Finite Group Databasebeta

← #13138671

Subgroups — #13138671

Order 1728.

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

normal subgroups only

This group has 682 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
2C2×C8×C3⋊D9C211yesmaximal, characteristic, complemented
3#11721000C211yesmaximal, characteristic, complemented
4C9×#821C211yesmaximal, characteristic, complemented
5C3×#63380C211yesmaximal, characteristic, complemented
6#11720294C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic, complemented
8#10590940—31nomaximal
9D9×#102—31nomaximal
10C22×D3×D9—21no—
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×C3×D36C2211yescharacteristic
14C2×#6194—21no—
15C2×(C3⋊D9)⋊C4—21no—
16#88646—22no—
17#88646C414yescomplemented
18C2×C3×C8×C9C2211yescharacteristic, complemented
19C2×C3⋊(C9⋊C8)C2211yescharacteristic, complemented
20C8×C3⋊D9—22no—
21C9×C4⋊Dic3C2211yescharacteristic, complemented
22C3×C4⋊Dic9C2211yescharacteristic, complemented
23#88442—21no—
24C2×#3802—31no—
25C2×C8×C3⋊D3—31no—
26#63900—31no—
27C2×D4×D9—31no—
28C3×#821—31no—
29C3×#821D311yescharacteristic
30#63562—31no—
31#63562—31no—
32C2×C8×D9—31no—
33C2×C8×D9—61no—
34C4⋊C4×D9—31no—
35C9×#102—31no—
36#63392—31no—
37#63380D311yescharacteristic, complemented
38#63369—31no—
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×C9C2311yescharacteristic
46C2×C3⋊Dic9D411yescharacteristic
47C4×C3⋊D9C2×C412yes—
48C4×C3⋊D9D412yes—
49C2×C9×Dic3—21no—
50C2×C3×Dic9—21no—
51C9×D12—21no—
52C9×D12C2×C412yescomplemented
53C3×D36—21no—
54C3×D36C2×C412yescomplemented
55#6194—24no—
56(C3⋊D9)⋊C4—42no—
57C3×C8×C9—21no—
58C3⋊(C9⋊C8)—21no—
59D3×#102—91no—
60D3×#102—91no—
61C22×D32—61no—
62C2×C4×C3⋊D3—31no—
63C2×C3×D12—31no—
64C2×C3×D12C2×D311yescharacteristic
65C2×#625—61no—
66C2×#624—61no—
67#3802—34no—
68#3802—62no—
69C23×D9—61no—
70C2×C32×C8C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)—31no—
72C8×C3⋊D3—62no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3C2×D311yescharacteristic
75#3724—61no—
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×C8×C9—21no—
85C2×C8×C9C2×D311yescharacteristic, complemented
86C9×C4⋊C4—31no—
87C4⋊Dic9C2×D311yescharacteristic, complemented
88Dic9⋊C4—61no—
89C2×C9⋊C8—31no—
90C2×C9⋊C8—61no—
91C8×D9—62no—
92C8×D9—64no—
93C22×C3×C9C2×D411yescharacteristic
94C2×C3⋊D9C22⋊C412yes—
95C2×C3⋊D9D812yes—
96C2×C3⋊D9SD1612yes—
97C2×D3×C9—22no—
98C2×D3×C9—41no—
99C2×D3×C9—41no—
100C2×C3×D9—22no—
101C2×C3×D9—41no—
102C2×C3×D9—41no—
103D3×D9—48no—
104C3×C4×C9C22×C411yescharacteristic
105C3×C4×C9C2×D411yescharacteristic
106C3⋊Dic9C22⋊C412yes—
107C9×Dic3—41no—
108C3×Dic9—41no—
109C2×D3×D4—91no—
110C2×D3×D4—91no—
111C2×D3×C8—91no—
112C2×D3×C8—91no—
113C2×D3×C8—181no—
114D3×C4⋊C4—91no—
115D3×C4⋊C4—91no—
116C3×#102—31no—
117C3×#102—91no—
118#832—91no—
119#832—91no—
120#821—31no—
121#821D911yescharacteristic, complemented
122#809—91no—
123#809—91no—
124C22×C3⋊D3—31no—
125C22×C3×D3—21no—
126C22×C3×D3—61no—
127C2×D32—64no—
128C2×D32—122no—
129C2×D32—122no—
130C2×C32×C4C22×D311yescharacteristic
131C2×C3⋊Dic3—31no—
132C4×C3⋊D3—32no—
133C4×C3⋊D3—32no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3—61no—
136C3×D12—21no—
137C3×D12—32no—
138C3×D12—61no—
139C3×D12C4×D312yes—
140#625—64no—
141#624—122no—
142C23×C9—61no—
143C22×D9—21no—
144C22×D9—31no—
145C22×D9—61no—
146C22×D9—64no—
