Finite Group Databasebeta

← #13138707

Subgroups — #13138707

Order 1728.

554 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×#88634C211yesmaximal, characteristic
2C2×#88523C211yesmaximal, characteristic, complemented
3#11721000C211yesmaximal, characteristic, complemented
4C3×#63380C211yesmaximal, characteristic, complemented
5C9×#816C211yesmaximal, characteristic, complemented
6#11720297C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic, complemented
8#10590957—31nomaximal
9#10588483—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C2×#6192—21no—
14C2×(C3⋊D9)⋊C4—21no—
15C2×Dic3×D9—21no—
16#88634—22no—
17#88634C414yescomplemented
18C2×C3×C8×C9C2211yescharacteristic
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20#88523—22no—
21C9×C4⋊Dic3C2211yescharacteristic
22C3×C4⋊Dic9C2211yescharacteristic
23#88442—21no—
24C2×#3796—31no—
25C2×#3743—31no—
26#63900—31no—
27C2×#3701—31no—
28C3×#821—31no—
29C3×#816D311yescharacteristic
30#63565—31no—
31#63562—31no—
32C2×#3663—31no—
33C2×#3663—61no—
34C4⋊C4×D9—31no—
35C9×#103—31no—
36#63395—31no—
37#63380D311yescharacteristic, complemented
38#63369—31no—
39C22×C3⋊D9D411yescharacteristic
40C22×C3×D9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9D411yescharacteristic
43C4×C3⋊D9C2×C412yes—
44C4×C3⋊D9D412yes—
45C2×C9×Dic3—21no—
46C2×C9×Dic3—21no—
47C2×C3×Dic9—21no—
48C3×D36—21no—
49C3×D36C2×C412yescomplemented
50C9×Dic6—21no—
51C9×Dic6C2×C412yescomplemented
52#6192—24no—
53(C3⋊D9)⋊C4—42no—
54Dic3×D9—24no—
55C3×C8×C9—21no—
56C3⋊(C9⋊C8)—21no—
57#4803—91no—
58#4771—91no—
59C2×C4×C3⋊D3—31no—
60C2×C3×D12—31no—
61C2×C3×Dic6C2×D311yescharacteristic
62C2×#626—61no—
63C2×#624—61no—
64C2×D3×Dic3—61no—
65#3796—34no—
66#3796—62no—
67C2×C32×C8C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)—31no—
69#3743—62no—
70C3×C4⋊Dic3—31no—
71C3×C4⋊Dic3C2×D311yescharacteristic
72#3724—61no—
73C2×Q8×C9—31no—
74#3701—34no—
75#3701—62no—
76C2×D36—61no—
77C2×D36C2×D311yescharacteristic
78C2×C4×D9—31no—
79C2×C4×D9—61no—
80C2×C4×D9—61no—
81C2×C4×D9—61no—
82C2×C8×C9—21no—
83C2×C8×C9C2×D311yescharacteristic
84C9×C4⋊C4—31no—
85C4⋊Dic9C2×D311yescharacteristic
86Dic9⋊C4—61no—
87C2×C9⋊C8—31no—
88C2×C9⋊C8—61no—
89#3663—62no—
90#3663—64no—
91C22×C3×C9C2×D411yescharacteristic
92C2×C3⋊D9—21no—
93C2×C3⋊D9—21no—
94C2×C3⋊D9C22⋊C412yes—
95C2×C3×D9—22no—
96C2×C3×D9—41no—
97C2×C3×D9—41no—
98C3×C4×C9C22×C411yescharacteristic
99C3×C4×C9C2×D411yescharacteristic
100C3⋊Dic9C22⋊C412yes—
101C9×Dic3—22no—
102C9×Dic3—41no—
103C3×Dic9—41no—
104C2×#239—91no—
105C2×#237—91no—
106C2×ΓL(1,𝔽25)—91no—
107C2×ΓL(1,𝔽25)—91no—
108C2×ΓL(1,𝔽25)—181no—
109D3×C4⋊C4—91no—
110D3×C4⋊C4—91no—
111C3×#103—31no—
112C3×#102—91no—
113#835—91no—
114#832—91no—
115#821—31no—
116#816D911yescharacteristic, complemented
117#810—91no—
118#809—91no—
119C22×C3⋊D3—31no—
120C22×C3×D3—61no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C3⋊Dic3—31no—
123C4×C3⋊D3—32no—
124C4×C3⋊D3—32no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—21no—
127C2×C3×Dic3—61no—
128C3×D12—32no—
