Finite Group Databasebeta

← #13140365

Subgroups — #13140365

Order 1728.

620 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 93 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×#88742C211yesmaximal, characteristic
2C2×C3×#3672C211yesmaximal, characteristic, complemented
3#11720936C211yesmaximal, characteristic, complemented
4C2×#88418C211yesmaximal, characteristic, complemented
5C9×#832C211yesmaximal, characteristic, complemented
6#11720282C211yesmaximal, characteristic, complemented
7#11720269C211yesmaximal, characteristic, complemented
8#10591466—31nomaximal
9#10588463—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C22×C3⋊Dic9—21no—
13#88742—22no—
14#88742—22no—
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×#6226C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18C3×#3672—22no—
19C9×C4⋊Dic3C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21C2×C3×C9⋊C8C2211yescharacteristic
22#88445—21no—
23Dic3⋊Dic9—21no—
24#88418—22no—
25C2×#3830—31no—
26C2×C3×#214—31no—
27#63871—31no—
28C2×#3719—31no—
29C2×#3699—31no—
30C2×#3699—61no—
31C3×#832D311yescharacteristic
32#63567—31no—
33#63557—31no—
34C2×#3672D311yescharacteristic, complemented
35C2×Dic36—31no—
36#63459—31no—
37C9×#102—31no—
38(C2×D9)⋊C8—31no—
39#63368—31no—
40C23×C3×C9—21no—
41C22×C3⋊D9D411yescharacteristic
42C3×D4×C9—21no—
43C3×D4×C9D412yes—
44C2×C3×C4×C9C2311yescharacteristic
45#6233—24no—
46C2×C3⋊Dic9—24no—
47C2×C3⋊Dic9—21no—
48C2×C3⋊Dic9D411yescharacteristic
49C4×C3⋊D9—21no—
50C4×C3⋊D9—21no—
51#6226—21no—
52#6226D412yes—
53C2×C9×Dic3—21no—
54C2×C3×Dic9—21no—
55C3×Dic18—21no—
56C3×Dic18—21no—
57C9×C3⋊C8—21no—
58C3×C9⋊C8—21no—
59#5173—91no—
60#4783—91no—
61C2×C32×D4C2×D311yescharacteristic
62C2×#638—61no—
63C22×C3⋊Dic3—61no—
64#3830—62no—
65#3830—62no—
66C2×C4×C3⋊D3—31no—
67C2×#634—31no—
68C2×C3×Dic6—31no—
69C3×#214—62no—
70C3×C4⋊Dic3C2×D311yescharacteristic
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8C2×D311yescharacteristic
73Dic3⋊Dic3—61no—
74#3722—61no—
75#3719—62no—
76C2×D4×C9—21no—
77C2×D4×C9C2×D311yescharacteristic
78C2×#611—61no—
79C2×#611—62no—
80C22×Dic9—61no—
81C22×Dic9—62no—
82#3699—62no—
83#3699—64no—
84#3699—64no—
85#3699—62no—
86C2×C4×D9—31no—
87C2×C4×D9—61no—
88C2×Dic18—31no—
89C2×Dic18—61no—
90C2×Dic18C2×D311yescharacteristic
91C2×C8×C9—31no—
92C9×C4⋊C4—31no—
93#3672—22no—
94(C2×D9)⋊C4—61no—
95C4⋊Dic9—61no—
96C2×C9⋊C8C2×D311yescharacteristic
97Dic36—62no—
98C22×C3×C9—22no—
99C22×C3×C9—41no—
100C22×C3×C9—41no—
101C22×C3×C9C2×D411yescharacteristic
102C2×C3⋊D9—21no—
103C2×C3⋊D9—21no—
104C2×C3⋊D9—21no—
105C3×C4×C9C2×D411yescharacteristic
106C3×C4×C9C2×D411yescharacteristic
107C3⋊Dic9—22no—
108C3⋊Dic9—21no—
109C9×Dic3—41no—
110C3×Dic9—41no—
111C2×#237—91no—
112C2×#237—91no—
