Finite Group Databasebeta

← #13144025

Subgroups — #13144025

Order 1728.

770 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 147 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#41×C3⋊D9C211yesmaximal, characteristic
2M16×C3⋊D9C211yesmaximal, characteristic, complemented
3#11721709C211yesmaximal, characteristic, complemented
4C3×C9×#104C211yesmaximal, characteristic, complemented
5#11720585C211yesmaximal, characteristic, complemented
6#11720558C211yesmaximal, characteristic, complemented
7#11720518C211yesmaximal, characteristic, complemented
8#10592520—31nomaximal
9#10588501—33nomaximal
10C3×C9×#41C2211yescharacteristic
11#88746C411yescharacteristic, complemented
12Q8×C3⋊D9C411yescharacteristic, complemented
13#88742—21no—
14#88742C411yescharacteristic, complemented
15D4×C3⋊D9—21no—
16D4×C3⋊D9C411yescharacteristic, complemented
17#88736—21no—
18#88736C2211yescharacteristic
19C2×C4×C3⋊D9—21no—
20C2×C4×C3⋊D9C2211yescharacteristic
21C3×C9×M16C2211yescharacteristic, complemented
22C3×C42×C9C2211yescharacteristic, complemented
23#88536—21no—
24#88534—21no—
25C4×C3⋊Dic9C2211yescharacteristic, complemented
26#88532C2211yescharacteristic, complemented
27#88523—21no—
28C8×C3⋊D9—21no—
29#41×C3⋊D3—31no—
30M16×C3⋊D3—31no—
31#64081—31no—
32#41×D9—33no—
33C32×#104D311yescharacteristic
34#63664—31no—
35#63652—31no—
36#63632—31no—
37M16×D9—33no—
38#63434—33no—
39C9×#104D313yescomplemented
40#63396—33no—
41#63384—33no—
42#63364—33no—
43C22×C3⋊D9—21no—
44C22×C3⋊D9D411yescharacteristic
45C3×Q8×C9C2×C411yescharacteristic
46C3×D4×C9—21no—
47C3×D4×C9C2×C411yescharacteristic
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9—21no—
50C2×C3×C4×C9C2311yescharacteristic
51#6233—21no—
52#6233—21no—
53#6233—21no—
54C2×C3⋊Dic9—21no—
55C2×C3⋊Dic9—21no—
56C2×C3⋊Dic9D411yescharacteristic
57C4⋊(C3⋊D9)—21no—
58C4⋊(C3⋊D9)C2×C411yescharacteristic
59C4×C3⋊D9—21no—
60C4×C3⋊D9—21no—
61C4×C3⋊D9—21no—
62C4×C3⋊D9C2×C411yescharacteristic
63C4×C3⋊D9C2×C411yescharacteristic
64C4×C3⋊D9D411yescharacteristic
65C4×C3⋊D9D411yescharacteristic
66#6226—21no—
67#6226C2×C411yescharacteristic
68C3×C8×C9—21no—
69C3⋊(C9⋊C8)—21no—
70#4821—93no—
71#4821—91no—
72C32×#41C2×D311yescharacteristic
73#3832—31no—
74Q8×C3⋊D3—31no—
75#3830—31no—
76#3830—61no—
77D4×C3⋊D3—31no—
78D4×C3⋊D3—61no—
79#3828—31no—
80#3828—61no—
81C2×C4×C3⋊D3—31no—
82C2×C4×C3⋊D3—61no—
83C32×M16C2×D311yescharacteristic
84C32×C42C2×D311yescharacteristic
85#3752—61no—
86#3750—61no—
87C4×C3⋊Dic3—31no—
88#3748—31no—
89#3743—61no—
90C8×C3⋊D3—61no—
91C9×#41C2×D313yes—
92#3701—33no—
93Q8×D9—33no—
94#3699—33no—
95#3699—63no—
96D4×D9—33no—