129C3×D12—61no—
130C3×Dic6—21no—
131C3×Dic6C4×D312yes—
132#626—64no—
133#624—122no—
134D3×Dic3—64no—
135C22×D9—21no—
136C22×D9—31no—
137C22×D9—61no—
138C32×C8—21no—
139C3⋊(C3⋊C8)—61no—
140Q8×C9—32no—
141Q8×C9—61no—
142C2×C4×C9—21no—
143C2×C4×C9—61no—
144C2×C4×C9—61no—
145C2×C4×C9C22×D311yescharacteristic
146C2×Dic9—21no—
147C2×Dic9—31no—
148C2×Dic9—61no—
149D36—21no—
150D36—64no—
151D36C4×D312yescomplemented
152C4×D9—32no—
153C4×D9—32no—
154C4×D9—62no—
155C4×D9—62no—
156C4×D9—64no—
157C4×D9—122no—
158C8×C9—22no—
159C8×C9—21no—
160C9⋊C8—62no—
161C9⋊C8—61no—
162#417—271noSylow
163C2×C3×C9C2×C22⋊C411yescharacteristic
164C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
165C2×C3×C9#13711yescharacteristic
166C3⋊D9—22no—
167C3×D9—42no—
168C2×C3×Q8—31no—
169C2×C3×D4—91no—
170C2×#67—181no—
171C22×Dic3—181no—
172#239—94no—
173#239—182no—
174#237—94no—
175#237—182no—
176C2×D12—31no—
177C2×D12—181no—
178C2×C4×D3—91no—
179C2×C4×D3—91no—
180C2×C4×D3—181no—
181C2×C4×D3—181no—
182C2×C4×D3—181no—
183C2×C4×D3—181no—
184C2×Dic6C2×D911yescharacteristic
185C2×C3×C8—21no—
186C2×C3×C8C2×D911yescharacteristic
187C2×C3×C8D3211yescharacteristic
188C3×C4⋊C4—31no—
189C3×C4⋊C4—91no—
190C4⋊Dic3—31no—
191C4⋊Dic3C2×D911yescharacteristic
192Dic3⋊C4—181no—
193Dic3⋊C4—181no—
194C2×C3⋊C8—91no—
195C2×C3⋊C8—91no—
196C2×C3⋊C8—181no—
197ΓL(1,𝔽25)—182no—
198ΓL(1,𝔽25)—182no—
199ΓL(1,𝔽25)—184no—
200C22×C32D3×D411yescharacteristic
201C2×C3⋊D3—32no—
202C2×C3⋊D3—61no—
203C2×C3⋊D3—61no—
204C2×C3×D3—62no—
205C2×C3×D3—121no—
206C2×C3×D3—121no—
207C32×C4C2×C4×D311yescharacteristic
208C32×C4D3×D411yescharacteristic
209C3⋊Dic3—32no—
210C3×Dic3—22no—
211C3×Dic3—41no—
212C3×Dic3—121no—
213C22×C9—21no—
214C22×C9D3×D411yescharacteristic
215C2×D9—22no—
216C2×D9—32no—
217C2×D9—41no—
218C2×D9—41no—
219C2×D9—62no—
220C2×D9—61no—
221C2×D9—61no—
222C2×D9—121no—
223C2×D9—121no—
224C4×C9—21no—
225C4×C9—21no—
226C4×C9—62no—
227C4×C9—121no—
228C4×C9C2×C4×D311yescharacteristic
229C4×C9D3×D411yescharacteristic
230Dic9—32no—
231Dic9—41no—
232Dic9—62no—
233C2×#41—271no—
234C2×M16—271no—
235C2×C4⋊C4—271no—
236#103—31no—
237#103—271no—
238#102—91no—
239#102—271no—
240C3×C9#41711yescharacteristic, complemented, Sylow
241C23×C3—181no—
242C22×D3—61no—
243C22×D3—91no—
244C22×D3—91no—
245C22×D3—181no—
246C3×Q8—32no—
247C3×Q8—61no—
248C3×D4—92no—
249C3×D4—181no—
250C2×C3×C4—21no—
251C2×C3×C4—61no—
252C2×C3×C4—61no—
253C2×C3×C4—181no—
254C2×C3×C4C22×D911yescharacteristic
255C2×C3×C4C2×D3211yescharacteristic
256#67—184no—
257C2×Dic3—21no—
258C2×Dic3—21no—
259C2×Dic3—61no—
260C2×Dic3—91no—
261C2×Dic3—91no—
262C2×Dic3—181no—
263C2×Dic3—184no—
264D12—32no—
265D12—61no—
266D12—184no—
267C4×D3—92no—
268C4×D3—92no—
269C4×D3—92no—
270C4×D3—92no—
271C4×D3—182no—
272C4×D3—182no—
273C4×D3—184no—
274C4×D3—362no—
275C4×D3—362no—
276Dic6—21no—
277Dic6C4×D912yescomplemented
278C3×C8—22no—
279C3×C8—21no—
280C3×C8—21no—
281C3⋊C8—182no—
282C3⋊C8—181no—
283C3⋊C8—181no—
284C2×C32D3×C22⋊C411yescharacteristic
285C2×C32#91411yescharacteristic
286C2×C32#91811yescharacteristic
