Finite Group Databasebeta

← #13143922

Subgroups — #13143922

Order 1728.

810 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 155 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×#88742C211yesmaximal, characteristic, complemented
2C2×#88544C211yesmaximal, characteristic, complemented
3C2×#88524C211yesmaximal, characteristic, complemented
4#11721751C211yesmaximal, characteristic, complemented
5C3×C9×#102C211yesmaximal, characteristic, complemented
6#11720542C211yesmaximal, characteristic, complemented
7#11720527C211yesmaximal, characteristic, complemented
8#10592480—31nomaximal
9#10588461—33nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C22×C3⋊Dic9—21no—
13#88742—22no—
14#88742—22no—
15C2×C4⋊(C3⋊D9)—21no—
16C2×C4⋊(C3⋊D9)C2211yescharacteristic
17C2×C4×C3⋊D9C2211yescharacteristic
18C2×#6226C2211yescharacteristic
19C2×C3×C8×C9C2211yescharacteristic
20C3×C9×C4⋊C4C2211yescharacteristic
21#88544—22no—
22#88536—21no—
23C2×C3⋊(C9⋊C8)C2211yescharacteristic
24#88524—22no—
25C2×#3830—31no—
26C2×#3753—31no—
27C2×#3744—31no—
28#64104—31no—
29C2×#3699—33no—
30C32×#102D311yescharacteristic
31#63647—31no—
32#63637—31no—
33C2×#3673—33no—
34C2×#3664—33no—
35#63457—33no—
36C9×#102D313yescomplemented
37(C2×D9)⋊C8—33no—
38#63368—33no—
39C23×C3×C9—21no—
40C22×C3⋊D9—21no—
41C22×C3⋊D9D411yescharacteristic
42C3×D4×C9—21no—
43C3×D4×C9D412yescomplemented
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9C2311yescharacteristic
46#6233—24no—
47C2×C3⋊Dic9—24no—
48C2×C3⋊Dic9—21no—
49C2×C3⋊Dic9D411yescharacteristic
50C4⋊(C3⋊D9)—21no—
51C4⋊(C3⋊D9)—21no—
52C4⋊(C3⋊D9)—42no—
53C4×C3⋊D9—21no—
54C4×C3⋊D9—21no—
55#6226—21no—
56#6226D412yescomplemented
57C3×C8×C9—21no—
58C3⋊(C9⋊C8)—21no—
59#4781—93no—
60#4781—91no—
61C2×C32×D4C2×D311yescharacteristic
62C2×#638—61no—
63C22×C3⋊Dic3—61no—
64#3830—62no—
65#3830—62no—
66C2×C4⋊(C3⋊D3)—31no—
67C2×C4⋊(C3⋊D3)—61no—
68C2×C4×C3⋊D3—31no—
69C2×#634—31no—
70C2×C32×C8C2×D311yescharacteristic
71C32×C4⋊C4C2×D311yescharacteristic
72#3753—62no—
73#3752—61no—
74C2×C3⋊(C3⋊C8)—31no—
75#3744—62no—
76C2×D4×C9C2×D313yes—
77C2×#611—63no—
78C22×Dic9—63no—
79#3699—66no—
80#3699—66no—
81C2×D36—33no—
82C2×D36—63no—
83C2×C4×D9—33no—
84C2×Dic18—33no—
85C2×C8×C9C2×D313yes—
86C9×C4⋊C4C2×D313yes—
87#3673—66no—
88(C2×D9)⋊C4—63no—
89C2×C9⋊C8—33no—
90#3664—66no—
91C22×C3×C9—22no—
92C22×C3×C9—41no—
93C22×C3×C9—41no—
94C22×C3×C9C2×D411yescharacteristic
95C2×C3⋊D9—21no—
96C2×C3⋊D9—21no—
97C2×C3⋊D9—21no—
98C2×C3⋊D9—41no—
99C2×C3⋊D9—41no—
100C2×C3⋊D9—41no—
101C3×C4×C9—41no—
102C3×C4×C9C2×D411yescharacteristic
103C3×C4×C9C2×D411yescharacteristic
104C3⋊Dic9—22no—
105C3⋊Dic9—21no—
106C2×#237—93no—
107C2×#237—91no—
108C2×#213—93no—
109C2×#213—91no—
110C2×#204—93no—
111C2×#204—91no—
112#895—93no—
113#895—91no—
114C3×#102D913yescomplemented
115C3×#102C3⋊D311yescharacteristic
116(C2×D3)⋊C8—93no—
117(C2×D3)⋊C8—91no—
118#810—93no—
119#810—91no—
120C23×C32—21no—
121C22×C3⋊D3—31no—
122C22×C3⋊D3—61no—
123C32×D4—21no—
