C3C4semidirect
28. #63644D3semidirect
47. C8×H(3,𝔽3)C2×C4semidirect
48. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
62. #3743Dic3semidirect
136. C32×C8C4×D3semidirect
137. C32×C8C2×Dic3semidirect
138. C3⋊(C3⋊C8)C2×Dic3semidirect
218. H(3,𝔽3)#428semidirect
321. C32#4861semidirect
322. C32#5121semidirect
334. C8((C3⋊D3)⋊C3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3743⋊C32
#117211292
#117210812
#117205512
#63644⋊C32
#117203072
#117203062
#105906883
#105906833