Finite Group Databasebeta

← #13140426

Subgroups — #13140426

Order 1728.

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

normal subgroups only

This group has 528 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#11721840C211yesmaximal, characteristic, complemented
2C2×C9×#213C211yesmaximal, characteristic, complemented
3#11721025C211yesmaximal, characteristic, complemented
4C2×#88422C211yesmaximal, characteristic, complemented
5C3×#63392C211yesmaximal, characteristic, complemented
6#11720284C211yesmaximal, characteristic, complemented
7#11720273C211yesmaximal, characteristic, complemented
8#10591433—31nomaximal
9#10588830—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×#6233—21no—
13C2×C4⋊(C3⋊D9)C2211yescharacteristic
14C2×C9×D12C2211yescharacteristic
15C2×D3×Dic9—21no—
16C4×C3⋊Dic9C2211yescharacteristic
17C9×#213—22no—
18C3×C4⋊Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21(C2×D3)⋊Dic9—21no—
22#88422—22no—
23#64149—31no—
24C2×C3×#213D311yescharacteristic
25#63898—31no—
26C2×#3714—31no—
27C3×#832—31no—
28#63572—31no—
29#63564—31no—
30C2×C9×D8—31no—
31#63502—31no—
32#63502—61no—
33D4×Dic9—31no—
34C2×#3673—31no—
35#63392D311yescharacteristic, complemented
36#63380—31no—
37Dic9⋊C8—31no—
38C23×C3×C9—21no—
39C22×C3⋊D9—21no—
40C22×D3×C9—21no—
41C3×D4×C9—21no—
42C3×D4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44#6233—42no—
45#6233—42no—
46C2×C3⋊Dic9D411yescharacteristic
47C2×C3⋊Dic9D411yescharacteristic
48C4⋊(C3⋊D9)—21no—
49C4⋊(C3⋊D9)—21no—
50C2×C3×Dic9—21no—
51C9×D12—21no—
52C9×D12D412yescomplemented
53D3×Dic9—42no—
54C9×C3⋊C8—21no—
55C3×C9⋊C8—21no—
56#5150—91no—
57#4786—91no—
58C2×C32×D4C2×D311yescharacteristic
59C2×#638—61no—
60C2×#638—61no—
61C2×C4⋊(C3⋊D3)—31no—
62C2×C3×D12C2×D311yescharacteristic
63C2×D3×Dic3—61no—
64C4×C3⋊Dic3—31no—
65C3×#213—22no—
66C3×C4⋊Dic3—31no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8C2×D311yescharacteristic
69(C2×D3)⋊Dic3—61no—
70#3714—62no—
71C2×D4×C9—21no—
72C2×D4×C9—31no—
73C2×D4×C9C2×D311yescharacteristic
74C2×#611—61no—
75C2×#611—61no—
76C2×#611—62no—
77C2×#611—62no—
78C22×Dic9—61no—
79C2×D36—31no—
80C2×D36—61no—
81C9×D8—62no—
82C2×C8×C9—31no—
83#3676—61no—
84#3673—62no—
85C4⋊Dic9C2×D311yescharacteristic
86C4×Dic9—31no—
87C4×Dic9—61no—
88C2×C9⋊C8C2×D311yescharacteristic
89C22×C3×C9—41no—
90C22×C3×C9—41no—
91C22×C3×C9—41no—
92C22×C3×C9C2×D411yescharacteristic
93C2×C3⋊D9—41no—
94C2×C3⋊D9—41no—
95C2×C3⋊D9—41no—
96C2×D3×C9—22no—
97C2×D3×C9—41no—
98C2×D3×C9—41no—
99C3×C4×C9C2×D411yescharacteristic
100C3×C4×C9#4111yescharacteristic
101C3⋊Dic9—21no—
102C3⋊Dic9D812yescomplemented
103C3×Dic9—41no—
104C2×C3×D8—31no—
105#940—91no—
106#940—91no—
107#940—181no—
108D4×Dic3—91no—
109C2×#213—91no—
110C2×#213D911yescharacteristic, complemented
111C2×D24—91no—
112#893—91no—
113C3×#102—91no—
114#832—31no—
115#821—91no—
116Dic3⋊C8—91no—
117Dic3⋊C8—91no—
118#809—91no—
119C23×C32—21no—
120C22×C3⋊D3—61no—
121C22×C3×D3—21no—
122C32×D4—21no—
123C32×D4—21no—
124C2×C32×C4C22×D311yescharacteristic
125#638—122no—
126#638—122no—
127C2×C3⋊Dic3—31no—
128C2×C3⋊Dic3—31no—
129C4⋊(C3⋊D3)—61no—
130C4⋊(C3⋊D3)—61no—
131C2×C3×Dic3—61no—
132C3×D12—21no—
