Finite Group Databasebeta

← #13138964

Subgroups — #13138964

Order 1728.

407 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721149C211yesmaximal, characteristic
2#11721093C211yesmaximal, characteristic, complemented
3#11720554C211yesmaximal, characteristic
4#63644⋊C3C211yesmaximal, characteristic, complemented
5#63641⋊C3C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic
7#11720306C211yesmaximal, characteristic, complemented
8#10590782—31nomaximal
9#10590772—31nomaximal
10C2×C4⋊#272C2211yescharacteristic
11C2×#6199—21no—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C4⋊#1084C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#88451—21no—
19#88449—21no—
20#88448—21no—
21#63898—31no—
22Dic3×D12—31no—
23#63872—31no—
24#63869—31no—
25#63644D311yescharacteristic, complemented
26#63641D311yescharacteristic, complemented
27C3×#821—31no—
28C3×#821—31no—
29C3×#821—61no—
30C3×#817—31no—
31C3×Dic3⋊C8—31no—
32#63576—31no—
33#63575—31no—
34#63563—31no—
35#63562—31no—
36C22×#272—21no—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C4⋊#272—21no—
40C4⋊#272Q812yes—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3D411yescharacteristic
44C2×(C3⋊Dic3)⋊C3D411yescharacteristic
45#6199—42no—
46C8×H(3,𝔽3)—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48#4871—91no—
49#4847—91no—
50#4763—91no—
51C2×C3×D12—31no—
52C2×C3×D12—31no—
53C2×C3×D12—61no—
54C2×D3×Dic3—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
61C4×C3⋊Dic3C2×D311yescharacteristic
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—31no—
65C3×C4⋊Dic3—31no—
66C3×C4⋊Dic3—31no—
67C3×C4⋊Dic3—61no—
68C3×C4×Dic3—31no—
69C2×C3×C3⋊C8—31no—
70#3724—61no—
71Dic3⋊Dic3—61no—
72Dic3⋊Dic3—61no—
73Dic3⋊Dic3—61no—
74(C2×D3)⋊Dic3—61no—
75(C2×D3)⋊Dic3—61no—
76C22×H(3,𝔽3)C2×D411yescharacteristic
77C2×#272—22no—
78C2×#272—41no—
79C2×#272—41no—
80C4×H(3,𝔽3)C2×Q811yescharacteristic
81C4×H(3,𝔽3)#4111yescharacteristic
82#1084—41no—
83(C3⋊Dic3)⋊C3—21no—
84(C3⋊Dic3)⋊C3—21no—
85(C3⋊Dic3)⋊C3—41no—
86(C3⋊Dic3)⋊C3—41no—
87D4×Dic3—91no—
88#893—91no—
89#890—91no—
90#890—91no—
91C4×D12—91no—
92#870—91no—
93C3×#106—91no—
94C3×C4⋊C8—91no—
95C3×#102—91no—
96#832—91no—
97#821—31no—
98#821—31no—
99#821—61no—
100#821—91no—
101#818—91no—
102#817—31no—
103#817—91no—
104Dic3⋊C8—31no—
105Dic3⋊C8—91no—
106#809—91no—
107#807—91no—
108#807—91no—
109C22×C3×D3—61no—
110C22×C3×D3—61no—
111C22×C3×D3—121no—
112C2×C32×C4—21no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—21no—
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3D1211yescharacteristic
118C2×C3⋊Dic3D1211yescharacteristic
119C2×C3×Dic3—31no—
120C2×C3×Dic3—31no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—121no—
126C3×D12—32no—
127C3×D12—32no—
128C3×D12—62no—
129C3×D12—61no—
130C3×D12—61no—
131C3×D12—121no—
132D3×Dic3—122no—
133D3×Dic3—122no—
134C32×C8—21no—
135C32×C8—21no—
136C32×C8—41no—
137C3⋊(C3⋊C8)—21no—
138C3×C3⋊C8—61no—
139#478—271noSylow
140C2×H(3,𝔽3)C22⋊Q811yescharacteristic
141C2×H(3,𝔽3)#13511yescharacteristic
142C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
143#272—42no—
144C2×C3×D4—91no—
145C22×Dic3—181no—
146C2×D12—31no—
147C2×D12—31no—
148C2×D12—61no—
149C2×C4×D3—181no—
150C2×C4×D3—181no—
151C2×C3×C8—31no—
152C2×C3×C8—31no—
153C2×C3×C8—61no—
154C2×C3×C8—91no—
155C2×C3×C8D3211yescharacteristic
156C3×C4⋊C4—91no—
157C3×C4⋊C4—91no—
158C3×C4⋊C4—91no—
159C3×C42—91no—
160#217—181no—
161(C2×D3)⋊C4—181no—
162(C2×D3)⋊C4—181no—
163C4⋊Dic3—31no—
164C4⋊Dic3—31no—
165C4⋊Dic3—31no—
166C4⋊Dic3—31no—
167C4⋊Dic3—61no—
168C4⋊Dic3—91no—
169C4⋊Dic3—91no—
170C4⋊Dic3—181no—
171C4⋊Dic3—181no—
172Dic3⋊C4—181no—
173Dic3⋊C4—181no—
174Dic3⋊C4—181no—
175Dic3⋊C4—181no—
176C4×Dic3—31no—
177C4×Dic3—91no—
178C2×C3⋊C8—31no—
179C2×C3⋊C8—91no—
180C22×C32—21no—
181C22×C32C2×D1211yescharacteristic
182C22×C32D3×D411yescharacteristic
183C2×C3×D3—62no—
184C2×C3×D3—62no—
185C2×C3×D3—122no—
186C2×C3×D3—121no—
187C2×C3×D3—121no—
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C2×C3×D3—241no—