287C3⋊D3—62no—
288C3×D3—122no—
289C2×C9—21no—
290C2×C9—21no—
291C2×C9—21no—
292C2×C9D3×C22⋊C411yescharacteristic
293C2×C9#91111yescharacteristic
294C2×C9#91511yescharacteristic
295D9—42no—
296D9—62no—
297D9—122no—
298#41—274no—
299#41—542no—
300C2×Q8—31no—
301C2×D4—91no—
302C2×D4—541no—
303C22×C4—271no—
304C22×C4—541no—
305C22×C4—541no—
306M16—542no—
307C2×C8—271no—
308C2×C8D3×D911yescharacteristic
309C4⋊C4—31no—
310C4⋊C4—91no—
311C4⋊C4—541no—
312C22×C3—21no—
313C22×C3—182no—
314C22×C3—361no—
315C22×C3—361no—
316C22×C3D4×D911yescharacteristic
317C22×C3#380211yescharacteristic
318C2×D3—62no—
319C2×D3—92no—
320C2×D3—92no—
321C2×D3—121no—
322C2×D3—121no—
323C2×D3—182no—
324C2×D3—181no—
325C2×D3—181no—
326C2×D3—181no—
327C2×D3—181no—
328C2×D3—361no—
329C2×D3—361no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—62no—
333C3×C4—121no—
334C3×C4—361no—
335C3×C4C2×C4×D911yescharacteristic
336C3×C4D4×D911yescharacteristic
337C3×C4C4×D3211yescharacteristic
338C3×C4#381111yescharacteristic
339Dic3—22no—
340Dic3—41no—
341Dic3—92no—
342Dic3—92no—
343Dic3—121no—
344Dic3—182no—
345C32#480311yescharacteristic
346C9—21no—
347C9#477111yescharacteristic, complemented
348C23—181no—
349C23—271no—
350Q8—32no—
351Q8—61no—
352D4—92no—
353D4—181no—
354D4—544no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—181no—
358C2×C4—271no—
359C2×C4—272no—
360C2×C4—272no—
361C2×C4—544no—
362C2×C4—1082no—
363C2×C4C2×D3×D911yescharacteristic
364C8—21no—
365C8—541no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—362no—
370C2×C3C22⋊C4×D911yescharacteristic
371C2×C3#6347611yescharacteristic
372C2×C3#6348011yescharacteristic
373C2×C3#6379811yescharacteristic
374C2×C3#6381111yescharacteristic
375C2×C3#6392111yescharacteristic
376D3—122no—
377D3—182no—
378D3—182no—
379D3—362no—
380C22—182no—
381C22—272no—
382C22—361no—
383C22—361no—
384C22—541no—
385C22—541no—
386C22#8864611yescharacteristic
387C4—62no—
388C4—121no—
389C4—272no—
390C4—361no—
391C4C4×D3×D911yescharacteristic
392C4#8866811yescharacteristic
393C3—21no—
394C3#1058848311yescharacteristic, complemented
395C3#1059095711yescharacteristic
396C2—362no—
397C2—542no—
398C2#1172082011yescharacteristic
399C2#1172083211yescharacteristic
400C2#1172104811yescharacteristic
subgroupG/H
centerC22#88646
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×#88634C2
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9#417
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#88523C2semidirect
3. #11721000C2semidirect
4. C3×#63380C2semidirect
5. C9×#816C2semidirect
6. #11720297C2semidirect
7. #11720266C2semidirect
17. #88634C4semidirect
37. #63380D3semidirect
49. C3×D36C2×C4semidirect
51. C9×Dic6C2×C4semidirect
116. #816D9semidirect
151. D36C4×D3semidirect
240. C3×C9#417semidirect
277. Dic6C4×D9semidirect
347. C9#4771semidirect
394. C3#10588483semidirect

Maximal subgroups

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

subgroupindex
C2×#886342
C2×#885232
#117210002
C3×#633802
C9×#8162
#117202972
#117202662
#105909573
#105884833