Finite Group Databasebeta

← #13137622

Subgroups — #13137622

Order 1728.

496 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 87 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#11721751C211yesmaximal, characteristic, complemented
2#11721007C211yesmaximal, characteristic, complemented
3C2×#88417C211yesmaximal, characteristic, complemented
4C2×(C3⋊D9)⋊C8C211yesmaximal, characteristic, complemented
5C3×#63369C211yesmaximal, characteristic, complemented
6C9×#807C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic, complemented
8#10590509—31nomaximal
9#10588563—31nomaximal
10C2×C4⋊(C3⋊D9)—21no—
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×C3⋊D9C2211yescharacteristic
13C2×C3×D36C2211yescharacteristic
14C2×#6192—21no—
15C3×C9×C4⋊C4C2211yescharacteristic
16#88536—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20(C2×D9)⋊Dic3—21no—
21#88417—22no—
22(C3⋊D9)⋊C8—22no—
23#64104—31no—
24#63916—31no—
25C2×#3714—31no—
26C2×#3710—31no—
27C3×#809—31no—
28C3×#807D311yescharacteristic
29#63564—31no—
30C2×D72—31no—
31C2×C8×D9—31no—
32#63457—31no—
33#63457—31no—
34#63457—61no—
35C9×#107—31no—
36#63369—31no—
37#63369D311yescharacteristic, complemented
38C22×C3⋊D9—21no—
39C22×C3⋊D9D411yescharacteristic
40C22×C3×D9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9D411yescharacteristic
44C4⋊(C3⋊D9)—21no—
45C4⋊(C3⋊D9)—21no—
46C4⋊(C3⋊D9)—42no—
47C4×C3⋊D9—21no—
48C4×C3⋊D9—21no—
49C2×C9×Dic3—21no—
50C3×D36—21no—
51C3×D36—21no—
52#6192—42no—
53C9×C3⋊C8D412yescomplemented
54C3×C9⋊C8—21no—
55#4883—91no—
56#4775—91no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×C4⋊(C3⋊D3)—61no—
59C2×C4×C3⋊D3—31no—
60C2×C3×D12—31no—
61C2×#626—61no—
62C32×C4⋊C4C2×D311yescharacteristic
63#3752—61no—
64C3×C4⋊Dic3C2×D311yescharacteristic
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8C2×D311yescharacteristic
67(C2×D3)⋊Dic3—61no—
68#3714—62no—
69#3710—62no—
70C2×D36—31no—
71C2×D36—61no—
72C2×D36—61no—
73C2×D36—62no—
74C2×D36—61no—
75C2×D36C2×D311yescharacteristic
76C2×C4×D9—31no—
77C2×C4×D9—61no—
78C2×C8×C9—31no—
79C9×C4⋊C4—21no—
80C9×C4⋊C4—31no—
81C9×C4⋊C4C2×D311yescharacteristic
82(C2×D9)⋊C4—61no—
83(C2×D9)⋊C4—61no—
84(C2×D9)⋊C4—62no—
85C2×C9⋊C8C2×D311yescharacteristic
86D72—62no—
87C8×D9—62no—
88C22×C3×C9C2×D411yescharacteristic
89C2×C3⋊D9—21no—
90C2×C3⋊D9—21no—
91C2×C3⋊D9—41no—
92C2×C3⋊D9—41no—
93C2×C3⋊D9—41no—
94C2×C3⋊D9D812yescomplemented
95C2×C3×D9—41no—
96C2×C3×D9—41no—
97C2×C3×D9—41no—
98C3×C4×C9—41no—
99C3×C4×C9C2×D411yescharacteristic
100C3×C4×C9#4111yescharacteristic
101C3⋊Dic9—21no—
102C9×Dic3—41no—
103#938—91no—
104C2×#213—91no—
105C2×D24—91no—
106C2×D3×C8—91no—
107C2×D3×C8—91no—
108#895—91no—
109#895—91no—
110#895—91no—
111#895—181no—
112C3×#107—31no—
113C3×#102—91no—
114#821—91no—
115#809—31no—
116#809—91no—
117#807D911yescharacteristic, complemented
118C22×C3⋊D3—31no—
119C22×C3⋊D3—61no—
120C22×C3×D3—61no—
121C2×C32×C4—21no—
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—31no—
124C4⋊(C3⋊D3)—61no—
125C4⋊(C3⋊D3)—61no—
126C4⋊(C3⋊D3)—122no—
127C4×C3⋊D3—61no—
128C4×C3⋊D3—61no—
129C2×C3×Dic3—21no—
130C3×D12—61no—
131C3×D12—61no—