113C2×#237—181no—
114C2×C3×SD16—91no—
115#946—91no—
116C2×#216—91no—
117C2×#214—31no—
118C2×Dic12—91no—
119#897—91no—
120C3×#102—31no—
121#832D911yescharacteristic, complemented
122(C2×D3)⋊C8—91no—
123(C2×D3)⋊C8—91no—
124#816—91no—
125#810—91no—
126C23×C32—21no—
127C22×C3⋊D3—31no—
128C32×D4—21no—
129C32×D4D1212yes—
130C2×C32×C4C22×D311yescharacteristic
131#638—64no—
132C2×C3⋊Dic3—31no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—61no—
135C4×C3⋊D3—61no—
136C4×C3⋊D3—61no—
137#634—32no—
138#634—61no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—61no—
141C3×Dic6—61no—
142C3×Dic6—61no—
143C23×C9—22no—
144C23×C9—21no—
145C22×D9—31no—
146C22×D9—61no—
147C3×C3⋊C8—21no—
148C3×C3⋊C8—61no—
149D4×C9—22no—
150D4×C9—21no—
151D4×C9—22no—
152D4×C9#6712yes—
153C2×C4×C9—21no—
154C2×C4×C9—61no—
155C2×C4×C9C22×D311yescharacteristic
156#611—68no—
157#611—64no—
158C2×Dic9—21no—
159C2×Dic9—31no—
160C2×Dic9—61no—
161C2×Dic9—62no—
162C2×Dic9—68no—
163C2×Dic9—64no—
164C2×Dic9—61no—
165C4×D9—62no—
166C4×D9—62no—
167C4×D9—61no—
168C4×D9—61no—
169Dic18—21no—
170Dic18—21no—
171Dic18—32no—
172Dic18—62no—
173Dic18—62no—
174Dic18—61no—
175C8×C9—61no—
176C9⋊C8—21no—
177#452—271noSylow
178C2×C3×C9—42no—
179C2×C3×C9#12011yescharacteristic
180C2×C3×C9#13511yescharacteristic
181C2×C3×C9#13711yescharacteristic
182C3⋊D9—41no—
183C2×C3×Q8—91no—
184C2×C3×D4—21no—
185C2×C3×D4C2×D911yescharacteristic
186C2×C3×D4D3211yescharacteristic
187C2×#67—181no—
188C2×#67—181no—
189C2×#67—182no—
190C22×Dic3—181no—
191C22×Dic3—181no—
192C22×Dic3—182no—
193#237—182no—
194#237—182no—
195#237—184no—
196#237—184no—
197#237—182no—
198#237—182no—
199C2×C4×D3—91no—
200C2×C4×D3—91no—
201C2×C4×D3—181no—
202C2×Dic6—31no—
203C2×Dic6—91no—
204C2×Dic6—91no—
205C2×Dic6—181no—
206C3×SD16—182no—
207C2×C3×C8—31no—
208C2×C3×C8—91no—
209C3×C4⋊C4—31no—
210#216—182no—
211#214—62no—
212(C2×D3)⋊C4—181no—
213(C2×D3)⋊C4—181no—
214C4⋊Dic3—181no—
215C4⋊Dic3C2×D911yescharacteristic
216Dic3⋊C4—181no—
217C2×C3⋊C8—31no—
218C2×C3⋊C8C2×D911yescharacteristic
219Dic12—182no—
220C22×C32—22no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32D3×D411yescharacteristic
224C2×C3⋊D3—61no—
225C2×C3⋊D3—61no—
226C2×C3⋊D3—61no—
227C32×C4C2×D1211yescharacteristic
228C32×C4D3×D411yescharacteristic
229C3⋊Dic3—62no—
230C3⋊Dic3—61no—
231C3×Dic3—41no—
232C3×Dic3—121no—
233C22×C9—24no—
234C22×C9—22no—
235C22×C9—21no—
236C22×C9—42no—
237C22×C9—42no—
238C22×C9—41no—
239C22×C9—41no—
240C22×C9D3×D411yescharacteristic
241C2×D9—62no—
242C2×D9—61no—
243C2×D9—61no—
244C2×D9—61no—
245C2×D9—121no—
246C2×D9—121no—
247C4×C9—21no—
248C4×C9—21no—
249C4×C9—121no—
250C4×C9D3×D411yescharacteristic
251C4×C9C2×#6711yescharacteristic
252Dic9—41no—
253Dic9—64no—
254Dic9—62no—
255Dic9—62no—
256Dic9—61no—
