Finite Group Databasebeta

← #13139513

Subgroups — #13139513

Order 1728.

484 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 95 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#11721819C211yesmaximal, characteristic, complemented
2C3×#63492C211yesmaximal, characteristic, complemented
3C2×C9×#208C211yesmaximal, characteristic, complemented
4C2×#88417C211yesmaximal, characteristic, complemented
5#11720284C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic, complemented
7#11720278C211yesmaximal, characteristic, complemented
8#10591165—31nomaximal
9#10588817—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×C4⋊(C3⋊D9)C2211yescharacteristic
13C2×C3×#611—21no—
14C2×C3×D36C2211yescharacteristic
15#88536—21no—
16C4⋊(C3⋊Dic9)C2211yescharacteristic
17C3×(C2×D9)⋊C4—21no—
18C3×C4⋊Dic9C2211yescharacteristic
19C9×#208—22no—
20C2×C9×C3⋊C8C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22#88417—22no—
23#64139—31no—
24C3×#930—31no—
25C2×C3×#208D311yescharacteristic
26C2×#3714—31no—
27#63573—31no—
28#63564—31no—
29#63563—31no—
30C2×C9×M16—31no—
31#63492—31no—
32#63492—61no—
33#63492D311yescharacteristic, complemented
34C2×D72—31no—
35#63380—31no—
36#63380—31no—
37#63377—31no—
38C23×C3×C9D411yescharacteristic
39C22×C3⋊D9—21no—
40C22×C3×D9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9D411yescharacteristic
44C2×C3×C4×C9C2311yescharacteristic
45#6233—42no—
46C2×C3⋊Dic9—21no—
47C4⋊(C3⋊D9)—21no—
48C4⋊(C3⋊D9)—21no—
49C3×#611—42no—
50C2×C3×Dic9—21no—
51C3×D36—21no—
52C3×D36—21no—
53C9×C3⋊C8—21no—
54C9×C3⋊C8D412yescomplemented
55#5135—91no—
56#5039—91no—
57C22×C32×C4C2×D311yescharacteristic
58C2×#638—61no—
59C2×C4⋊(C3⋊D3)—31no—
60C2×C3×#67—61no—
61C2×C3×D12—31no—
62#3752—61no—
63C4⋊(C3⋊Dic3)—31no—
64C3×(C2×D3)⋊C4—61no—
65C3×C4⋊Dic3—31no—
66C3×#208—22no—
67C2×C3×C3⋊C8C2×D311yescharacteristic
68C2×C3×C3⋊C8C2×D311yescharacteristic
69#3714—62no—
70C22×C4×C9—21no—
71C22×C4×C9C2×D311yescharacteristic
72C2×#611—21no—
73C2×#611—62no—
74C2×#611—61no—
75C2×D36—31no—
76C2×D36—61no—
77C2×D36C2×D311yescharacteristic
78C9×M16—62no—
79C2×C8×C9—31no—
80C2×C8×C9—31no—
81(C2×D9)⋊C4—21no—
82(C2×D9)⋊C4—62no—
83(C2×D9)⋊C4—61no—
84C4⋊Dic9—31no—
85C4⋊Dic9—61no—
86C4⋊Dic9C2×D311yescharacteristic
87D72—62no—
88C22×C3×C9—21no—
89C22×C3×C9—21no—
90C22×C3×C9—21no—
91C22×C3×C9C2×D411yescharacteristic
92C2×C3⋊D9—41no—
93C2×C3⋊D9—41no—
94C2×C3⋊D9—41no—
95C2×C3×D9—41no—
96C2×C3×D9—41no—
97C2×C3×D9—41no—
98C3×C4×C9—21no—
99C3×C4×C9C2×D411yescharacteristic
100C3×C4×C9#4111yescharacteristic
101C3⋊Dic9—41no—
102C3×Dic9—41no—
103C2×C3×M16—31no—
104C3×#121—91no—
105C2×#213—91no—
106#930—31no—
107#930—91no—
108#930—91no—
109#930—181no—
110C2×#208D911yescharacteristic, complemented
111C2×D24—91no—
112#832—91no—
113#821—91no—
114#821—91no—
115#817—91no—
116#809—91no—
117#808—91no—
118C23×C32D1211yescharacteristic
119C22×C3⋊D3—61no—
120C22×C3×D3—61no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4D1211yescharacteristic
124C2×C32×C4C22×D311yescharacteristic
125#638—122no—
126C2×C3⋊Dic3—61no—
127C4⋊(C3⋊D3)—61no—
