Finite Group Databasebeta

← #13144790

Subgroups — #13144790

Order 1728.

760 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 135 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#11722672C211yesmaximal, characteristic
2C3×C9×#137C211yesmaximal, characteristic, complemented
3#11721857C211yesmaximal, characteristic, complemented
4#11721853C211yesmaximal, characteristic, complemented
5#11720585C211yesmaximal, characteristic, complemented
6#11720558C211yesmaximal, characteristic, complemented
7#11720557C211yesmaximal, characteristic, complemented
8#10592901—31nomaximal
9#10588883—33nomaximal
10C3×C9×#41C2211yescharacteristic
11C2×C3×Q8×C9C2211yescharacteristic, complemented
12#88746—21no—
13D4×C3⋊D9—22no—
14D4×C3⋊D9—21no—
15#88736—21no—
16#88736C2211yescharacteristic
17C2×C4⋊(C3⋊D9)—21no—
18C2×C4⋊(C3⋊D9)C2211yescharacteristic
19C3×C9×Q16—21no—
20C3×C9×SD16—21no—
21C3×C9×M16C2211yescharacteristic, complemented
22#88548—21no—
23#88547—21no—
24#88536—22no—
25C4×C3⋊Dic9C2211yescharacteristic, complemented
26#88532C2211yescharacteristic, complemented
27#64366—31no—
28C32×#137D311yescharacteristic
29#64156—31no—
30#64151—31no—
31#63715—33no—
32#63664—31no—
33#63652—31no—
34#63650—31no—
35C9×#137D313yescomplemented
36#63509—33no—
37#63505—33no—
38#63396—33no—
39#63384—33no—
40#63383—33no—
41C22×C3⋊D9—22no—
42C22×C3⋊D9D412yes—
43C3×Q8×C9—21no—
44C3×Q8×C9—21no—
45C3×Q8×C9D411yescharacteristic
46C3×D4×C9—21no—
47C3×D4×C9D411yescharacteristic
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9—21no—
50C2×C3×C4×C9C2311yescharacteristic
51#6233—22no—
52#6233—21no—
53C2×C3⋊Dic9—21no—
54C4⋊(C3⋊D9)—22no—
55C4⋊(C3⋊D9)—21no—
56C4⋊(C3⋊D9)—21no—
57C4⋊(C3⋊D9)D411yescharacteristic
58C4×C3⋊D9—22no—
59C4×C3⋊D9—21no—
60#6226D411yescharacteristic
61C3×C8×C9—21no—
62C3⋊(C9⋊C8)—21no—
63#5202—93no—
64#5202—91no—
65C32×#41C2×D311yescharacteristic
66C2×C32×Q8C2×D311yescharacteristic
67#3832—61no—
68D4×C3⋊D3—62no—
69D4×C3⋊D3—61no—
70#3828—31no—
71#3828—61no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×C4⋊(C3⋊D3)—61no—
74C32×Q16—21no—
75C32×SD16—21no—
76C32×M16C2×D311yescharacteristic
77#3756—61no—
78#3755—61no—
79#3752—62no—
80C4×C3⋊Dic3—31no—
81#3748—31no—
82C9×#41C2×D313yes—
83C2×Q8×C9C2×D313yescomplemented
84#3701—63no—
85D4×D9—66no—
86D4×D9—63no—
87#3697—33no—
88#3697—63no—
89C2×D36—33no—
90C2×D36—63no—
91C9×Q16—23no—
92C9×SD16—23no—
93C9×M16C2×D313yescomplemented
94#3675—63no—
95#3674—63no—
96(C2×D9)⋊C4—66no—
97C4×Dic9—33no—
98#3667—33no—
99C22×C3×C9—21no—
100C22×C3×C9C2×D411yescharacteristic
101C2×C3⋊D9—22no—
102C2×C3⋊D9—21no—
103C2×C3⋊D9—42no—
104C2×C3⋊D9—82no—
105C3×C4×C9—21no—
106C3×C4×C9—41no—
107C3×C4×C9C2×D411yescharacteristic
108C3×C4×C9C2×D411yescharacteristic
109C3⋊Dic9—21no—
110C3⋊Dic9—41no—
111#1009—93no—
112#1009—91no—
113C3×#137D913yescomplemented
114C3×#137C3⋊D311yescharacteristic
115#947—93no—
116#947—91no—
117#942—93no—
118#942—91no—
119#837—93no—
120#837—91no—
121#825—93no—
122#825—91no—