124C32×D4D1212yes—
125C2×C32×C4—21no—
126C2×C32×C4C22×D311yescharacteristic
127#638—64no—
128C2×C3⋊Dic3—31no—
129C2×C3⋊Dic3—64no—
130C2×C3⋊Dic3—61no—
131C4⋊(C3⋊D3)—61no—
132C4⋊(C3⋊D3)—61no—
133C4⋊(C3⋊D3)—122no—
134C4×C3⋊D3—61no—
135C4×C3⋊D3—61no—
136#634—32no—
137#634—61no—
138C23×C9—23no—
139C22×D9—33no—
140C22×D9—63no—
141C32×C8—21no—
142C3⋊(C3⋊C8)—61no—
143D4×C9—23no—
144D4×C9D1216yescomplemented
145C2×C4×C9—23no—
146C2×C4×C9C22×D313yes—
147#611—612no—
148C2×Dic9—33no—
149C2×Dic9—612no—
150C2×Dic9—63no—
151D36—63no—
152D36—63no—
153D36—126no—
154C4×D9—63no—
155C4×D9—63no—
156Dic18—36no—
157Dic18—63no—
158C8×C9—23no—
159C9⋊C8—63no—
160#449—271noSylow
161C2×C3×C9—42no—
162C2×C3×C9#12011yescharacteristic
163C2×C3×C9#13511yescharacteristic
164C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
165C3⋊D9—41no—
166C3⋊D9—81no—
167C2×C3×D4C2×D913yes—
168C2×C3×D4C2×C3⋊D311yescharacteristic
169C2×#67—181no—
170C2×#67—183no—
171C22×Dic3—181no—
172C22×Dic3—183no—
173#237—182no—
174#237—186no—
175#237—186no—
176#237—182no—
177C2×D12—93no—
178C2×D12—91no—
179C2×D12—181no—
180C2×D12—183no—
181C2×C4×D3—93no—
182C2×C4×D3—91no—
183C2×Dic6—93no—
184C2×Dic6—91no—
185C2×C3×C8C2×D913yes—
186C2×C3×C8C2×C3⋊D311yescharacteristic
187C3×C4⋊C4C2×D913yes—
188C3×C4⋊C4C2×C3⋊D311yescharacteristic
189#213—186no—
190#213—182no—
191(C2×D3)⋊C4—181no—
192(C2×D3)⋊C4—183no—
193C2×C3⋊C8—93no—
194C2×C3⋊C8—91no—
195#204—182no—
196#204—186no—
197C22×C32—22no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32D3×D411yescharacteristic
201C2×C3⋊D3—61no—
202C2×C3⋊D3—61no—
203C2×C3⋊D3—61no—
204C2×C3⋊D3—121no—
205C2×C3⋊D3—121no—
206C2×C3⋊D3—121no—
207C32×C4—41no—
208C32×C4C2×D1211yescharacteristic
209C32×C4D3×D411yescharacteristic
210C3⋊Dic3—62no—
211C3⋊Dic3—61no—
212C22×C9—26no—
213C22×C9—43no—
214C22×C9—43no—
215C22×C9D3×D413yes—
216C2×D9—63no—
217C2×D9—63no—
218C2×D9—63no—
219C2×D9—123no—
220C2×D9—123no—
221C2×D9—123no—
222C4×C9—43no—
223C4×C9C2×D1213yes—
224C4×C9D3×D413yes—
225Dic9—66no—
226Dic9—63no—
227C2×#41—271no—
228C2×SD16—271no—
229C2×D8—271no—
230#121—271no—
231#103—271no—
232#102C3⋊D911yescharacteristic, complemented
233C22⋊C8—271no—
234C3×C9#44911yescharacteristic, complemented, Sylow
235C23×C3—23no—
236C23×C3—21no—
237C22×D3—93no—
238C22×D3—91no—
239C22×D3—183no—
240C22×D3—181no—
241C3×D4—23no—
242C3×D4—21no—
243C3×D4D3616yescomplemented
244C3×D4C4⋊(C3⋊D3)12yes—
245C2×C3×C4—23no—
246C2×C3×C4—21no—
247C2×C3×C4C22×D913yes—
248C2×C3×C4C22×C3⋊D311yescharacteristic
249#67—184no—
250#67—1812no—
251C2×Dic3—93no—
252C2×Dic3—91no—
253C2×Dic3—184no—
254C2×Dic3—1812no—
255C2×Dic3—183no—
256C2×Dic3—181no—
257D12—183no—
258D12—183no—
259D12—181no—
260D12—181no—
261D12—362no—
262D12—366no—
263C4×D3—183no—
264C4×D3—183no—
265C4×D3—181no—
266C4×D3—181no—
267Dic6—96no—
268Dic6—92no—
269Dic6—183no—
270Dic6—181no—
271C3×C8—23no—
272C3×C8—21no—
273C3⋊C8—183no—
274C3⋊C8—181no—
275C2×C32—42no—
276C2×C32#88211yescharacteristic
277C2×C32#91111yescharacteristic