133C3×D12#6712yes—
134D3×Dic3—122no—
135C23×C9—22no—
136C23×C9—21no—
137C23×C9—61no—
138C22×D9—62no—
139C22×D9—61no—
140C3×C3⋊C8—21no—
141C3×C3⋊C8—61no—
142D4×C9—21no—
143D4×C9—21no—
144D4×C9—32no—
145D4×C9—41no—
146D4×C9—41no—
147D4×C9—61no—
148C2×C4×C9—21no—
149C2×C4×C9C22×D311yescharacteristic
150#611—122no—
151#611—124no—
152#611—124no—
153#611—122no—
154C2×Dic9—21no—
155C2×Dic9—31no—
156C2×Dic9—31no—
157C2×Dic9—61no—
158C2×Dic9—61no—
159C2×Dic9—122no—
160D36—61no—
161D36—61no—
162D36—121no—
163D36—121no—
164C8×C9—61no—
165C9⋊C8—21no—
166#459—271noSylow
167C2×C3×C9—81no—
168C2×C3×C9#12111yescharacteristic
169C2×C3×C9C2×D811yescharacteristic
170C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
171C3⋊D9—81no—
172D3×C9—42no—
173C2×C3×D4—21no—
174C2×C3×D4—31no—
175C2×C3×D4C2×D911yescharacteristic
176C2×C3×D4D3211yescharacteristic
177C2×#67—181no—
178C2×#67—181no—
179C2×#67—181no—
180C2×#67—181no—
181C2×#67—182no—
182C2×#67—182no—
183C22×Dic3—181no—
184C2×D12—91no—
185C2×D12—91no—
186C2×D12—181no—
187C2×D12C2×D911yescharacteristic
188C2×C4×D3—181no—
189C3×D8—62no—
190C2×C3×C8—31no—
191C2×C3×C8—91no—
192C3×C4⋊C4—91no—
193#217—181no—
194#213—22no—
195#213—182no—
196(C2×D3)⋊C4—181no—
197C4⋊Dic3—31no—
198C4×Dic3—91no—
199C4×Dic3—91no—
200C4×Dic3—181no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8C2×D911yescharacteristic
203D24—182no—
204C22×C32—41no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32D3×D411yescharacteristic
208C2×C3⋊D3—121no—
209C2×C3⋊D3—121no—
210C2×C3⋊D3—121no—
211C2×C3×D3—22no—
212C2×C3×D3—41no—
213C2×C3×D3—41no—
214C32×C4#23711yescharacteristic
215C32×C4C2×#6711yescharacteristic
216C3⋊Dic3—32no—
217C3⋊Dic3—61no—
218C3×Dic3—121no—
219C22×C9—21no—
220C22×C9—42no—
221C22×C9—42no—
222C22×C9—42no—
223C22×C9—41no—
224C22×C9—41no—
225C22×C9—41no—
226C22×C9—62no—
227C22×C9—121no—
228C22×C9—121no—
229C22×C9D3×D411yescharacteristic
230C2×D9—122no—
231C2×D9—122no—
232C2×D9—122no—
233C2×D9—121no—
234C2×D9—121no—
235C2×D9—121no—
236C4×C9—21no—
237C4×C9—21no—
238C4×C9#23511yescharacteristic
239C4×C9D3×D411yescharacteristic
240Dic9—32no—
241Dic9—41no—
242Dic9—62no—
243Dic9—62no—
244Dic9—61no—
245C2×D8—31no—
246C2×D8—271no—
247#127—271no—
248C4×D4—271no—
249C4⋊C8—271no—
250#102—91no—
251#102—271no—
252C3×C9#45911yescharacteristic, complemented, Sylow
253C23×C3—22no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—61no—
257C22×D3—21no—
258C22×D3—182no—
259C22×D3—181no—
260C22×D3—181no—
261C3×D4—21no—
262C3×D4—21no—
263C3×D4—21no—
264C3×D4—21no—
265C3×D4—32no—
266C3×D4—41no—
267C3×D4—41no—
268C3×D4—61no—
269C2×C3×C4—21no—
270C2×C3×C4—181no—
271C2×C3×C4C22×D911yescharacteristic
272C2×C3×C4C2×D3211yescharacteristic
273#67—362no—
274#67—362no—
275#67—362no—
276#67—364no—
277#67—364no—
278#67—362no—
279C2×Dic3—61no—
280C2×Dic3—91no—
281C2×Dic3—91no—
282C2×Dic3—91no—
283C2×Dic3—91no—
284C2×Dic3—181no—
285C2×Dic3—181no—
286C2×Dic3—362no—
287D12—21no—
288D12—181no—
289D12—181no—
290D12—181no—
291D12—181no—
292D12—361no—
293D12—361no—
294D12#61112yescomplemented
295C4×D3—362no—
296C3×C8—61no—
297C3×C8—181no—
298C3⋊C8—21no—
299C3⋊C8—61no—
300C2×C32—81no—
301C2×C32D3×D811yescharacteristic