123#823—93no—
124#823—91no—
125C22×C3⋊D3—32no—
126C22×C3⋊D3—62no—
127C32×Q8—21no—
128C32×Q8—21no—
129C32×Q8#6711yescharacteristic
130C32×D4—21no—
131C32×D4#6711yescharacteristic
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4C22×D311yescharacteristic
135#638—62no—
136#638—61no—
137C2×C3⋊Dic3—61no—
138C4⋊(C3⋊D3)—31no—
139C4⋊(C3⋊D3)—62no—
140C4⋊(C3⋊D3)—61no—
141C4⋊(C3⋊D3)—61no—
142C4×C3⋊D3—62no—
143C4×C3⋊D3—61no—
144#634—31no—
145C22×D9—36no—
146C22×D9—66no—
147C32×C8—21no—
148C3⋊(C3⋊C8)—61no—
149Q8×C9—23no—
150Q8×C9—23no—
151Q8×C9#6713yes—
152D4×C9—23no—
153D4×C9#6713yes—
154C2×C4×C9—23no—
155C2×C4×C9—23no—
156C2×C4×C9C22×D313yes—
157#611—66no—
158#611—63no—
159C2×Dic9—63no—
160D36—33no—
161D36—66no—
162D36—63no—
163D36—63no—
164C4×D9—66no—
165C4×D9—63no—
166Dic18—33no—
167C8×C9—23no—
168C9⋊C8—63no—
169#455—271noSylow
170C2×C3×C9—21no—
171C2×C3×C9—41no—
172C2×C3×C9#12011yescharacteristic
173C3⋊D9—42no—
174C3⋊D9—41no—
175C3×#41C2×D913yes—
176C3×#41C2×C3⋊D311yescharacteristic
177C2×C3×Q8C2×D913yescomplemented
178C2×C3×Q8C2×C3⋊D311yescharacteristic
179#239—183no—
180#239—181no—
181D3×D4—182no—
182D3×D4—186no—
183D3×D4—183no—
184D3×D4—181no—
185#235—93no—
186#235—91no—
187#235—181no—
188#235—183no—
189C2×D12—93no—
190C2×D12—91no—
191C2×D12—181no—
192C2×D12—183no—
193C3×Q16—23no—
194C3×Q16—21no—
195C3×SD16—23no—
196C3×SD16—21no—
197C3×M16C2×D913yescomplemented
198C3×M16C2×C3⋊D311yescharacteristic
199#216—181no—
200#216—183no—
201#215—181no—
202#215—183no—
203(C2×D3)⋊C4—182no—
204(C2×D3)⋊C4—186no—
205C4×Dic3—93no—
206C4×Dic3—91no—
207#208—93no—
208#208—91no—
209C22×C32—21no—
210C22×C32D3×D411yescharacteristic
211C2×C3⋊D3—61no—
212C2×C3⋊D3—62no—
213C2×C3⋊D3—122no—
214C2×C3⋊D3—242no—
215C32×C4—21no—
216C32×C4—41no—
217C32×C4D3×D411yescharacteristic
218C32×C4C2×#6711yescharacteristic
219C3⋊Dic3—61no—
220C3⋊Dic3—121no—
221C22×C9—23no—
222C22×C9D3×D413yes—
223C2×D9—63no—
224C2×D9—66no—
225C2×D9—126no—
226C2×D9—246no—
227C4×C9—23no—
228C4×C9—43no—
229C4×C9D3×D413yes—
230C4×C9C2×#6713yes—
231Dic9—63no—
232Dic9—123no—
233H(5,𝔽2)—271no—
234#137—271no—
235#137C3⋊D911yescharacteristic, complemented
236#124—271no—
237#104—271no—
238#104—271no—
239#100—271no—
240C3×C9#45511yescharacteristic, complemented, Sylow
241C22×D3—96no—
242C22×D3—92no—
243C22×D3—182no—
244C22×D3—186no—
245C3×Q8—23no—
246C3×Q8—23no—
247C3×Q8—21no—
248C3×Q8—21no—
249C3×Q8#61113yes—
250C3×Q8#63811yescharacteristic
251C3×D4—23no—
252C3×D4—21no—
253C3×D4#61113yes—
254C3×D4#63811yescharacteristic
255C2×C3×C4—23no—
256C2×C3×C4—21no—
257C2×C3×C4—23no—
258C2×C3×C4—21no—
259C2×C3×C4C22×D913yes—
260C2×C3×C4C22×C3⋊D311yescharacteristic
261#67—182no—
262#67—186no—
263#67—183no—
264#67—181no—
265C2×Dic3—183no—
266C2×Dic3—181no—
267D12—93no—
268D12—91no—
269D12—182no—
270D12—186no—
271D12—183no—
272D12—183no—
273D12—181no—
274D12—181no—
275C4×D3—182no—
276C4×D3—186no—
277C4×D3—183no—
278C4×D3—181no—
279Dic6—93no—
280Dic6—91no—
281C3×C8—23no—
282C3×C8—21no—