147C22×D9—122no—
148C22×D9—122no—
149C32×C8—21no—
150C3⋊(C3⋊C8)—61no—
151D4×C9—32no—
152D4×C9—61no—
153C2×C4×C9—21no—
154C2×C4×C9—61no—
155C2×C4×C9C22×D311yescharacteristic
156#611—64no—
157C2×Dic9—21no—
158C2×Dic9—31no—
159C2×Dic9—61no—
160D36—21no—
161D36C4×D312yescomplemented
162C4×D9—32no—
163C4×D9—32no—
164C4×D9—62no—
165C4×D9—62no—
166C4×D9—122no—
167C8×C9—22no—
168C8×C9—21no—
169C9⋊C8—62no—
170C9⋊C8—61no—
171C2×#102—271noSylow
172C2×C3×C9C2×C22⋊C411yescharacteristic
173C2×C3×C9C2×D811yescharacteristic
174C2×C3×C9C2×SD1611yescharacteristic
175C3⋊D9#10214yescomplemented
176D3×C9—42no—
177C3×D9—42no—
178C23×D3—181no—
179C23×D3—181no—
180C2×C3×D4—31no—
181C2×C3×D4—91no—
182C2×#67—181no—
183C2×#67—181no—
184D3×D4—94no—
185D3×D4—94no—
186D3×D4—182no—
187D3×D4—182no—
188C2×D12—31no—
189C2×D12C2×D911yescharacteristic
190C2×C4×D3—91no—
191C2×C4×D3—91no—
192C2×C4×D3—181no—
193C2×C4×D3—181no—
194C2×C4×D3—181no—
195C2×C3×C8—21no—
196C2×C3×C8C2×D911yescharacteristic, complemented
197C2×C3×C8D3211yescharacteristic
198C3×C4⋊C4—31no—
199C3×C4⋊C4—91no—
200C4⋊Dic3—31no—
201C4⋊Dic3C2×D911yescharacteristic, complemented
202Dic3⋊C4—181no—
203Dic3⋊C4—181no—
204C2×C3⋊C8—91no—
205C2×C3⋊C8—91no—
206C2×C3⋊C8—181no—
207D3×C8—182no—
208D3×C8—182no—
209D3×C8—184no—
210C22×C32D3×D411yescharacteristic
211C2×C3⋊D3—32no—
212C2×C3⋊D3—32no—
213C2×C3⋊D3—32no—
214C2×C3×D3—22no—
215C2×C3×D3—41no—
216C2×C3×D3—41no—
217C2×C3×D3—62no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220D32—128no—
221C32×C4C2×C4×D311yescharacteristic
222C32×C4D3×D411yescharacteristic
223C3⋊Dic3—32no—
224C3×Dic3—41no—
225C3×Dic3—121no—
226C22×C9—21no—
227C22×C9—62no—
228C22×C9—121no—
229C22×C9—121no—
230C22×C9D3×D411yescharacteristic
231C2×D9—22no—
232C2×D9—32no—
233C2×D9—32no—
234C2×D9—32no—
235C2×D9—41no—
236C2×D9—41no—
237C2×D9—62no—
238C2×D9—62no—
239C2×D9—62no—
240C2×D9—128no—
241C4×C9—21no—
242C4×C9—21no—
243C4×C9—121no—
244C4×C9C2×C4×D311yescharacteristic
245C4×C9D3×D411yescharacteristic
246Dic9—32no—
247Dic9—41no—
248Dic9—62no—
249C22×D4—271no—
250C22×C8—271no—
251C2×C4⋊C4—271no—
252#102—31no—
253#102—91no—
254#102—271no—
255#102—271no—
256C3×C9C2×#10211yescharacteristic, complemented, Sylow
257C23×C3—61no—
258C23×C3—181no—
259C22×D3—21no—
260C22×D3—61no—
261C22×D3—91no—
262C22×D3—91no—
263C22×D3—181no—
264C22×D3—184no—
265C22×D3—184no—
266C22×D3—362no—
267C22×D3—362no—
268C22×D3—362no—
269C22×D3—362no—
270C3×D4—32no—
271C3×D4—61no—
272C3×D4—92no—
273C3×D4—181no—
274C2×C3×C4—21no—
275C2×C3×C4—61no—
276C2×C3×C4—181no—
277C2×C3×C4C22×D911yescharacteristic
278C2×C3×C4C2×D3211yescharacteristic
279#67—184no—
280#67—184no—
281C2×Dic3—21no—
282C2×Dic3—61no—
283C2×Dic3—91no—
284C2×Dic3—91no—
285C2×Dic3—181no—
286D12—21no—
287D12—32no—
288D12—61no—
289D12C4×D912yescomplemented
290C4×D3—92no—
291C4×D3—92no—
292C4×D3—92no—
293C4×D3—92no—
294C4×D3—182no—
295C4×D3—182no—
296C4×D3—362no—
297C4×D3—362no—
298C3×C8—22no—
299C3×C8—21no—
300C3×C8—21no—
301C3⋊C8—182no—
302C3⋊C8—181no—
303C3⋊C8—181no—
304C2×C32D3×C22⋊C411yescharacteristic
305C2×C32D3×D811yescharacteristic
306C2×C32D3×SD1611yescharacteristic
307C3⋊D3—34no—
308C3×D3—42no—
309C3×D3—122no—
310C2×C9—21no—
311C2×C9—21no—
312C2×C9—21no—
313C2×C9—122no—
314C2×C9D3×C22⋊C411yescharacteristic
315C2×C9D3×D811yescharacteristic
316C2×C9D3×SD1611yescharacteristic
