Finite Group Databasebeta

← #13137855

Subgroups — #13137855

Order 1728.

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

normal subgroups only

This group has 550 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
2#11721748C211yesmaximal, characteristic, complemented
3C2×(C3⋊D9)⋊C8C211yesmaximal, characteristic, complemented
4C9×#810C211yesmaximal, characteristic, complemented
5C3×#63369C211yesmaximal, characteristic, complemented
6#11720298C211yesmaximal, characteristic, complemented
7#11720283C211yesmaximal, characteristic, complemented
8#10590611—31nomaximal
9#10588484—31nomaximal
10C2×C4×C3⋊D9C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C2×#6192—21no—
14C2×Dic3×D9—21no—
15#88634—22no—
16#88634C414yescomplemented
17C3×C9×C4⋊C4C2211yescharacteristic
18#88536—21no—
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20C4×C3⋊Dic9—21no—
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23(C3⋊D9)⋊C8—22no—
24C2×#3796—31no—
25#64101—31no—
26C2×#3710—31no—
27C2×#3701—31no—
28C3×#810D311yescharacteristic
29C3×#809—31no—
30#63566—31no—
31#63563—31no—
32C2×C8×D9—31no—
33#63454—31no—
34#63454—61no—
35C9×#103—31no—
36#63395—31no—
37#63380—31no—
38#63369D311yescharacteristic, complemented
39C22×C3⋊D9D411yescharacteristic
40C22×C3×D9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44C2×C3⋊Dic9D411yescharacteristic
45C4×C3⋊D9C2×C412yes—
46C4×C3⋊D9D412yes—
47C2×C9×Dic3—21no—
48C3×D36—21no—
49C3×D36C2×C412yes—
50C9×Dic6—21no—
51C9×Dic6C2×C412yes—
52#6192—24no—
53Dic3×D9—24no—
54C9×C3⋊C8—21no—
55C3×C9⋊C8—21no—
56#4804—91no—
57#4772—91no—
58C2×C4×C3⋊D3—31no—
59C2×C3×D12—31no—
60C2×C3×Dic6C2×D311yescharacteristic
61C2×#626—61no—
62C2×D3×Dic3—61no—
63#3796—34no—
64#3796—62no—
65C32×C4⋊C4C2×D311yescharacteristic
66#3752—61no—
67C4⋊(C3⋊Dic3)—31no—
68C4×C3⋊Dic3—61no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8C2×D311yescharacteristic
71#3710—62no—
72C2×Q8×C9—31no—
73#3701—34no—
74#3701—62no—
75C2×D36—61no—
76C2×D36C2×D311yescharacteristic
77C2×C4×D9—31no—
78C2×C4×D9—61no—
79C2×C4×D9—61no—
80C2×C8×C9—31no—
81C9×C4⋊C4—21no—
82C9×C4⋊C4C2×D311yescharacteristic
83(C2×D9)⋊C4—61no—
84(C2×D9)⋊C4—62no—
85C4⋊Dic9—31no—
86C4⋊Dic9—61no—
87C4×Dic9—61no—
88C4×Dic9—62no—
89C2×C9⋊C8C2×D311yescharacteristic
90C8×D9—62no—
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×C9—41no—
99C3×C4×C9C22×C411yescharacteristic
100C3×C4×C9C2×D411yescharacteristic
101C3⋊Dic9—41no—
102C3⋊Dic9C22⋊C412yes—
103C9×Dic3—22no—
104C2×#239—91no—
105C2×#237—91no—
106C2×D3×C8—91no—
107C2×D3×C8—91no—
108#892—91no—
109#892—91no—
110#892—181no—
111C3×#103—31no—
112C3×#102—91no—
113#835—91no—
114#832—91no—
115#821—91no—
116#816—91no—
117#810D911yescharacteristic, complemented
118#809—31no—
119C22×C3⋊D3—31no—
120C22×C3×D3—61no—
121C2×C32×C4—21no—
122C2×C32×C4C22×D311yescharacteristic
123C2×C3⋊Dic3—31no—
124C2×C3⋊Dic3—61no—
125C4×C3⋊D3—32no—
126C4×C3⋊D3—32no—
127C2×C3×Dic3—21no—
128C3×D12—32no—
129C3×D12—61no—
130C3×Dic6—21no—
