Finite Group Databasebeta

← #13137831

Subgroups — #13137831

Order 1728.

698 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 109 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×#88646C211yesmaximal, characteristic
2C4⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
3C2×(C3⋊D9)⋊C8C211yesmaximal, characteristic, complemented
4C9×#809C211yesmaximal, characteristic, complemented
5C3×#63369C211yesmaximal, characteristic, complemented
6#11720295C211yesmaximal, characteristic, complemented
7#11720283C211yesmaximal, characteristic, complemented
8#10590593—31nomaximal
9D9×#102—31nomaximal
10C22×D3×D9—21no—
11C2×C4×C3⋊D9—21no—
12C2×C4×C3⋊D9C2211yescharacteristic
13C2×C9×D12C2211yescharacteristic
14C2×C3×D36C2211yescharacteristic
15C2×#6194—21no—
16#88646—22no—
17#88646C414yescomplemented
18C3×C9×C4⋊C4C2211yescharacteristic, complemented
19C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
20#88534—21no—
21C2×C9×C3⋊C8C2211yescharacteristic, complemented
22C2×C3×C9⋊C8C2211yescharacteristic, complemented
23(C3⋊D9)⋊C8—22no—
24C2×#3802—31no—
25C4⋊C4×C3⋊D3—31no—
26C2×#3710—31no—
27C2×D4×D9—31no—
28C3×#809—31no—
29C3×#809D311yescharacteristic
30#63563—31no—
31#63563—31no—
32C2×C8×D9—31no—
33C4⋊C4×D9—31no—
34C4⋊C4×D9—61no—
35C9×#102—31no—
36#63392—31no—
37#63380—31no—
38#63369D311yescharacteristic, complemented
39C22×C3⋊D9D411yescharacteristic
40C22×D3×C9—21no—
41C22×C3×D9—21no—
42C2×D3×D9—24no—
43C2×D3×D9—42no—
44C2×D3×D9—42no—
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9C2311yescharacteristic
47C2×C3⋊Dic9—21no—
48C2×C3⋊Dic9D411yescharacteristic
49C4×C3⋊D9—42no—
50C4×C3⋊D9C2×C412yes—
51C4×C3⋊D9D412yes—
52C9×D12—21no—
53C9×D12C2×C412yescomplemented
54C3×D36—21no—
55C3×D36C2×C412yescomplemented
56#6194—24no—
57C9×C3⋊C8—21no—
58C3×C9⋊C8—21no—
59D3×#102—91no—
60D3×#102—91no—
61C22×D32—61no—
62C2×C4×C3⋊D3—31no—
63C2×C4×C3⋊D3—61no—
64C2×C3×D12—31no—
65C2×C3×D12C2×D311yescharacteristic
66C2×#625—61no—
67#3802—34no—
68#3802—62no—
69C23×D9—61no—
70C32×C4⋊C4C2×D311yescharacteristic
71C4⋊(C3⋊Dic3)—31no—
72#3750—61no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8C2×D311yescharacteristic
75#3710—62no—
76C2×D4×C9—31no—
77C2×#611—61no—
78D4×D9—34no—
79D4×D9—62no—
80C2×D36C2×D311yescharacteristic
81C2×C4×D9—31no—
82C2×C4×D9—61no—
83C2×C4×D9—61no—
84C2×C4×D9—62no—
85C2×C8×C9—31no—
86C9×C4⋊C4—21no—
87C9×C4⋊C4C2×D311yescharacteristic, complemented
88C4⋊Dic9—31no—
89C4⋊Dic9—61no—
90Dic9⋊C4—61no—
91Dic9⋊C4—62no—
92C2×C9⋊C8C2×D311yescharacteristic, complemented
93C8×D9—62no—
94C22×C3×C9C2×D411yescharacteristic
95C2×C3⋊D9C22⋊C412yes—
96C2×C3⋊D9D812yes—