97D4×D9—63no—
98#3697—33no—
99#3697—63no—
100C2×C4×D9—33no—
101C2×C4×D9—63no—
102C9×M16C2×D313yescomplemented
103C42×C9C2×D313yescomplemented
104(C2×D9)⋊C4—63no—
105Dic9⋊C4—63no—
106C4×Dic9—33no—
107#3667—33no—
108#3663—63no—
109C8×D9—63no—
110C22×C3×C9—21no—
111C22×C3×C9C2×D411yescharacteristic
112C2×C3⋊D9—21no—
113C2×C3⋊D9—41no—
114C2×C3⋊D9—42no—
115C2×C3⋊D9C22⋊C411yescharacteristic
116C2×C3⋊D9C22⋊C411yescharacteristic
117C3×C4×C9—21no—
118C3×C4×C9—41no—
119C3×C4×C9C22×C411yescharacteristic
120C3×C4×C9C2×D411yescharacteristic
121C3⋊Dic9—21no—
122C3⋊Dic9—41no—
123C3⋊Dic9C22⋊C411yescharacteristic
124C3⋊Dic9C22⋊C411yescharacteristic
125D3×#41—93no—
126D3×#41—91no—
127D3×M16—93no—
128D3×M16—91no—
129#872—93no—
130#872—91no—
131C3×#104D913yescomplemented
132C3×#104C3⋊D311yescharacteristic
133#837—93no—
134#837—91no—
135#825—93no—
136#825—91no—
137#805—93no—
138#805—91no—
139C22×C3⋊D3—31no—
140C22×C3⋊D3—61no—
141C32×Q8C4×D311yescharacteristic
142C32×D4—21no—
143C32×D4C4×D311yescharacteristic
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4C22×D311yescharacteristic
147#638—61no—
148#638—61no—
149#638—61no—
150C2×C3⋊Dic3—31no—
151C2×C3⋊Dic3—61no—
152C2×C3⋊Dic3—61no—
153C4⋊(C3⋊D3)—31no—
154C4⋊(C3⋊D3)—61no—
155C4×C3⋊D3—31no—
156C4×C3⋊D3—31no—
157C4×C3⋊D3—31no—
158C4×C3⋊D3—31no—
159C4×C3⋊D3—61no—
160C4×C3⋊D3—61no—
161C4×C3⋊D3—61no—
162#634—31no—
163#634—61no—
164C22×D9—33no—
165C22×D9—63no—
166C32×C8—21no—
167C3⋊(C3⋊C8)—61no—
168Q8×C9C4×D313yes—
169D4×C9—23no—
170D4×C9C4×D313yes—
171C2×C4×C9—23no—
172C2×C4×C9—23no—
173C2×C4×C9C22×D313yes—
174#611—63no—
175#611—63no—
176#611—63no—
177C2×Dic9—33no—
178C2×Dic9—63no—
179C2×Dic9—63no—
180D36—33no—
181D36—63no—
182C4×D9—33no—
183C4×D9—33no—
184C4×D9—33no—
185C4×D9—33no—
186C4×D9—63no—
187C4×D9—63no—
188C4×D9—63no—
189Dic18—33no—
190Dic18—63no—
191C8×C9—23no—
192C9⋊C8—63no—
193#421—271noSylow
194C2×C3×C9—21no—
195C2×C3×C9—41no—
196C2×C3×C9C2×C22⋊C411yescharacteristic
197C3⋊D9—21no—
198C3⋊D9—21no—
199C3⋊D9—41no—
200C3×#41C2×D913yes—
201C3×#41C2×C3⋊D311yescharacteristic
202#239—93no—
203#239—91no—
204D3×Q8—93no—
205D3×Q8—91no—
206#237—93no—
207#237—91no—
208#237—181no—
209#237—183no—
210D3×D4—93no—
211D3×D4—91no—
212D3×D4—181no—
213D3×D4—183no—
214#235—93no—
215#235—91no—
216#235—181no—
217#235—183no—
218C2×C4×D3—93no—
219C2×C4×D3—91no—
220C2×C4×D3—181no—
221C2×C4×D3—183no—
222C3×M16C2×D913yescomplemented
223C3×M16C2×C3⋊D311yescharacteristic
224C3×C42C2×D913yescomplemented
225C3×C42C2×C3⋊D311yescharacteristic
226(C2×D3)⋊C4—181no—
227(C2×D3)⋊C4—183no—
228Dic3⋊C4—181no—
229Dic3⋊C4—183no—
230C4×Dic3—93no—
231C4×Dic3—91no—