131C3×Dic6C4×D312yes—
132#626—64no—
133D3×Dic3—64no—
134C22×D9—21no—
135C22×D9—31no—
136C22×D9—61no—
137C3×C3⋊C8—21no—
138C3×C3⋊C8—61no—
139Q8×C9—32no—
140Q8×C9—61no—
141C2×C4×C9—22no—
142C2×C4×C9—21no—
143C2×C4×C9—21no—
144C2×C4×C9—61no—
145C2×C4×C9C22×D311yescharacteristic
146C2×Dic9—31no—
147C2×Dic9—61no—
148C2×Dic9—62no—
149C2×Dic9—61no—
150D36—21no—
151D36—64no—
152D36C4×D312yes—
153C4×D9—32no—
154C4×D9—32no—
155C4×D9—62no—
156C4×D9—62no—
157C4×D9—64no—
158C8×C9—61no—
159C9⋊C8—21no—
160#416—271noSylow
161C2×C3×C9C2×C22⋊C411yescharacteristic
162C2×C3×C9#13511yescharacteristic
163C2×C3×C9#13511yescharacteristic
164C3⋊D9—22no—
165C3×D9—42no—
166C2×C3×Q8—31no—
167C2×C3×D4—91no—
168C2×#67—181no—
169C22×Dic3—181no—
170#239—94no—
171#239—182no—
172#237—94no—
173#237—182no—
174C2×D12—31no—
175C2×D12—181no—
176C2×C4×D3—91no—
177C2×C4×D3—91no—
178C2×C4×D3—181no—
179C2×C4×D3—181no—
180C2×Dic6C2×D911yescharacteristic
181C2×C3×C8—31no—
182C2×C3×C8—91no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4C2×D911yescharacteristic
185C3×C4⋊C4D3211yescharacteristic
186(C2×D3)⋊C4—181no—
187(C2×D3)⋊C4—181no—
188(C2×D3)⋊C4—182no—
189C4⋊Dic3—91no—
190C4⋊Dic3—91no—
191C4⋊Dic3—181no—
192C4×Dic3—181no—
193C4×Dic3—181no—
194C4×Dic3—182no—
195C2×C3⋊C8—31no—
196C2×C3⋊C8C2×D911yescharacteristic
197D3×C8—182no—
198D3×C8—182no—
199C22×C32D3×D411yescharacteristic
200C2×C3⋊D3—32no—
201C2×C3⋊D3—61no—
202C2×C3⋊D3—61no—
203C2×C3×D3—62no—
204C2×C3×D3—121no—
205C2×C3×D3—121no—
206C32×C4—41no—
207C32×C4C2×C4×D311yescharacteristic
208C32×C4D3×D411yescharacteristic
209C3⋊Dic3—32no—
210C3⋊Dic3—121no—
211C3×Dic3—22no—
212C22×C9—21no—
213C22×C9D3×D411yescharacteristic
214C2×D9—22no—
215C2×D9—32no—
216C2×D9—41no—
217C2×D9—41no—
218C2×D9—62no—
219C2×D9—61no—
220C2×D9—61no—
221C2×D9—121no—
222C2×D9—121no—
223C4×C9—21no—
224C4×C9—21no—
225C4×C9—42no—
226C4×C9—41no—
227C4×C9—62no—
228C4×C9C2×C4×D311yescharacteristic
229C4×C9D3×D411yescharacteristic
230Dic9—32no—
231Dic9—62no—
232Dic9—122no—
233Dic9—121no—
234C2×#41—271no—
235C22×C8—271no—
236#117—271no—
237#103—31no—
238#103—271no—
239#102—91no—
240#102—271no—
241C3×C9#41611yescharacteristic, complemented, Sylow
242C23×C3—181no—
243C22×D3—61no—
244C22×D3—91no—
245C22×D3—91no—
246C22×D3—181no—
247C3×Q8—32no—
248C3×Q8—61no—
249C3×D4—92no—
250C3×D4—181no—
251C2×C3×C4—22no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—61no—
256C2×C3×C4C22×D911yescharacteristic
257C2×C3×C4C2×D3211yescharacteristic
258#67—184no—
259C2×Dic3—21no—
260C2×Dic3—91no—
261C2×Dic3—91no—
262C2×Dic3—181no—
263C2×Dic3—184no—
264C2×Dic3—182no—
265C2×Dic3—181no—
266C2×Dic3—181no—
267D12—32no—
268D12—61no—
269D12—184no—
270C4×D3—92no—
271C4×D3—92no—
272C4×D3—92no—
273C4×D3—92no—
274C4×D3—182no—
275C4×D3—182no—
276C4×D3—184no—
277Dic6—21no—
278Dic6C4×D912yes—
279C3×C8—61no—
280C3×C8—181no—
281C3⋊C8—21no—
282C3⋊C8—61no—
283C2×C32D3×C22⋊C411yescharacteristic
284C2×C32#91611yescharacteristic
285C2×C32#91911yescharacteristic