191C2×C3×D3—241no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4#23511yescharacteristic
195C32×C4#23711yescharacteristic
196C32×C4D3×Q811yescharacteristic
197C32×C4D3×Q811yescharacteristic
198C3⋊Dic3—21no—
199C3⋊Dic3—21no—
200C3⋊Dic3—41no—
201C3⋊Dic3—41no—
202C3×Dic3—61no—
203C3×Dic3—61no—
204C3×Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207C3×Dic3—121no—
208C3×Dic3—241no—
209#125—271no—
210C4×D4—271no—
211#107—271no—
212#106—91no—
213C4⋊C8—91no—
214#102—91no—
215#102—271no—
216H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
217C23×C3—181no—
218C22×D3—61no—
219C22×D3—61no—
220C22×D3—121no—
221C3×D4—92no—
222C3×D4—181no—
223C2×C3×C4—31no—
224C2×C3×C4—31no—
225C2×C3×C4—61no—
226C2×C3×C4—91no—
227C2×C3×C4—91no—
228C2×C3×C4—181no—
229C2×C3×C4—181no—
230C2×C3×C4—181no—
231C2×C3×C4C2×D3211yescharacteristic
232C2×Dic3—31no—
233C2×Dic3—31no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236C2×Dic3—61no—
237C2×Dic3—61no—
238C2×Dic3—91no—
239C2×Dic3—91no—
240C2×Dic3—121no—
241C2×Dic3—181no—
242C2×Dic3—181no—
243C2×Dic3—362no—
244D12—32no—
245D12—32no—
246D12—62no—
247D12—61no—
248D12—61no—
249D12—121no—
250C4×D3—362no—
251C4×D3—362no—
252C3×C8—21no—
253C3×C8—61no—
254C3×C8—61no—
255C3×C8—121no—
256C3×C8—181no—
257C3⋊C8—61no—
258C3⋊C8—181no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32#89611yescharacteristic
263C2×C32#89711yescharacteristic
264C2×C32#90411yescharacteristic
265C2×C32#90811yescharacteristic
266C2×C32#91411yescharacteristic
267C2×C32#91611yescharacteristic
268C3×D3—122no—
269C3×D3—122no—
270C3×D3—242no—
271C2×D4—91no—
272C22×C4—541no—
273C2×C8—91no—
274C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
275C4⋊C4—91no—
276C4⋊C4—91no—
277C4⋊C4—91no—
278C4⋊C4—541no—
279C4⋊C4—541no—
280C22⋊C4—541no—
281C42—91no—
282C22×C3—31no—
283C22×C3—31no—
284C22×C3—61no—
285C22×C3—182no—
286C22×C3—361no—
287C22×C3—361no—
288C22×C3D3×D1211yescharacteristic
289C2×D3—62no—
290C2×D3—62no—
291C2×D3—122no—
292C2×D3—121no—
293C2×D3—121no—
294C2×D3—121no—
295C2×D3—121no—
296C2×D3—241no—
297C2×D3—241no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—61no—
303C3×C4—61no—
304C3×C4—181no—
305C3×C4—181no—
306C3×C4—361no—
307C3×C4—361no—
308C3×C4—361no—
309C3×C4#379711yescharacteristic
310C3×C4#380411yescharacteristic
311Dic3—61no—
312Dic3—61no—
313Dic3—121no—
314Dic3—121no—
315Dic3—121no—
316Dic3—121no—
317Dic3—181no—
318Dic3—181no—
319Dic3—241no—
320Dic3—361no—
321Dic3—361no—
322C32—21no—
323C32#484711yescharacteristic, complemented
324C32#487111yescharacteristic, complemented
325C23—181no—
326D4—92no—
327D4—181no—
328C2×C4—91no—
329C2×C4—91no—
330C2×C4—181no—
331C2×C4—181no—
332C2×C4—181no—
333C2×C4—1082no—
334C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
335C8—21no—
336C8—181no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—61no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3—362no—
347C2×C3#6379411yescharacteristic
348C2×C3#6380611yescharacteristic
349C2×C3#6389511yescharacteristic
350D3—122no—
351D3—122no—
352D3—242no—
353C22—182no—
354C22—361no—
355C22—361no—
356C22#8865411yescharacteristic
357C4—181no—
358C4—181no—
359C4—361no—
360C4—361no—
361C4—361no—
362C4#8865111yescharacteristic
363C4#8867111yescharacteristic
364C3—31no—
365C3—31no—
366C3—61no—
367C3#1059076911yescharacteristic
368C2—362no—
369C2#1172084811yescharacteristic
370C2#1172085411yescharacteristic
371C2#1172115011yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC2×C4⋊#272C22
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#478
LCSG ▷ C2×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. #11721093C2semidirect
4. #63644⋊C3C2semidirect
5. #63641⋊C3C2semidirect
7. #11720306C2semidirect
25. #63644D3semidirect
26. #63641D3semidirect
216. H(3,𝔽3)#478semidirect
323. C32#4847semidirect
324. C32#4871semidirect

Maximal subgroups

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

subgroupindex
#117211492
#117210932
#117205542
#63644⋊C32
#63641⋊C32
#117203242
#117203062
#105907823
#105907723