128C4⋊(C3⋊D3)—61no—
129C3×#67—122no—
130C2×C3×Dic3—61no—
131C3×D12—61no—
132C3×D12—61no—
133C23×C9—21no—
134C23×C9#6711yescharacteristic
135C22×D9—21no—
136C22×D9—62no—
137C22×D9—61no—
138C3×C3⋊C8—21no—
139C3×C3⋊C8#6712yes—
140C2×C4×C9—21no—
141C2×C4×C9—21no—
142C2×C4×C9—21no—
143C2×C4×C9—21no—
144C2×C4×C9—41no—
145C2×C4×C9—41no—
146C2×C4×C9#6711yescharacteristic
147C2×C4×C9C22×D311yescharacteristic
148#611—42no—
149#611—124no—
150#611—122no—
151C2×Dic9—21no—
152C2×Dic9—62no—
153C2×Dic9—61no—
154D36—21no—
155D36—21no—
156D36—61no—
157D36—61no—
158D36—121no—
159D36—121no—
160C8×C9—32no—
161C8×C9—61no—
162#469—271noSylow
163C2×C3×C9—41no—
164C2×C3×C9#12111yescharacteristic
165C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
166C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
167C3⋊D9—81no—
168C3×D9—81no—
169C2×C3×D4—91no—
170C2×C3×D4—181no—
171C22×C3×C4—21no—
172C22×C3×C4C2×D911yescharacteristic
173C22×C3×C4D3211yescharacteristic
174C2×#67—61no—
175C2×#67—182no—
176C2×#67—181no—
177C2×#67—181no—
178C2×D12—31no—
179C2×D12—91no—
180C2×D12—91no—
181C2×D12—181no—
182C3×M16—62no—
183C2×C3×C8—31no—
184C2×C3×C8—31no—
185C3×C4⋊C4—91no—
186C3×C22⋊C4—181no—
187#213—182no—
188(C2×D3)⋊C4—61no—
189(C2×D3)⋊C4—182no—
190(C2×D3)⋊C4—181no—
191(C2×D3)⋊C4—181no—
192C4⋊Dic3—31no—
193C4⋊Dic3—91no—
194C4⋊Dic3—91no—
195C4⋊Dic3—181no—
196#208—22no—
197C2×C3⋊C8C2×D911yescharacteristic
198C2×C3⋊C8C2×D911yescharacteristic
199D24—182no—
200C22×C32—21no—
201C22×C32—21no—
202C22×C32—21no—
203C22×C32C2×D1211yescharacteristic
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—21no—
211C32×C4#23511yescharacteristic
212C32×C4C2×#6711yescharacteristic
213C3⋊Dic3—121no—
214C3×Dic3—121no—
215C22×C9—22no—
216C22×C9—21no—
217C22×C9—21no—
218C22×C9—21no—
219C22×C9—21no—
220C22×C9—41no—
221C22×C9—41no—
222C22×C9C2×#6711yescharacteristic
223C2×D9—41no—
224C2×D9—41no—
225C2×D9—41no—
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—21no—
235C4×C9—41no—
236C4×C9D3×D411yescharacteristic
237C4×C9#23711yescharacteristic
238Dic9—41no—
239Dic9—122no—
240Dic9—121no—
241C2×D8—271no—
242C2×M16—31no—
243#121—91no—
244#121—271no—
245#106—271no—
246#102—271no—
247#102—271no—
248C3×C9#46911yescharacteristic, complemented, Sylow
249C23×C3—21no—
250C23×C3—181no—
251C23×C3D3611yescharacteristic
252C23×C3#62611yescharacteristic
253C22×D3—61no—
254C22×D3—182no—
255C22×D3—181no—
256C22×D3—181no—
257C3×D4—181no—
258C3×D4—181no—
259C3×D4—362no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—181no—
269C2×C3×C4D3611yescharacteristic
270C2×C3×C4C22×D911yescharacteristic
271C2×C3×C4#62611yescharacteristic
272C2×C3×C4C2×D3211yescharacteristic
273#67—122no—
274#67—362no—
275#67—364no—
276#67—362no—
277C2×Dic3—61no—
278C2×Dic3—182no—
279C2×Dic3—181no—
280C2×Dic3—181no—
281D12—61no—
282D12—61no—
283D12—181no—
284D12—181no—
285D12—181no—
286D12—181no—
287D12—361no—
288D12—361no—
289C3×C8—32no—
290C3×C8—61no—
291C3⋊C8—21no—
292C3⋊C8#61112yescomplemented
293C2×C32—41no—
294C2×C32#90811yescharacteristic
295C2×C32#90811yescharacteristic