132#626—122no—
133C22×D9—21no—
134C22×D9—31no—
135C22×D9—61no—
136C22×D9—62no—
137C22×D9—61no—
138C3×C3⋊C8—61no—
139C3×C3⋊C8D1212yes—
140C2×C4×C9—22no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
143C2×C4×C9—61no—
144C2×C4×C9C22×D311yescharacteristic
145C2×Dic9—31no—
146C2×Dic9—61no—
147D36—21no—
148D36—21no—
149D36—61no—
150D36—61no—
151D36—121no—
152D36—121no—
153D36—122no—
154D36—124no—
155D36—122no—
156C4×D9—61no—
157C4×D9—61no—
158C4×D9—121no—
159C4×D9—121no—
160C8×C9—32no—
161C9⋊C8—21no—
162#466—271noSylow
163C2×C3×C9#12111yescharacteristic
164C2×C3×C9C2×D811yescharacteristic
165C2×C3×C9#13511yescharacteristic
166C3⋊D9—22no—
167C3⋊D9—81no—
168C3×D9—81no—
169C2×C3×D4—91no—
170C2×#67—181no—
171C2×D12—31no—
172C2×D12—91no—
173C2×D12—91no—
174C2×D12—181no—
175C2×D12—181no—
176C2×D12—181no—
177C2×D12—182no—
178C2×D12—181no—
179C2×C4×D3—91no—
180C2×C4×D3—91no—
181C2×C4×D3—181no—
182C2×C3×C8—31no—
183C2×C3×C8—91no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—31no—
186C3×C4⋊C4C2×D911yescharacteristic
187C3×C4⋊C4D3211yescharacteristic
188#217—181no—
189#213—182no—
190(C2×D3)⋊C4—181no—
191(C2×D3)⋊C4—181no—
192(C2×D3)⋊C4—181no—
193(C2×D3)⋊C4—182no—
194C4⋊Dic3C2×D911yescharacteristic
195C2×C3⋊C8—31no—
196C2×C3⋊C8C2×D911yescharacteristic
197D24—182no—
198D3×C8—182no—
199D3×C8—182no—
200C22×C32D3×D411yescharacteristic
201C2×C3⋊D3—32no—
202C2×C3⋊D3—61no—
203C2×C3⋊D3—61no—
204C2×C3⋊D3—121no—
205C2×C3⋊D3—121no—
206C2×C3⋊D3—121no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C32×C4—41no—
211C32×C4C2×D1211yescharacteristic
212C32×C4#23911yescharacteristic
213C3⋊Dic3—61no—
214C3×Dic3—41no—
215C22×C9—21no—
216C22×C9D3×D411yescharacteristic
217C2×D9—32no—
218C2×D9—41no—
219C2×D9—41no—
220C2×D9—41no—
221C2×D9—62no—
222C2×D9—61no—
223C2×D9—61no—
224C2×D9—121no—
225C2×D9—121no—
226C2×D9—122no—
227C2×D9—122no—
228C2×D9—122no—
229C2×D9—121no—
230C2×D9—121no—
231C2×D9—121no—
232C4×C9—21no—
233C4×C9—21no—
234C4×C9—42no—
235C4×C9—41no—
236C4×C9—121no—
237C4×C9#23511yescharacteristic
238C4×C9D3×D411yescharacteristic
239Dic9—61no—
240Dic9—121no—
241C2×D8—271no—
242C22×C8—271no—
243#121—271no—
244#121—271no—
245#107—31no—
246#102—91no—
247#102—271no—
248C3×C9#46611yescharacteristic, complemented, Sylow
249C23×C3—181no—
250C22×D3—61no—
251C22×D3—91no—
252C22×D3—91no—
253C22×D3—181no—
254C22×D3—182no—
255C22×D3—181no—
256C22×D3—181no—
257C3×D4—181no—
258C3×D4—181no—
259C2×C3×C4—22no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—61no—
264C2×C3×C4C22×D911yescharacteristic
265C2×C3×C4C2×D3211yescharacteristic
266#67—362no—
267C2×Dic3—21no—
268C2×Dic3—91no—
269C2×Dic3—91no—
270C2×Dic3—181no—
271D12—61no—
272D12—61no—
273D12—181no—
274D12—181no—
275D12—181no—
276D12—181no—
277D12—361no—
278D12—361no—
279D12—362no—
280D12—362no—
281D12—364no—
282D12—362no—
283C4×D3—181no—
284C4×D3—181no—
285C4×D3—181no—
286C4×D3—181no—
287C4×D3—361no—
288C4×D3—361no—
289C3×C8—32no—
290C3×C8—181no—
291C3⋊C8—61no—
292C3⋊C8D3612yescomplemented
293C2×C32#89511yescharacteristic
294C2×C32D3×D811yescharacteristic
295C2×C32#91911yescharacteristic
296C3⋊D3—62no—