283C3⋊C8—183no—
284C3⋊C8—181no—
285C2×C32—21no—
286C2×C32—41no—
287C2×C32#93711yescharacteristic
288C3⋊D3—122no—
289C3⋊D3—121no—
290C2×C9—23no—
291C2×C9—43no—
292C2×C9#93713yes—
293D9—126no—
294D9—123no—
295#41—271no—
296#41—541no—
297#41—541no—
298#41C2×C3⋊D911yescharacteristic
299C2×Q8C2×C3⋊D911yescharacteristic, complemented
300C2×D4—271no—
301C2×D4—541no—
302C2×D4—542no—
303C2×D4—541no—
304Q16—21no—
305Q16—541no—
306SD16—21no—
307SD16—541no—
308M16—271no—
309M16C2×C3⋊D911yescharacteristic, complemented
310C22⋊C4—542no—
311C42—271no—
312C22×C3—23no—
313C22×C3—21no—
314C22×C3D4×D913yes—
315C22×C3D4×C3⋊D311yescharacteristic
316C2×D3—183no—
317C2×D3—186no—
318C2×D3—181no—
319C2×D3—182no—
320C2×D3—362no—
321C2×D3—366no—
322C2×D3—722no—
323C2×D3—726no—
324C3×C4—23no—
325C3×C4—21no—
326C3×C4—43no—
327C3×C4—41no—
328C3×C4D4×D913yes—
329C3×C4C2×#61113yes—
330C3×C4D4×C3⋊D311yescharacteristic
331C3×C4C2×#63811yescharacteristic
332Dic3—183no—
333Dic3—181no—
334Dic3—361no—
335Dic3—363no—
336C32#520211yescharacteristic
337C9#520213yescomplemented
338C23—272no—
339C23—542no—
340Q8—21no—
341Q8—21no—
342Q8—271no—
343Q8#623311yescharacteristic
344D4—21no—
345D4—271no—
346D4—542no—
347D4—542no—
348D4—541no—
349D4—541no—
350D4—541no—
351D4#623311yescharacteristic
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—542no—
355C2×C4—541no—
356C2×C4—541no—
357C2×C4C22×C3⋊D911yescharacteristic
358C8—21no—
359C8—541no—
360C2×C3—21no—
361C2×C3—23no—
362C2×C3—41no—
363C2×C3—43no—
364C2×C3#6349913yes—
365C2×C3#6414611yescharacteristic
366D3—362no—
367D3—361no—
368D3—366no—
369D3—363no—
370C22—21no—
371C22—541no—
372C22—542no—
373C22—1082no—
374C22—2162no—
375C22D4×C3⋊D911yescharacteristic
376C4—21no—
377C4—41no—
378C4—541no—
379C4—1081no—
380C4D4×C3⋊D911yescharacteristic
381C4C2×#623311yescharacteristic
382C3#1058888313yescomplemented
383C3#1059290111yescharacteristic
384C2—21no—
385C2—41no—
386C2—1082no—
387C2—1081no—
388C2#1172183711yescharacteristic
subgroupG/H
centerC2#11721837
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#137C2
socleC2×C32#937
prime radical#11722672C2
hypercenter#137C3⋊D9
hypocenterC3×C9#455
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. C3×C9×#137C2semidirect
3. #11721857C2semidirect
4. #11721853C2semidirect
5. #11720585C2semidirect
6. #11720558C2semidirect
7. #11720557C2semidirect
11. C2×C3×Q8×C9C22semidirect
21. C3×C9×M16C22semidirect
25. C4×C3⋊Dic9C22semidirect
26. #88532C22semidirect
35. C9×#137D3semidirect
83. C2×Q8×C9C2×D3semidirect
93. C9×M16C2×D3semidirect
113. C3×#137D9semidirect
177. C2×C3×Q8C2×D9semidirect
197. C3×M16C2×D9semidirect
235. #137C3⋊D9semidirect
240. C3×C9#455semidirect
299. C2×Q8C2×C3⋊D9semidirect
309. M16C2×C3⋊D9semidirect
337. C9#5202semidirect
382. C3#10588883semidirect

Maximal subgroups

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

subgroupindex
#117226722
C3×C9×#1372
#117218572
#117218532
#117205852
#117205582
#117205572
#105929013
#105888833