97C2×C3⋊D9SD1612yes—
98C2×D3×C9—22no—
99C2×D3×C9—41no—
100C2×D3×C9—41no—
101C2×C3×D9—22no—
102C2×C3×D9—41no—
103C2×C3×D9—41no—
104D3×D9—48no—
105C3×C4×C9—41no—
106C3×C4×C9C22×C411yescharacteristic
107C3×C4×C9C2×D411yescharacteristic
108C3⋊Dic9—41no—
109C3⋊Dic9C22⋊C412yes—
110C2×D3×D4—91no—
111C2×D3×D4—91no—
112C2×D3×C8—91no—
113C2×D3×C8—91no—
114D3×C4⋊C4—91no—
115D3×C4⋊C4—91no—
116D3×C4⋊C4—181no—
117C3×#102—31no—
118C3×#102—91no—
119#832—91no—
120#832—91no—
121#821—91no—
122#821—91no—
123#809—31no—
124#809D911yescharacteristic, complemented
125C22×C3⋊D3—31no—
126C22×C3×D3—21no—
127C22×C3×D3—61no—
128C2×D32—64no—
129C2×D32—122no—
130C2×D32—122no—
131C2×C32×C4—21no—
132C2×C32×C4C22×D311yescharacteristic
133C2×C3⋊Dic3—31no—
134C2×C3⋊Dic3—61no—
135C4×C3⋊D3—32no—
136C4×C3⋊D3—32no—
137C4×C3⋊D3—122no—
138C3×D12—21no—
139C3×D12—32no—
140C3×D12—61no—
141C3×D12C4×D312yes—
142#625—64no—
143C23×C9—61no—
144C22×D9—21no—
145C22×D9—31no—
146C22×D9—61no—
147C22×D9—64no—
148C22×D9—122no—
149C22×D9—122no—
150C3×C3⋊C8—21no—
151C3×C3⋊C8—61no—
152D4×C9—32no—
153D4×C9—61no—
154C2×C4×C9—22no—
155C2×C4×C9—21no—
156C2×C4×C9—21no—
157C2×C4×C9C22×D311yescharacteristic
158#611—64no—
159C2×Dic9—31no—
160C2×Dic9—61no—
161C2×Dic9—62no—
162C2×Dic9—61no—
163D36—21no—
164D36C4×D312yescomplemented
165C4×D9—32no—
166C4×D9—32no—
167C4×D9—62no—
168C4×D9—62no—
169C4×D9—122no—
170C4×D9—124no—
171C8×C9—61no—
172C9⋊C8—21no—
173C2×#102—271noSylow
174C2×C3×C9C2×C22⋊C411yescharacteristic
175C2×C3×C9C2×D811yescharacteristic
176C2×C3×C9C2×SD1611yescharacteristic
177C3⋊D9#10214yescomplemented
178D3×C9—42no—
179C3×D9—42no—
180C23×D3—181no—
181C23×D3—181no—
182C2×C3×D4—31no—
183C2×C3×D4—91no—
184C2×#67—181no—
185C2×#67—181no—
186D3×D4—94no—
187D3×D4—94no—
188D3×D4—182no—
189D3×D4—182no—
190C2×D12—31no—
191C2×D12C2×D911yescharacteristic
192C2×C4×D3—91no—
193C2×C4×D3—91no—
194C2×C4×D3—181no—
195C2×C4×D3—181no—
196C2×C4×D3—181no—
197C2×C4×D3—182no—
198C2×C3×C8—31no—
199C2×C3×C8—91no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4C2×D911yescharacteristic, complemented
202C3×C4⋊C4D3211yescharacteristic
203C4⋊Dic3—91no—
204C4⋊Dic3—91no—
205C4⋊Dic3—181no—
206Dic3⋊C4—181no—
207Dic3⋊C4—181no—
208Dic3⋊C4—182no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8C2×D911yescharacteristic, complemented
211D3×C8—182no—
212D3×C8—182no—
213C22×C32D3×D411yescharacteristic
214C2×C3⋊D3—32no—
215C2×C3⋊D3—32no—
216C2×C3⋊D3—32no—
217C2×C3×D3—22no—
218C2×C3×D3—41no—
219C2×C3×D3—41no—
220C2×C3×D3—62no—
221C2×C3×D3—121no—
222C2×C3×D3—121no—
223D32—128no—
224C32×C4—41no—
225C32×C4C2×C4×D311yescharacteristic