302C2×C32#91111yescharacteristic
303C2×C32#93911yescharacteristic
304C3⋊D3—241no—
305C3×D3—42no—
306C2×C9—21no—
307C2×C9—21no—
308C2×C9—21no—
309C2×C9—82no—
310C2×C9—81no—
311C2×C9—122no—
312C2×C9#88411yescharacteristic
313C2×C9D3×D811yescharacteristic
314C2×C9#91411yescharacteristic
315D9—242no—
316D9—241no—
317C2×D4—31no—
318C2×D4—271no—
319C2×D4—541no—
320C2×D4—541no—
321C2×D4D3×D911yescharacteristic
322C22×C4—541no—
323D8—62no—
324D8—542no—
325C2×C8—31no—
326C2×C8—91no—
327C4⋊C4—91no—
328C22⋊C4—541no—
329C42—271no—
330C22×C3—21no—
331C22×C3—42no—
332C22×C3—42no—
333C22×C3—42no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—62no—
341C22×C3—121no—
342C22×C3—121no—
343C22×C3D4×D911yescharacteristic
344C22×C3#381111yescharacteristic
345C2×D3—22no—
346C2×D3—41no—
347C2×D3—41no—
348C2×D3—362no—
349C2×D3—362no—
350C2×D3—362no—
351C2×D3—361no—
352C2×D3—361no—
353C2×D3—361no—
354C2×D3—361no—
355C2×D3—361no—
356C2×D3—361no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—361no—
360C3×C4#369911yescharacteristic
361C3×C4C2×#61111yescharacteristic
362C3×C4#380411yescharacteristic
363C3×C4D3×#6711yescharacteristic
364Dic3—92no—
365Dic3—92no—
366Dic3—121no—
367Dic3—182no—
368Dic3—182no—
369Dic3—181no—
370Dic3—181no—
371C32#515011yescharacteristic
372C9—21no—
373C9#478611yescharacteristic, complemented
374C23—21no—
375C23—61no—
376C23—541no—
377D4—21no—
378D4—21no—
379D4—32no—
380D4—61no—
381D4—541no—
382D4—541no—
383D4—1082no—
384D4—1082no—
385C2×C4—181no—
386C2×C4—271no—
387C2×C4—271no—
388C2×C4—1082no—
389C2×C4C2×D3×D911yescharacteristic
390C8—61no—
391C8—181no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—82no—
396C2×C3—81no—
397C2×C3—81no—
398C2×C3—122no—
399C2×C3D8×D911yescharacteristic
400C2×C3#6347311yescharacteristic
401C2×C3#6350111yescharacteristic
402C2×C3#6392611yescharacteristic
403C2×C3#6393711yescharacteristic
404C2×C3#6397111yescharacteristic
405D3—42no—
406D3—722no—
407D3—721no—
408D3—721no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—62no—
413C22—121no—
414C22—121no—
415C22—1081no—
416C22—1081no—
417C22—1081no—
418C22#8866811yescharacteristic
419C4—272no—
420C4—361no—
421C4—541no—
422C4#8866211yescharacteristic
423C4D3×#61111yescharacteristic
424C3—21no—
425C3#1058883011yescharacteristic, complemented
426C3#1059143311yescharacteristic
427C2—81no—
428C2—122no—
429C2—2161no—
430C2#1172117411yescharacteristic
431C2#1172117911yescharacteristic
432C2#1172128211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
hypercenterC2×D4D3×D9
hypocenterC3×C9#459
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11721840C2semidirect
2. C2×C9×#213C2semidirect
3. #11721025C2semidirect
4. C2×#88422C2semidirect
5. C3×#63392C2semidirect
6. #11720284C2semidirect
7. #11720273C2semidirect
35. #63392D3semidirect
52. C9×D12D4semidirect
102. C3⋊Dic9D8semidirect
110. C2×#213D9semidirect
252. C3×C9#459semidirect
294. D12#611semidirect
373. C9#4786semidirect
425. C3#10588830semidirect

Maximal subgroups

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

subgroupindex
#117218402
C2×C9×#2132
#117210252
C2×#884222
C3×#633922
#117202842
#117202732
#105914333
#105888303