297C3⋊D3—241no—
298C3×D3—241no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9—21no—
302C2×C9#88411yescharacteristic
303C2×C9D3×D811yescharacteristic
304C2×C9#91611yescharacteristic
305D9—62no—
306D9—81no—
307D9—122no—
308D9—242no—
309D9—241no—
310C2×D4—91no—
311C2×D4—271no—
312C2×D4—541no—
313C2×D4—541no—
314C22×C4—271no—
315D8—542no—
316C2×C8—31no—
317C2×C8—91no—
318C2×C8—542no—
319C4⋊C4—31no—
320C4⋊C4D3×D911yescharacteristic
321C22⋊C4—541no—
322C22⋊C4—541no—
323C22×C3—21no—
324C22×C3—361no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3D4×D911yescharacteristic
328C22×C3#381111yescharacteristic
329C2×D3—92no—
330C2×D3—92no—
331C2×D3—121no—
332C2×D3—121no—
333C2×D3—121no—
334C2×D3—182no—
335C2×D3—181no—
336C2×D3—181no—
337C2×D3—181no—
338C2×D3—181no—
339C2×D3—361no—
340C2×D3—361no—
341C2×D3—362no—
342C2×D3—362no—
343C2×D3—362no—
344C2×D3—361no—
345C2×D3—361no—
346C2×D3—361no—
347C2×D3—361no—
348C2×D3—361no—
349C2×D3—361no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—42no—
353C3×C4—41no—
354C3×C4—41no—
355C3×C4—121no—
356C3×C4C2×D3611yescharacteristic
357C3×C4#370111yescharacteristic
358C3×C4#379911yescharacteristic
359C3×C4D3×D1211yescharacteristic
360Dic3—41no—
361Dic3—181no—
362Dic3—181no—
363Dic3—361no—
364C32#488311yescharacteristic
365C9—21no—
366C9#477511yescharacteristic, complemented
367C23—181no—
368C23—271no—
369C23—541no—
370D4—181no—
371D4—181no—
372D4—541no—
373D4—541no—
374D4—1082no—
375D4—1082no—
376C2×C4—21no—
377C2×C4—61no—
378C2×C4—271no—
379C2×C4—541no—
380C2×C4—541no—
381C2×C4C2×D3×D911yescharacteristic
382C8—32no—
383C8—181no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—721no—
388C2×C3#6345711yescharacteristic
389C2×C3D8×D911yescharacteristic
390C2×C3#6348111yescharacteristic
391C2×C3#6391011yescharacteristic
392C2×C3#6392611yescharacteristic
393C2×C3#6395011yescharacteristic
394D3—182no—
395D3—182no—
396D3—241no—
397D3—362no—
398D3—722no—
399D3—721no—
400D3—721no—
401C22—272no—
402C22—361no—
403C22—361no—
404C22—361no—
405C22—541no—
406C22—541no—
407C22—1081no—
408C22—1081no—
409C22—1081no—
410C22#8866811yescharacteristic
411C4—41no—
412C4—121no—
413C4—541no—
414C4#8863511yescharacteristic
415C4D3×D3611yescharacteristic
416C3—21no—
417C3#1058856311yescharacteristic, complemented
418C3#1059050911yescharacteristic
419C2—542no—
420C2—721no—
421C2—2161no—
422C2#1172100511yescharacteristic
423C2#1172117411yescharacteristic
424C2#1172122411yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#466
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721751C2semidirect
2. #11721007C2semidirect
3. C2×#88417C2semidirect
4. C2×(C3⋊D9)⋊C8C2semidirect
5. C3×#63369C2semidirect
6. C9×#807C2semidirect
7. #11720267C2semidirect
37. #63369D3semidirect
53. C9×C3⋊C8D4semidirect
94. C2×C3⋊D9D8semidirect
117. #807D9semidirect
248. C3×C9#466semidirect
292. C3⋊C8D36semidirect
366. C9#4775semidirect
417. C3#10588563semidirect

Maximal subgroups

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

subgroupindex
#117217512
#117210072
C2×#884172
C2×(C3⋊D9)⋊C82
C3×#633692
C9×#8072
#117202672
#105905093
#105885633