317D9—34no—
318D9—42no—
319D9—64no—
320C24—541no—
321C2×D4—31no—
322C2×D4—91no—
323C2×D4—274no—
324C2×D4—542no—
325C2×D4—541no—
326C22×C4—271no—
327C22×C4—541no—
328C2×C8—271no—
329C2×C8—542no—
330C2×C8D3×D911yescharacteristic, complemented
331C4⋊C4—31no—
332C4⋊C4—91no—
333C4⋊C4—541no—
334C22×C3—21no—
335C22×C3—62no—
336C22×C3—121no—
337C22×C3—121no—
338C22×C3—182no—
339C22×C3—361no—
340C22×C3—361no—
341C22×C3D4×D911yescharacteristic
342C22×C3#380211yescharacteristic
343C2×D3—22no—
344C2×D3—41no—
345C2×D3—41no—
346C2×D3—62no—
347C2×D3—92no—
348C2×D3—92no—
349C2×D3—92no—
350C2×D3—92no—
351C2×D3—92no—
352C2×D3—92no—
353C2×D3—121no—
354C2×D3—121no—
355C2×D3—182no—
356C2×D3—182no—
357C2×D3—182no—
358C2×D3—368no—
359C2×D3—368no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—121no—
363C3×C4—361no—
364C3×C4C2×C4×D911yescharacteristic
365C3×C4D4×D911yescharacteristic
366C3×C4C4×D3211yescharacteristic
367C3×C4#381111yescharacteristic
368Dic3—41no—
369Dic3—92no—
370Dic3—92no—
371Dic3—121no—
372Dic3—182no—
373C32D3×#10211yescharacteristic
374C9—21no—
375C9D3×#10211yescharacteristic, complemented
376C23—61no—
377C23—181no—
378C23—271no—
379C23—544no—
380C23—1082no—
381C23—1082no—
382D4—32no—
383D4—61no—
384D4—92no—
385D4—181no—
386D4—544no—
387C2×C4—61no—
388C2×C4—181no—
389C2×C4—271no—
390C2×C4—272no—
391C2×C4—272no—
392C2×C4—1082no—
393C2×C4C2×D3×D911yescharacteristic
394C8—21no—
395C8—541no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—122no—
400C2×C3—362no—
401C2×C3C22⋊C4×D911yescharacteristic
402C2×C3D8×D911yescharacteristic
403C2×C3SD16×D911yescharacteristic
404C2×C3#6379611yescharacteristic
405C2×C3#6379711yescharacteristic
406C2×C3#6392111yescharacteristic
407D3—42no—
408D3—94no—
409D3—94no—
410D3—122no—
411D3—184no—
412C22—62no—
413C22—121no—
414C22—121no—
415C22—182no—
416C22—272no—
417C22—272no—
418C22—272no—
419C22—361no—
420C22—361no—
421C22—1088no—
422C22#8864611yescharacteristic
423C4—121no—
424C4—272no—
425C4—361no—
426C4C4×D3×D911yescharacteristic
427C4#8866811yescharacteristic
428C3—21no—
429C3D9×#10211yescharacteristic, complemented
430C3#1059094011yescharacteristic
431C2—122no—
432C2—274no—
433C2—362no—
434C2#1172081711yescharacteristic
435C2#1172081811yescharacteristic
436C2#1172104811yescharacteristic
subgroupG/H
centerC22#88646
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×#88646C2
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9C2×#102
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×C8×C3⋊D9C2semidirect
3. #11721000C2semidirect
4. C9×#821C2semidirect
5. C3×#63380C2semidirect
6. #11720294C2semidirect
7. #11720266C2semidirect
17. #88646C4semidirect
18. C2×C3×C8×C9C22semidirect
19. C2×C3⋊(C9⋊C8)C22semidirect
21. C9×C4⋊Dic3C22semidirect
22. C3×C4⋊Dic9C22semidirect
37. #63380D3semidirect
52. C9×D12C2×C4semidirect
54. C3×D36C2×C4semidirect
85. C2×C8×C9C2×D3semidirect
87. C4⋊Dic9C2×D3semidirect
121. #821D9semidirect
161. D36C4×D3semidirect
175. C3⋊D9#102semidirect
196. C2×C3×C8C2×D9semidirect
201. C4⋊Dic3C2×D9semidirect
256. C3×C9C2×#102semidirect
289. D12C4×D9semidirect
330. C2×C8D3×D9semidirect
375. C9D3×#102semidirect
429. C3D9×#102semidirect

Maximal subgroups

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

subgroupindex
C2×#886462
C2×C8×C3⋊D92
#117210002
C9×#8212
C3×#633802
#117202942
#117202662
#105909403
D9×#1023