Finite Group Databasebeta

← #13137922

Subgroups — #13137922

Order 1728.

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

normal subgroups only

This group has 414 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#64097⋊C3C211yesmaximal, characteristic, complemented
2#11721132C211yesmaximal, characteristic
3#11720530C211yesmaximal, characteristic, complemented
4#63636⋊C3C211yesmaximal, characteristic
5#11720318C211yesmaximal, characteristic
6#11720311C211yesmaximal, characteristic, complemented
7#11720307C211yesmaximal, characteristic, complemented
8#10590472—31nomaximal
9#10590466—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×#634⋊C3—21no—
12C2×((C3⋊D3)⋊C3)⋊C4—21no—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C4⋊#1084C2211yescharacteristic
15C2×#6180C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20#88450—21no—
21#64097D311yescharacteristic, complemented
22C3×#888—31no—
23#63898—31no—
24#63894—31no—
25#63636D311yescharacteristic
26C3×#809—31no—
27C3×#808—31no—
28C3×#808—31no—
29C3×#808—61no—
30#63574—31no—
31#63573—31no—
32#63572—31no—
33Dic3⋊(C3⋊C8)—31no—
34#63564—31no—
35#63563—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39C2×#1084—21no—
40C2×(C3⋊Dic3)⋊C3—21no—
41C2×(C3⋊Dic3)⋊C3D411yescharacteristic
42C2×(C3⋊Dic3)⋊C3D411yescharacteristic
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)Q812yescomplemented
45#634⋊C3—42no—
46((C3⋊D3)⋊C3)⋊C4—42no—
47#6180—21no—
48(C3⋊(C3⋊C8))⋊C3—21no—
49#5084—91no—
50#4870—91no—
51#4761—91no—
52C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
53C2×#634—21no—
54C2×C3×D12—31no—
55C2×C3×Dic6—61no—
56C2×#624—61no—
57C2×D3×Dic3—61no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4C2×D311yescharacteristic
60C32×C4⋊C4C2×D311yescharacteristic
61C4⋊(C3⋊Dic3)C2×D311yescharacteristic
62#3750—21no—
63C4×C3⋊Dic3C2×D311yescharacteristic
64C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—61no—
69C3×Dic3⋊C4—61no—
70C3×C4×Dic3—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—61no—
75#3722—61no—
76(C2×D3)⋊Dic3—61no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C2×(C3⋊D3)⋊C3—22no—
79C2×(C3⋊D3)⋊C3—41no—
80C2×(C3⋊D3)⋊C3—41no—
81C4×H(3,𝔽3)—41no—
82C4×H(3,𝔽3)C2×Q811yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84#1084—41no—
85(C3⋊Dic3)⋊C3—21no—
86(C3⋊Dic3)⋊C3—41no—
87(C3⋊Dic3)⋊C3SD1612yes—
88C3×C4⋊Q8—91no—
89D4×Dic3—91no—
90#893—91no—
91#888—31no—
92#888—91no—
93C4×D12—91no—
94C3×#106—91no—
95C3×#102—91no—
96#832—91no—
97#821—91no—
98#817—91no—
99Dic3⋊C8—91no—
100Dic3⋊C8—91no—
101#809—31no—
102#809—91no—
103#809—91no—
104#808—31no—
105#808—31no—
106#808—61no—
107#808—91no—
108#808—91no—
109C4⋊(C3⋊C8)—91no—
110C22×C3⋊D3—21no—
111C22×C3×D3—61no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—41no—
116C2×C32×C4C22×D311yescharacteristic
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3#6711yescharacteristic
120C2×C3⋊Dic3#6711yescharacteristic
121C4⋊(C3⋊D3)—21no—
122C4⋊(C3⋊D3)Dic612yescomplemented
123#634—42no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—121no—
130C3×D12—32no—
131C3×D12—61no—
132C3×Dic6—122no—
133#624—122no—
134D3×Dic3—122no—
135C3⋊(C3⋊C8)—21no—
136C3×C3⋊C8—62no—
137C3×C3⋊C8—61no—
138C3×C3⋊C8—61no—
139C3×C3⋊C8—61no—
140#476—271noSylow
141C2×H(3,𝔽3)C22⋊Q811yescharacteristic
142C2×H(3,𝔽3)C2×SD1611yescharacteristic
143C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
144(C3⋊D3)⋊C3—42no—
145C2×C3×Q8—181no—
146C2×C3×D4—91no—
147C22×Dic3—181no—
148C2×D12—31no—
149C2×D12—91no—
150C2×C4×D3—181no—
151C2×C4×D3—181no—
152C2×Dic6—61no—
153C2×Dic6—181no—
154C2×C3×C8—91no—
155C2×C3×C8—91no—
156C3×C4⋊C4—31no—
157C3×C4⋊C4—31no—
158C3×C4⋊C4—61no—
159C3×C4⋊C4—91no—
160C3×C4⋊C4—91no—
161C3×C4⋊C4—181no—
162C3×C4⋊C4D3211yescharacteristic
163C3×C42—91no—
164#217—181no—
165(C2×D3)⋊C4—181no—
166(C2×D3)⋊C4—181no—
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—31no—
170C4⋊Dic3—61no—
171C4⋊Dic3—91no—
172Dic3⋊C4—61no—
173Dic3⋊C4—181no—
174C4×Dic3—31no—
175C4×Dic3—91no—
176C2×C3⋊C8—31no—
177C2×C3⋊C8—31no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—61no—
180C2×C3⋊C8—91no—