286C3⋊D3—62no—
287C3×D3—122no—
288C2×C9—21no—
289C2×C9—21no—
290C2×C9—21no—
291C2×C9D3×C22⋊C411yescharacteristic
292C2×C9#91211yescharacteristic
293C2×C9#91611yescharacteristic
294D9—42no—
295D9—62no—
296D9—122no—
297#41—274no—
298#41—542no—
299C2×Q8—31no—
300C2×D4—91no—
301C2×D4—541no—
302C22×C4—271no—
303C22×C4—541no—
304C2×C8—31no—
305C2×C8—91no—
306C2×C8—542no—
307C4⋊C4—271no—
308C4⋊C4D3×D911yescharacteristic
309C22⋊C4—541no—
310C42—541no—
311C22×C3—21no—
312C22×C3—182no—
313C22×C3—361no—
314C22×C3—361no—
315C22×C3D4×D911yescharacteristic
316C22×C3#381111yescharacteristic
317C2×D3—62no—
318C2×D3—92no—
319C2×D3—92no—
320C2×D3—121no—
321C2×D3—121no—
322C2×D3—182no—
323C2×D3—181no—
324C2×D3—181no—
325C2×D3—181no—
326C2×D3—181no—
327C2×D3—361no—
328C2×D3—361no—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—42no—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4—62no—
335C3×C4C2×C4×D911yescharacteristic
336C3×C4D4×D911yescharacteristic
337C3×C4C4×D3211yescharacteristic
338C3×C4#380211yescharacteristic
339Dic3—22no—
340Dic3—92no—
341Dic3—92no—
342Dic3—182no—
343Dic3—362no—
344Dic3—361no—
345Dic3—361no—
346C32#480411yescharacteristic
347C9—21no—
348C9#477211yescharacteristic, complemented
349C23—181no—
350C23—271no—
351Q8—32no—
352Q8—61no—
353D4—92no—
354D4—181no—
355D4—544no—
356C2×C4—21no—
357C2×C4—61no—
358C2×C4—271no—
359C2×C4—272no—
360C2×C4—272no—
361C2×C4—544no—
362C2×C4—541no—
363C2×C4C2×D3×D911yescharacteristic
364C8—61no—
365C8—181no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—362no—
370C2×C3C22⋊C4×D911yescharacteristic
371C2×C3#6347811yescharacteristic
372C2×C3#6348111yescharacteristic
373C2×C3#6392111yescharacteristic
374C2×C3#6393611yescharacteristic
375C2×C3#6395011yescharacteristic
376D3—122no—
377D3—182no—
378D3—182no—
379D3—362no—
380C22—182no—
381C22—272no—
382C22—361no—
383C22—361no—
384C22—541no—
385C22—541no—
386C22#8866811yescharacteristic
387C4—41no—
388C4—62no—
389C4—272no—
390C4—1081no—
391C4C4×D3×D911yescharacteristic
392C4#8864611yescharacteristic
393C3—21no—
394C3#1058848411yescharacteristic, complemented
395C3#1059061111yescharacteristic
396C2—362no—
397C2—542no—
398C2#1172104811yescharacteristic
399C2#1172118611yescharacteristic
400C2#1172122411yescharacteristic
subgroupG/H
centerC22#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×#88634C2
cosocleC2×C3×C4×C9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#416
LCSG ▷ 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
2. #11721748C2semidirect
3. C2×(C3⋊D9)⋊C8C2semidirect
4. C9×#810C2semidirect
5. C3×#63369C2semidirect
6. #11720298C2semidirect
7. #11720283C2semidirect
16. #88634C4semidirect
38. #63369D3semidirect
117. #810D9semidirect
241. C3×C9#416semidirect
348. C9#4772semidirect
394. C3#10588484semidirect

Maximal subgroups

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

subgroupindex
C2×#886342
#117217482
C2×(C3⋊D9)⋊C82
C9×#8102
C3×#633692
#117202982
#117202832
#105906113
#105884843