226C32×C4D3×D411yescharacteristic
227C3⋊Dic3—32no—
228C3⋊Dic3—121no—
229C22×C9—21no—
230C22×C9—62no—
231C22×C9—121no—
232C22×C9—121no—
233C22×C9D3×D411yescharacteristic
234C2×D9—22no—
235C2×D9—32no—
236C2×D9—32no—
237C2×D9—32no—
238C2×D9—41no—
239C2×D9—41no—
240C2×D9—62no—
241C2×D9—62no—
242C2×D9—62no—
243C2×D9—128no—
244C4×C9—21no—
245C4×C9—21no—
246C4×C9—42no—
247C4×C9—41no—
248C4×C9C2×C4×D311yescharacteristic
249C4×C9D3×D411yescharacteristic
250Dic9—32no—
251Dic9—62no—
252Dic9—122no—
253Dic9—121no—
254C22×D4—271no—
255C22×C8—271no—
256C2×C4⋊C4—271no—
257#102—31no—
258#102—91no—
259#102—271no—
260#102—271no—
261C3×C9C2×#10211yescharacteristic, complemented, Sylow
262C23×C3—61no—
263C23×C3—181no—
264C22×D3—21no—
265C22×D3—61no—
266C22×D3—91no—
267C22×D3—91no—
268C22×D3—181no—
269C22×D3—184no—
270C22×D3—184no—
271C22×D3—362no—
272C22×D3—362no—
273C22×D3—362no—
274C22×D3—362no—
275C3×D4—32no—
276C3×D4—61no—
277C3×D4—92no—
278C3×D4—181no—
279C2×C3×C4—22no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4C22×D911yescharacteristic
284C2×C3×C4C2×D3211yescharacteristic
285#67—184no—
286#67—184no—
287C2×Dic3—91no—
288C2×Dic3—91no—
289C2×Dic3—181no—
290C2×Dic3—182no—
291C2×Dic3—181no—
292C2×Dic3—181no—
293D12—21no—
294D12—32no—
295D12—61no—
296D12C4×D912yescomplemented
297C4×D3—92no—
298C4×D3—92no—
299C4×D3—92no—
300C4×D3—92no—
301C4×D3—182no—
302C4×D3—182no—
303C4×D3—362no—
304C4×D3—362no—
305C4×D3—364no—
306C3×C8—61no—
307C3×C8—181no—
308C3⋊C8—21no—
309C3⋊C8—61no—
310C2×C32D3×C22⋊C411yescharacteristic
311C2×C32D3×D811yescharacteristic
312C2×C32D3×SD1611yescharacteristic
313C3⋊D3—34no—
314C3×D3—42no—
315C3×D3—122no—
316C2×C9—21no—
317C2×C9—21no—
318C2×C9—21no—
319C2×C9—122no—
320C2×C9D3×C22⋊C411yescharacteristic
321C2×C9D3×D811yescharacteristic
322C2×C9D3×SD1611yescharacteristic
323D9—34no—
324D9—42no—
325D9—64no—
326C24—541no—
327C2×D4—31no—
328C2×D4—91no—
329C2×D4—274no—
330C2×D4—542no—
331C2×D4—541no—
332C22×C4—271no—
333C22×C4—541no—
334C2×C8—31no—
335C2×C8—91no—
336C2×C8—542no—
337C4⋊C4—271no—
338C4⋊C4—541no—
339C4⋊C4D3×D911yescharacteristic, complemented
340C22×C3—21no—
341C22×C3—62no—
342C22×C3—121no—
343C22×C3—121no—
344C22×C3—182no—
345C22×C3—361no—
346C22×C3—361no—
347C22×C3D4×D911yescharacteristic
348C22×C3#381111yescharacteristic
349C2×D3—22no—
350C2×D3—41no—
351C2×D3—41no—
352C2×D3—62no—
353C2×D3—92no—
354C2×D3—92no—
355C2×D3—92no—
356C2×D3—92no—
357C2×D3—92no—
358C2×D3—92no—
359C2×D3—121no—
360C2×D3—121no—
361C2×D3—182no—
362C2×D3—182no—
363C2×D3—182no—
364C2×D3—368no—
365C2×D3—368no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—42no—