232#208—93no—
233#208—91no—
234ΓL(1,𝔽25)—181no—
235ΓL(1,𝔽25)—183no—
236D3×C8—181no—
237D3×C8—183no—
238C22×C32—21no—
239C22×C32D3×D411yescharacteristic
240C2×C3⋊D3—31no—
241C2×C3⋊D3—31no—
242C2×C3⋊D3—61no—
243C2×C3⋊D3—121no—
244C2×C3⋊D3—122no—
245C32×C4—21no—
246C32×C4—41no—
247C32×C4C2×C4×D311yescharacteristic
248C32×C4D3×D411yescharacteristic
249C3⋊Dic3—31no—
250C3⋊Dic3—31no—
251C3⋊Dic3—61no—
252C3⋊Dic3—121no—
253C22×C9—23no—
254C22×C9D3×D413yes—
255C2×D9—33no—
256C2×D9—33no—
257C2×D9—63no—
258C2×D9—123no—
259C2×D9—126no—
260C4×C9—23no—
261C4×C9—43no—
262C4×C9C2×C4×D313yes—
263C4×C9D3×D413yes—
264Dic9—33no—
265Dic9—33no—
266Dic9—63no—
267Dic9—123no—
268C2×#41—271no—
269C2×M16—271no—
270#117—271no—
271#104—271no—
272#104—271no—
273#104—271no—
274#104C3⋊D911yescharacteristic, complemented
275C3×C9#42111yescharacteristic, complemented, Sylow
276C22×D3—93no—
277C22×D3—91no—
278C22×D3—181no—
279C22×D3—183no—
280C3×Q8C4×D913yes—
281C3×Q8C4×C3⋊D311yescharacteristic
282C3×D4—23no—
283C3×D4—21no—
284C3×D4C4×D913yes—
285C3×D4C4×C3⋊D311yescharacteristic
286C2×C3×C4—23no—
287C2×C3×C4—21no—
288C2×C3×C4—23no—
289C2×C3×C4—21no—
290C2×C3×C4C22×D913yes—
291C2×C3×C4C22×C3⋊D311yescharacteristic
292#67—181no—
293#67—183no—
294#67—181no—
295#67—183no—
296#67—183no—
297#67—181no—
298C2×Dic3—93no—
299C2×Dic3—91no—
300C2×Dic3—181no—
301C2×Dic3—183no—
302C2×Dic3—183no—
303C2×Dic3—181no—
304D12—93no—
305D12—91no—
306D12—181no—
307D12—183no—
308C4×D3—93no—
309C4×D3—93no—
310C4×D3—93no—
311C4×D3—93no—
312C4×D3—91no—
313C4×D3—91no—
314C4×D3—91no—
315C4×D3—91no—
316C4×D3—181no—
317C4×D3—183no—
318C4×D3—181no—
319C4×D3—183no—
320C4×D3—183no—
321C4×D3—181no—
322Dic6—93no—
323Dic6—91no—
324Dic6—181no—
325Dic6—183no—
326C3×C8—23no—
327C3×C8—21no—
328C3⋊C8—183no—
329C3⋊C8—181no—
330C2×C32—21no—
331C2×C32—41no—
332C2×C32D3×C22⋊C411yescharacteristic
333C3⋊D3—61no—
334C3⋊D3—61no—
335C3⋊D3—121no—
336C2×C9—23no—
337C2×C9—43no—
338C2×C9D3×C22⋊C413yes—
339D9—63no—
340D9—63no—
341D9—123no—
342#41—271no—
343#41—271no—
344#41—271no—
345#41—541no—
346#41—541no—
347#41C2×C3⋊D911yescharacteristic
348C2×Q8—271no—
349C2×D4—271no—
350C2×D4—541no—
351C22×C4—271no—
352C22×C4—541no—
353M16—271no—
354M16—541no—
355M16C2×C3⋊D911yescharacteristic, complemented
356C2×C8—541no—
357C4⋊C4—541no—
358C22⋊C4—541no—
359C42—271no—
360C42C2×C3⋊D911yescharacteristic, complemented
361C22×C3—21no—
362C22×C3—23no—
363C22×C3D4×D913yes—
364C22×C3D4×C3⋊D311yescharacteristic
365C2×D3—91no—
366C2×D3—91no—
367C2×D3—93no—
368C2×D3—93no—
369C2×D3—181no—
370C2×D3—183no—
371C2×D3—361no—
372C2×D3—363no—
373C2×D3—362no—
374C2×D3—366no—