181C22×C32—21no—
182C22×C32D3×D411yescharacteristic
183C22×C32C2×#6711yescharacteristic
184C2×C3⋊D3—22no—
185C2×C3⋊D3—41no—
186C2×C3⋊D3—41no—
187C2×C3×D3—62no—
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4—81no—
195C32×C4C2×Dic611yescharacteristic
196C32×C4#23711yescharacteristic
197C32×C4D3×Q811yescharacteristic
198C32×C4#23911yescharacteristic
199C3⋊Dic3—21no—
200C3⋊Dic3—41no—
201C3⋊Dic3#21512yes—
202C3×Dic3—32no—
203C3×Dic3—61no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207C3×Dic3—241no—
208C4⋊Q8—91no—
209C4×D4—271no—
210#106—91no—
211#106—271no—
212C4⋊C8—271no—
213#102—91no—
214#102—271no—
215H(3,𝔽3)#47611yescharacteristic, complemented, Sylow
216C23×C3—181no—
217C22×D3—61no—
218C22×D3—181no—
219C3×Q8—362no—
220C3×D4—92no—
221C3×D4—181no—
222C2×C3×C4—21no—
223C2×C3×C4—31no—
224C2×C3×C4—31no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—61no—
228C2×C3×C4—91no—
229C2×C3×C4—91no—
230C2×C3×C4—121no—
231C2×C3×C4—181no—
232C2×C3×C4—181no—
233C2×C3×C4C2×D3211yescharacteristic
234C2×Dic3—31no—
235C2×Dic3—31no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—61no—
239C2×Dic3—91no—
240C2×Dic3—91no—
241C2×Dic3—121no—
242C2×Dic3—181no—
243C2×Dic3—362no—
244D12—32no—
245D12—61no—
246D12—92no—
247D12—181no—
248C4×D3—362no—
249C4×D3—362no—
250Dic6—122no—
251Dic6—362no—
252C3×C8—181no—
253C3×C8—181no—
254C3⋊C8—61no—
255C3⋊C8—61no—
256C3⋊C8—62no—
257C3⋊C8—61no—
258C3⋊C8—181no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32#87811yescharacteristic
263C2×C32#91111yescharacteristic
264C2×C32D3×SD1611yescharacteristic
265C2×C32#93211yescharacteristic
266C2×C32C2×#21511yescharacteristic
267C2×C32#94611yescharacteristic
268C3⋊D3—42no—
269C3×D3—122no—
270C2×Q8—181no—
271C2×D4—91no—
272C22×C4—541no—
273C2×C8—91no—
274C2×C8—91no—
275C4⋊C4—91no—
276C4⋊C4—91no—
277C4⋊C4—181no—
278C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
279C22⋊C4—541no—
280C42—91no—
281C22×C3—31no—
282C22×C3—31no—
283C22×C3—61no—
284C22×C3—182no—
285C22×C3—361no—
286C22×C3—361no—
287C22×C3D3×#6711yescharacteristic
288C2×D3—62no—
289C2×D3—121no—
290C2×D3—121no—
291C2×D3—182no—
292C2×D3—361no—
293C2×D3—361no—
294C3×C4—31no—
295C3×C4—31no—
296C3×C4—31no—
297C3×C4—31no—
298C3×C4—41no—
299C3×C4—61no—
300C3×C4—61no—
301C3×C4—92no—
302C3×C4—121no—
303C3×C4—121no—
304C3×C4—181no—
305C3×C4—241no—
306C3×C4—361no—
307C3×C4—361no—
308C3×C4D3×Dic611yescharacteristic
309C3×C4#379611yescharacteristic
310Dic3—32no—
311Dic3—61no—
312Dic3—92no—
313Dic3—121no—
314Dic3—121no—
315Dic3—121no—
316Dic3—181no—
317Dic3—241no—
318Dic3—361no—
319C32—21no—
320C32#476111yescharacteristic, complemented
321C32#508411yescharacteristic, complemented
322C23—181no—
323Q8—362no—
324D4—92no—
325D4—181no—
326C2×C4—21no—
327C2×C4—91no—
328C2×C4—91no—
329C2×C4—181no—
330C2×C4—181no—
331C2×C4—1082no—
332C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
333C8—181no—
334C8—181no—
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—362no—
345C2×C3#6388711yescharacteristic
346C2×C3#6393211yescharacteristic
347C2×C3D3×#21511yescharacteristic
348D3—122no—
349D3—362no—
350C22—182no—
351C22—361no—
352C22—361no—
353C22#8867611yescharacteristic
354C4—41no—
355C4—92no—
356C4—181no—
357C4—361no—
358C4—361no—
359C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
360C4#8864911yescharacteristic
361C3—31no—
362C3—31no—
363C3—61no—
364C3#1059046511yescharacteristic
365C2—362no—
366C2#1172112811yescharacteristic
367C2#1172120011yescharacteristic
368C2#1172123211yescharacteristic
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)#476
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
1. #64097⋊C3C2semidirect
3. #11720530C2semidirect
6. #11720311C2semidirect
7. #11720307C2semidirect
21. #64097D3semidirect
44. C4⋊((C3⋊D3)⋊C3)Q8semidirect
122. C4⋊(C3⋊D3)Dic6semidirect
215. H(3,𝔽3)#476semidirect
320. C32#4761semidirect
321. C32#5084semidirect

Maximal subgroups

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

subgroupindex
#64097⋊C32
#117211322
#117205302
#63636⋊C32
#117203182
#117203112
#117203072
#105904723
#105904663