278C2×C32#91611yescharacteristic
279C3⋊D3—121no—
280C3⋊D3—241no—
281C2×C9—46no—
282C2×C9#88213yes—
283C2×C9#91113yes—
284C2×C9#91613yes—
285D9—123no—
286D9—243no—
287#41—542no—
288#41—542no—
289C2×Q8—271no—
290C2×D4—271no—
291C2×D4—541no—
292C2×D4—541no—
293C2×D4C2×C3⋊D911yescharacteristic
294C22×C4—271no—
295C22×C4—541no—
296SD16—542no—
297D8—542no—
298C2×C8—271no—
299C2×C8C2×C3⋊D911yescharacteristic
300C4⋊C4C2×C3⋊D911yescharacteristic
301C22⋊C4—541no—
302C22×C3—22no—
303C22×C3—26no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—43no—
307C22×C3—43no—
308C22×C3D4×D913yes—
309C22×C3D4×C3⋊D311yescharacteristic
310C2×D3—181no—
311C2×D3—181no—
312C2×D3—181no—
313C2×D3—183no—
314C2×D3—183no—
315C2×D3—183no—
316C2×D3—361no—
317C2×D3—361no—
318C2×D3—361no—
319C2×D3—363no—
320C2×D3—363no—
321C2×D3—363no—
322C3×C4—41no—
323C3×C4—43no—
324C3×C4C2×D3613yes—
325C3×C4D4×D913yes—
326C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
327C3×C4D4×C3⋊D311yescharacteristic
328Dic3—182no—
329Dic3—181no—
330Dic3—186no—
331Dic3—183no—
332C32#478111yescharacteristic
333C9#478113yescomplemented
334C23—21no—
335C23—271no—
336C23—541no—
337Q8—272no—
338Q8—541no—
339D4—21no—
340D4—544no—
341D4—541no—
342D4—541no—
343D4—1082no—
344D4C4⋊(C3⋊D9)12yescomplemented
345C2×C4—21no—
346C2×C4—271no—
347C2×C4—544no—
348C2×C4—541no—
349C2×C4—541no—
350C2×C4—541no—
351C2×C4C22×C3⋊D911yescharacteristic
352C8—21no—
353C8—541no—
354C2×C3—46no—
355C2×C3—42no—
356C2×C3#6344413yes—
357C2×C3#6347313yes—
358C2×C3#6347813yes—
359C2×C3#6409111yescharacteristic
360C2×C3#6412011yescharacteristic
361C2×C3#6412511yescharacteristic
362D3—363no—
363D3—361no—
364D3—723no—
365D3—721no—
366C22—22no—
367C22—41no—
368C22—41no—
369C22—541no—
370C22—541no—
371C22—541no—
372C22—1081no—
373C22—1081no—
374C22—1081no—
375C22D4×C3⋊D911yescharacteristic
376C4—41no—
377C4—541no—
378C4—542no—
379C4C2×C4⋊(C3⋊D9)11yescharacteristic
380C4D4×C3⋊D911yescharacteristic
381C3#1058846113yescomplemented
382C3#1059248011yescharacteristic
383C2—42no—
384C2—1081no—
385C2—2161no—
386C2#1172172811yescharacteristic
387C2#1172179011yescharacteristic
388C2#1172179811yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#102C2
socleC22×C32D3×D4
hypercenter#102C3⋊D9
hypocenterC3×C9#449
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
1. C2×#88742C2semidirect
2. C2×#88544C2semidirect
3. C2×#88524C2semidirect
4. #11721751C2semidirect
5. C3×C9×#102C2semidirect
6. #11720542C2semidirect
7. #11720527C2semidirect
36. C9×#102D3semidirect
43. C3×D4×C9D4semidirect
56. #6226D4semidirect
114. C3×#102D9semidirect
144. D4×C9D12semidirect
232. #102C3⋊D9semidirect
234. C3×C9#449semidirect
243. C3×D4D36semidirect
333. C9#4781semidirect
344. D4C4⋊(C3⋊D9)semidirect
381. C3#10588461semidirect

Maximal subgroups

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

subgroupindex
C2×#887422
C2×#885442
C2×#885242
#117217512
C3×C9×#1022
#117205422
#117205272
#105924803
#105884613