369C3×C4—41no—
370C3×C4—41no—
371C3×C4C2×C4×D911yescharacteristic
372C3×C4D4×D911yescharacteristic
373C3×C4C4×D3211yescharacteristic
374C3×C4#380211yescharacteristic
375Dic3—92no—
376Dic3—92no—
377Dic3—182no—
378Dic3—362no—
379Dic3—361no—
380Dic3—361no—
381C32D3×#10211yescharacteristic
382C9—21no—
383C9D3×#10211yescharacteristic, complemented
384C23—61no—
385C23—181no—
386C23—271no—
387C23—544no—
388C23—1082no—
389C23—1082no—
390D4—32no—
391D4—61no—
392D4—92no—
393D4—181no—
394D4—544no—
395C2×C4—21no—
396C2×C4—271no—
397C2×C4—272no—
398C2×C4—272no—
399C2×C4—541no—
400C2×C4—1082no—
401C2×C4C2×D3×D911yescharacteristic
402C8—61no—
403C8—181no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—122no—
408C2×C3—362no—
409C2×C3C22⋊C4×D911yescharacteristic
410C2×C3D8×D911yescharacteristic
411C2×C3SD16×D911yescharacteristic
412C2×C3#6392111yescharacteristic
413C2×C3#6392611yescharacteristic
414C2×C3#6394011yescharacteristic
415D3—42no—
416D3—94no—
417D3—94no—
418D3—122no—
419D3—184no—
420C22—62no—
421C22—121no—
422C22—121no—
423C22—182no—
424C22—272no—
425C22—272no—
426C22—272no—
427C22—361no—
428C22—361no—
429C22—1088no—
430C22#8866811yescharacteristic
431C4—41no—
432C4—272no—
433C4—1081no—
434C4C4×D3×D911yescharacteristic
435C4#8864611yescharacteristic
436C3—21no—
437C3D9×#10211yescharacteristic, complemented
438C3#1059059311yescharacteristic
439C2—122no—
440C2—274no—
441C2—362no—
442C2#1172104811yescharacteristic
443C2#1172117411yescharacteristic
444C2#1172121211yescharacteristic
subgroupG/H
centerC22#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×#88646C2
cosocleC2×C3×C4×C9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9C2×#102
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. C4⋊C4×C3⋊D9C2semidirect
3. C2×(C3⋊D9)⋊C8C2semidirect
4. C9×#809C2semidirect
5. C3×#63369C2semidirect
6. #11720295C2semidirect
7. #11720283C2semidirect
17. #88646C4semidirect
18. C3×C9×C4⋊C4C22semidirect
19. C4⋊(C3⋊Dic9)C22semidirect
21. C2×C9×C3⋊C8C22semidirect
22. C2×C3×C9⋊C8C22semidirect
38. #63369D3semidirect
53. C9×D12C2×C4semidirect
55. C3×D36C2×C4semidirect
87. C9×C4⋊C4C2×D3semidirect
92. C2×C9⋊C8C2×D3semidirect
124. #809D9semidirect
164. D36C4×D3semidirect
177. C3⋊D9#102semidirect
201. C3×C4⋊C4C2×D9semidirect
210. C2×C3⋊C8C2×D9semidirect
261. C3×C9C2×#102semidirect
296. D12C4×D9semidirect
339. C4⋊C4D3×D9semidirect
383. C9D3×#102semidirect
437. C3D9×#102semidirect

Maximal subgroups

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

subgroupindex
C2×#886462
C4⋊C4×C3⋊D92
C2×(C3⋊D9)⋊C82
C9×#8092
C3×#633692
#117202952
#117202832
#105905933
D9×#1023