Finite Group Databasebeta

← #13137757

Subgroups — #13137757

Order 1728.

514 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 83 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#11721749C211yesmaximal, characteristic, complemented
2#11721043C211yesmaximal, characteristic, complemented
3C2×#88422C211yesmaximal, characteristic, complemented
4C2×#88417C211yesmaximal, characteristic, complemented
5C9×#809C211yesmaximal, characteristic, complemented
6C3×#63369C211yesmaximal, characteristic, complemented
7#11720273C211yesmaximal, characteristic, complemented
8#10590551—31nomaximal
9#10588466—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C4×C3⋊D9—21no—
12C2×C9×D12C2211yescharacteristic
13C2×C3×D36C2211yescharacteristic
14C2×#6194—21no—
15C2×#6194—21no—
16C3×C9×C4⋊C4C2211yescharacteristic
17#88536—21no—
18C4×C3⋊Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21#88422—22no—
22#88417—22no—
23#64102—31no—
24#63914—31no—
25C2×#3714—31no—
26C2×#3714—31no—
27C3×#809—31no—
28C3×#809D311yescharacteristic
29#63572—31no—
30#63502—31no—
31C2×#3673—31no—
32C2×D72—31no—
33#63455—31no—
34#63455—61no—
35C9×#102—31no—
36Dic9⋊C8—31no—
37#63369D311yescharacteristic, complemented
38C22×C3⋊D9—21no—
39C22×D3×C9—21no—
40C22×C3×D9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9D411yescharacteristic
44C2×C3⋊Dic9D411yescharacteristic
45C4⋊(C3⋊D9)—21no—
46C4⋊(C3⋊D9)D412yescomplemented
47C4×C3⋊D9—42no—
48C9×D12—21no—
49C9×D12—21no—
50C3×D36—21no—
51C3×D36—21no—
52#6194—42no—
53#6194—42no—
54C9×C3⋊C8—21no—
55C3×C9⋊C8—21no—
56#4786—91no—
57#4786—91no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C4×C3⋊D3—61no—
60C2×C3×D12—31no—
61C2×C3×D12C2×D311yescharacteristic
62C2×#625—61no—
63C2×#625—61no—
64C32×C4⋊C4C2×D311yescharacteristic
65#3752—61no—
66C4×C3⋊Dic3—31no—
67C2×C3×C3⋊C8—31no—
68C2×C3×C3⋊C8C2×D311yescharacteristic
69#3714—62no—
70#3714—62no—
71C2×D4×C9—31no—
72C2×#611—61no—
73C2×#611—61no—
74C2×D36—31no—
75C2×D36—61no—
76C2×D36C2×D311yescharacteristic
77C2×C4×D9—61no—
78C2×C4×D9—62no—
79C2×C8×C9—31no—
80C9×C4⋊C4—21no—
81C9×C4⋊C4C2×D311yescharacteristic
82#3673—62no—
83(C2×D9)⋊C4—61no—
84(C2×D9)⋊C4—62no—
85C4×Dic9—31no—
86C4×Dic9—61no—
87C2×C9⋊C8C2×D311yescharacteristic
88D72—62no—
89C22×C3×C9C2×D411yescharacteristic
90C2×C3⋊D9—22no—
91C2×C3⋊D9—41no—
92C2×C3⋊D9—41no—
93C2×D3×C9—41no—
94C2×D3×C9—41no—
95C2×D3×C9—41no—
96C2×C3×D9—41no—
97C2×C3×D9—41no—
98C2×C3×D9—41no—
99C3×C4×C9—41no—
100C3×C4×C9C2×D411yescharacteristic
101C3×C4×C9#4111yescharacteristic
102C3⋊Dic9—21no—
103C3⋊Dic9D812yescomplemented
104#940—91no—
105#940—91no—
106C2×#213—91no—
107C2×#213—91no—
108C2×D24—91no—
109C2×D24—91no—
110#893—91no—
111#893—91no—
112#893—181no—
113C3×#102—31no—
114C3×#102—91no—
115Dic3⋊C8—91no—
116Dic3⋊C8—91no—
117#809—31no—
118#809D911yescharacteristic, complemented
119C22×C3⋊D3—61no—
120C22×C3×D3—21no—
121C22×C3×D3—61no—
122C2×C32×C4—21no—
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—31no—
125C2×C3⋊Dic3—31no—
126C4⋊(C3⋊D3)—32no—
127C4⋊(C3⋊D3)—61no—
128C4×C3⋊D3—122no—
129C3×D12—21no—
130C3×D12—21no—