296C2×C32#93011yescharacteristic
297C3⋊D3—241no—
298C3×D3—241no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9—21no—
302C2×C9—42no—
303C2×C9—41no—
304C2×C9#93211yescharacteristic
305C2×C9#93911yescharacteristic
306C2×C9#94911yescharacteristic
307D9—81no—
308D9—242no—
309D9—241no—
310C2×D4—91no—
311C2×D4—181no—
312C2×D4—271no—
313C2×D4—541no—
314C22×C4D3×D911yescharacteristic
315D8—542no—
316M16—62no—
317C2×C8—31no—
318C2×C8—31no—
319C4⋊C4—91no—
320C4⋊C4—271no—
321C22⋊C4—181no—
322C22⋊C4—541no—
323C22×C3—22no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—361no—
334C22×C3—361no—
335C22×C3—361no—
336C22×C3C2×D3611yescharacteristic
337C22×C3C2×#62611yescharacteristic
338C2×D3—121no—
339C2×D3—121no—
340C2×D3—121no—
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—21no—
353C3×C4—21no—
354C3×C4—41no—
355C3×C4—361no—
356C3×C4#369711yescharacteristic
357C3×C4C2×#61111yescharacteristic
358C3×C4#380411yescharacteristic
359C3×C4D3×#6711yescharacteristic
360Dic3—121no—
361Dic3—362no—
362Dic3—361no—
363Dic3—361no—
364C32#513511yescharacteristic
365C9—21no—
366C9#503911yescharacteristic, complemented
367C23—181no—
368C23—541no—
369C23#619211yescharacteristic
370D4—181no—
371D4—181no—
372D4—362no—
373D4—541no—
374D4—541no—
375D4—1082no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—181no—
379C2×C4—541no—
380C2×C4#619211yescharacteristic
381C2×C4C2×D3×D911yescharacteristic
382C8—32no—
383C8—61no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—42no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—721no—
391C2×C3#6347011yescharacteristic
392C2×C3#6347011yescharacteristic
393C2×C3#6349211yescharacteristic
394C2×C3#6382511yescharacteristic
395C2×C3#6383011yescharacteristic
396C2×C3#6397811yescharacteristic
397D3—241no—
398D3—722no—
399D3—721no—
400D3—721no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—361no—
405C22—361no—
406C22—361no—
407C22—1081no—
408C22—1081no—
409C22—1081no—
410C22C2×#619211yescharacteristic
411C4—21no—
412C4—361no—
413C4—1081no—
414C4#8865811yescharacteristic
415C4D3×#61111yescharacteristic
416C3—21no—
417C3#1058881711yescharacteristic, complemented
418C3#1059116511yescharacteristic
419C2—41no—
420C2—721no—
421C2—2161no—
422C2#1172086811yescharacteristic
423C2#1172087311yescharacteristic
424C2#1172127711yescharacteristic
subgroupG/H
centerC22C2×#6192
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×D12
hypercenterC22×C4D3×D9
hypocenterC3×C9#469
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11721819C2semidirect
2. C3×#63492C2semidirect
3. C2×C9×#208C2semidirect
4. C2×#88417C2semidirect
5. #11720284C2semidirect
6. #11720283C2semidirect
7. #11720278C2semidirect
33. #63492D3semidirect
54. C9×C3⋊C8D4semidirect
110. C2×#208D9semidirect
248. C3×C9#469semidirect
292. C3⋊C8#611semidirect
366. C9#5039semidirect
417. C3#10588817semidirect

Maximal subgroups

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

subgroupindex
#117218192
C3×#634922
C2×C9×#2082
C2×#884172
#117202842
#117202832
#117202782
#105911653
#105888173