375C3×C4—21no—
376C3×C4—23no—
377C3×C4—41no—
378C3×C4—43no—
379C3×C4C2×C4×D913yes—
380C3×C4D4×D913yes—
381C3×C4C2×C4×C3⋊D311yescharacteristic
382C3×C4D4×C3⋊D311yescharacteristic
383Dic3—91no—
384Dic3—91no—
385Dic3—93no—
386Dic3—93no—
387Dic3—181no—
388Dic3—183no—
389Dic3—361no—
390Dic3—363no—
391C32#482111yescharacteristic
392C9#482113yescomplemented
393C23—271no—
394C23—541no—
395Q8—271no—
396Q8—541no—
397Q8C4×C3⋊D911yescharacteristic
398D4—21no—
399D4—271no—
400D4—541no—
401D4—541no—
402D4—541no—
403D4—541no—
404D4C4×C3⋊D911yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—271no—
408C2×C4—271no—
409C2×C4—271no—
410C2×C4—271no—
411C2×C4—271no—
412C2×C4—541no—
413C2×C4—541no—
414C2×C4—541no—
415C2×C4—541no—
416C2×C4—541no—
417C2×C4C22×C3⋊D911yescharacteristic
418C8—21no—
419C8—541no—
420C2×C3—21no—
421C2×C3—23no—
422C2×C3—41no—
423C2×C3—43no—
424C2×C3C22⋊C4×D913yes—
425C2×C3C22⋊C4×C3⋊D311yescharacteristic
426D3—181no—
427D3—181no—
428D3—183no—
429D3—183no—
430D3—361no—
431D3—363no—
432C22—21no—
433C22—271no—
434C22—271no—
435C22—541no—
436C22—1081no—
437C22—1082no—
438C22D4×C3⋊D911yescharacteristic
439C4—21no—
440C4—41no—
441C4—271no—
442C4—271no—
443C4—541no—
444C4—1081no—
445C4C2×C4×C3⋊D911yescharacteristic
446C4D4×C3⋊D911yescharacteristic
447C3#1058850113yescomplemented
448C3#1059252011yescharacteristic
449C2—21no—
450C2—41no—
451C2—541no—
452C2—541no—
453C2—1081no—
454C2C22⋊C4×C3⋊D911yescharacteristic
subgroupG/H
centerC4D4×C3⋊D9
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#104C2
socleC2×C32D3×C22⋊C4
prime radical#41×C3⋊D9C2
cosocleC2×C3×C4×C9C23
hypercenter#104C3⋊D9
hypocenterC3×C9#421
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C4 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. M16×C3⋊D9C2semidirect
3. #11721709C2semidirect
4. C3×C9×#104C2semidirect
5. #11720585C2semidirect
6. #11720558C2semidirect
7. #11720518C2semidirect
11. #88746C4semidirect
12. Q8×C3⋊D9C4semidirect
14. #88742C4semidirect
16. D4×C3⋊D9C4semidirect
21. C3×C9×M16C22semidirect
22. C3×C42×C9C22semidirect
25. C4×C3⋊Dic9C22semidirect
26. #88532C22semidirect
39. C9×#104D3semidirect
102. C9×M16C2×D3semidirect
103. C42×C9C2×D3semidirect
131. C3×#104D9semidirect
222. C3×M16C2×D9semidirect
224. C3×C42C2×D9semidirect
274. #104C3⋊D9semidirect
275. C3×C9#421semidirect
355. M16C2×C3⋊D9semidirect
360. C42C2×C3⋊D9semidirect
392. C9#4821semidirect
447. C3#10588501semidirect

Maximal subgroups

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

subgroupindex
#41×C3⋊D92
M16×C3⋊D92
#117217092
C3×C9×#1042
#117205852
#117205582
#117205182
#105925203
#105885013