131C3×D12—61no—
132C3×D12—61no—
133#625—122no—
134#625—122no—
135C23×C9—61no—
136C22×D9—21no—
137C22×D9—62no—
138C22×D9—61no—
139C3×C3⋊C8—21no—
140C3×C3⋊C8—61no—
141D4×C9—61no—
142D4×C9—61no—
143C2×C4×C9—22no—
144C2×C4×C9—21no—
145C2×C4×C9—21no—
146C2×C4×C9C22×D311yescharacteristic
147#611—122no—
148#611—122no—
149C2×Dic9—31no—
150C2×Dic9—31no—
151C2×Dic9—61no—
152C2×Dic9—61no—
153D36—21no—
154D36—21no—
155D36—32no—
156D36—62no—
157D36—62no—
158D36—61no—
159C4×D9—122no—
160C4×D9—124no—
161C8×C9—61no—
162C9⋊C8—21no—
163#459—271noSylow
164C2×C3×C9#12111yescharacteristic
165C2×C3×C9C2×D811yescharacteristic
166C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
167C3⋊D9—42no—
168D3×C9—81no—
169C3×D9—81no—
170C2×C3×D4—31no—
171C2×C3×D4—91no—
172C2×#67—181no—
173C2×#67—181no—
174C2×#67—181no—
175C2×#67—181no—
176C2×D12—31no—
177C2×D12—91no—
178C2×D12—91no—
179C2×D12—181no—
180C2×D12C2×D911yescharacteristic
181C2×C4×D3—181no—
182C2×C4×D3—181no—
183C2×C4×D3—182no—
184C2×C3×C8—31no—
185C2×C3×C8—91no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4C2×D911yescharacteristic
188C3×C4⋊C4D3211yescharacteristic
189#213—182no—
190#213—182no—
191(C2×D3)⋊C4—181no—
192(C2×D3)⋊C4—181no—
193(C2×D3)⋊C4—182no—
194C4×Dic3—91no—
195C4×Dic3—91no—
196C4×Dic3—181no—
197C2×C3⋊C8—31no—
198C2×C3⋊C8C2×D911yescharacteristic
199D24—182no—
200D24—182no—
201C22×C32D3×D411yescharacteristic
202C2×C3⋊D3—62no—
203C2×C3⋊D3—121no—
204C2×C3⋊D3—121no—
205C2×C3×D3—41no—
206C2×C3×D3—41no—
207C2×C3×D3—41no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C2×C3×D3—121no—
211C32×C4—41no—
212C32×C4#23511yescharacteristic
213C32×C4D3×D411yescharacteristic
214C3⋊Dic3—32no—
215C3⋊Dic3—61no—
216C22×C9—21no—
217C22×C9—121no—
218C22×C9—121no—
219C22×C9—121no—
220C22×C9D3×D411yescharacteristic
221C2×D9—41no—
222C2×D9—41no—
223C2×D9—41no—
224C2×D9—64no—
225C2×D9—62no—
226C2×D9—122no—
227C2×D9—122no—
228C2×D9—121no—
229C2×D9—121no—
230C4×C9—21no—
231C4×C9—21no—
232C4×C9—42no—
233C4×C9—41no—
234C4×C9#23511yescharacteristic
235C4×C9D3×D411yescharacteristic
236Dic9—32no—
237Dic9—62no—
238Dic9—61no—
239Dic9—121no—
240C2×D8—271no—
241C2×D8—271no—
242#127—271no—
243C4×D4—271no—
244C4⋊C8—271no—
245#102—31no—
246#102—91no—
247C3×C9#45911yescharacteristic, complemented, Sylow
248C23×C3—61no—
249C23×C3—181no—
250C22×D3—21no—
251C22×D3—61no—
252C22×D3—182no—
253C22×D3—181no—
254C22×D3—181no—
255C3×D4—61no—
256C3×D4—61no—
257C3×D4—181no—
258C3×D4—181no—
259C2×C3×C4—22no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4C22×D911yescharacteristic
264C2×C3×C4C2×D3211yescharacteristic
265#67—362no—
266#67—362no—
267#67—362no—
268#67—362no—
269C2×Dic3—91no—
270C2×Dic3—91no—
271C2×Dic3—91no—
272C2×Dic3—91no—
273C2×Dic3—181no—
274C2×Dic3—181no—
275D12—21no—
276D12—21no—
277D12—61no—
278D12—61no—
279D12—92no—
280D12—92no—
281D12—182no—
282D12—182no—
283D12—181no—
284D12—181no—
285C4×D3—362no—
286C4×D3—362no—
287C4×D3—364no—
288C3×C8—61no—
289C3×C8—181no—
290C3⋊C8—21no—
291C3⋊C8—61no—
292C2×C32#88411yescharacteristic
293C2×C32D3×D811yescharacteristic
294C2×C32#91411yescharacteristic
295C3⋊D3—122no—
296C3×D3—81no—
297C3×D3—241no—
298C2×C9—21no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9—241no—
302C2×C9#88411yescharacteristic
303C2×C9D3×D811yescharacteristic
304C2×C9#91411yescharacteristic
305D9—81no—
306D9—124no—
307D9—122no—
308C2×D4—31no—
309C2×D4—91no—
310C2×D4—271no—
311C2×D4—541no—
312C2×D4—541no—
313C22×C4—541no—
314D8—542no—
315D8—542no—
316C2×C8—31no—
317C2×C8—91no—
318C4⋊C4D3×D911yescharacteristic
319C22⋊C4—541no—
320C42—271no—
321C22×C3—21no—
322C22×C3—121no—
323C22×C3—121no—
324C22×C3—121no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3—361no—
328C22×C3D4×D911yescharacteristic
329C22×C3#381111yescharacteristic
330C2×D3—41no—
331C2×D3—41no—
332C2×D3—41no—
333C2×D3—121no—
334C2×D3—121no—
335C2×D3—121no—
336C2×D3—184no—
337C2×D3—182no—
338C2×D3—182no—
339C2×D3—362no—
340C2×D3—362no—
341C2×D3—361no—
342C2×D3—361no—
343C2×D3—361no—
344C2×D3—361no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—42no—
348C3×C4—41no—
349C3×C4—41no—
350C3×C4#369711yescharacteristic
351C3×C4D4×D911yescharacteristic
352C3×C4#379811yescharacteristic
353C3×C4#380211yescharacteristic
354Dic3—92no—
355Dic3—92no—
356Dic3—182no—
357Dic3—181no—
358Dic3—181no—
359Dic3—361no—
360C32#478611yescharacteristic
361C9—21no—
362C9#478611yescharacteristic, complemented
363C23—61no—
364C23—181no—
365C23—541no—
366D4—61no—
367D4—61no—
368D4—181no—
369D4—181no—
370D4—272no—
371D4—541no—
372D4—1082no—
373D4—1082no—
374C2×C4—21no—
375C2×C4—271no—
376C2×C4—271no—
377C2×C4—1082no—
378C2×C4C2×D3×D911yescharacteristic
379C8—61no—
380C8—181no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—241no—
385C2×C3—721no—
386C2×C3#6344611yescharacteristic
387C2×C3D8×D911yescharacteristic
388C2×C3#6347611yescharacteristic
389C2×C3#6391211yescharacteristic
390C2×C3#6392611yescharacteristic
391C2×C3#6394111yescharacteristic
392D3—81no—
393D3—241no—
394D3—364no—
395D3—362no—
396D3—362no—
397C22—121no—
398C22—121no—
399C22—121no—
400C22—361no—
401C22—361no—
402C22—361no—
403C22—542no—
404C22—1081no—
405C22—1081no—
406C22#8866811yescharacteristic
407C4—41no—
408C4—272no—
409C4—541no—
410C4#8864011yescharacteristic
411C4#8864611yescharacteristic
412C3—21no—
413C3#1058846611yescharacteristic, complemented
414C3#1059055111yescharacteristic
415C2—241no—
416C2—721no—
417C2—1082no—
418C2#1172104111yescharacteristic
419C2#1172117411yescharacteristic
420C2#1172121311yescharacteristic
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#459
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. #11721749C2semidirect
2. #11721043C2semidirect
3. C2×#88422C2semidirect
4. C2×#88417C2semidirect
5. C9×#809C2semidirect
6. C3×#63369C2semidirect
7. #11720273C2semidirect
37. #63369D3semidirect
46. C4⋊(C3⋊D9)D4semidirect
103. C3⋊Dic9D8semidirect
118. #809D9semidirect
247. C3×C9#459semidirect
362. C9#4786semidirect
413. C3#10588466semidirect

Maximal subgroups

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

subgroupindex
#117217492
#117210432
C2×#884222
C2×#884172
C9×#8092
C3×#633692
#117202732
#105905513
#105884663