257C2×#41—271no—
258C2×Q16—271no—
259C2×SD16—91no—
260C22⋊Q8—271no—
261#103—271no—
262#102—31no—
263C22⋊C8—271no—
264C3×C9#45211yescharacteristic, complemented, Sylow
265C23×C3—22no—
266C23×C3—21no—
267C23×C3—21no—
268C22×D3—91no—
269C22×D3—91no—
270C22×D3—181no—
271C3×Q8—181no—
272C3×Q8—181no—
273C3×D4—22no—
274C3×D4—21no—
275C3×D4—21no—
276C3×D4—22no—
277C3×D4D3612yes—
278C3×D4#62612yes—
279C2×C3×C4—21no—
280C2×C3×C4—61no—
281C2×C3×C4—181no—
282C2×C3×C4C22×D911yescharacteristic
283C2×C3×C4C2×D3211yescharacteristic
284#67—184no—
285#67—188no—
286#67—184no—
287C2×Dic3—21no—
288C2×Dic3—61no—
289C2×Dic3—91no—
290C2×Dic3—91no—
291C2×Dic3—181no—
292C2×Dic3—184no—
293C2×Dic3—182no—
294C2×Dic3—188no—
295C2×Dic3—184no—
296C2×Dic3—181no—
297C2×Dic3—181no—
298C4×D3—182no—
299C4×D3—182no—
300C4×D3—181no—
301C4×D3—181no—
302C4×D3—181no—
303C4×D3—181no—
304Dic6—61no—
305Dic6—61no—
306Dic6—92no—
307Dic6—92no—
308Dic6—182no—
309Dic6—182no—
310Dic6—181no—
311Dic6—181no—
312C3×C8—61no—
313C3×C8—181no—
314C3⋊C8—21no—
315C3⋊C8—61no—
316C2×C32—42no—
317C2×C32#88211yescharacteristic
318C2×C32#91211yescharacteristic
319C2×C32#91511yescharacteristic
320C3⋊D3—121no—
321C2×C9—21no—
322C2×C9—21no—
323C2×C9—21no—
324C2×C9—44no—
325C2×C9—42no—
326C2×C9#91511yescharacteristic
327C2×C9#91611yescharacteristic
328C2×C9#93711yescharacteristic
329D9—122no—
330D9—121no—
331#41—542no—
332#41—542no—
333C2×Q8—91no—
334C2×Q8—271no—
335C2×D4—541no—
336C2×D4D3×D911yescharacteristic
337C22×C4—271no—
338C22×C4—541no—
339Q16—542no—
340SD16—182no—
341C2×C8—31no—
342C2×C8—91no—
343C4⋊C4—31no—
344C4⋊C4—541no—
345C22⋊C4—541no—
346C22×C3—24no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—21no—
350C22×C3—42no—
351C22×C3—42no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3D4×D911yescharacteristic
357C22×C3#381111yescharacteristic
358C2×D3—182no—
359C2×D3—181no—
360C2×D3—181no—
361C2×D3—181no—
362C2×D3—181no—
363C2×D3—181no—
364C2×D3—181no—
365C2×D3—361no—
366C2×D3—361no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—121no—
370C3×C4—361no—
371C3×C4C2×D3611yescharacteristic
372C3×C4D4×D911yescharacteristic
373C3×C4C2×#62611yescharacteristic
374C3×C4#381111yescharacteristic
375Dic3—41no—
376Dic3—121no—
377Dic3—184no—
378Dic3—182no—
379Dic3—182no—
380Dic3—181no—
381Dic3—182no—
382Dic3—181no—
383C32#478311yescharacteristic
384C9—21no—
385C9#517311yescharacteristic, complemented
386C23—21no—
387C23—271no—
388Q8—181no—
389Q8—181no—
390Q8—272no—
391Q8—541no—
392D4—21no—
393D4—544no—
394D4#619212yes—
395C2×C4—61no—
396C2×C4—181no—
397C2×C4—271no—
398C2×C4—544no—
399C2×C4—541no—
400C2×C4—541no—
401C2×C4—541no—
402C2×C4C2×D3×D911yescharacteristic
403C8—61no—
404C8—181no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—44no—
409C2×C3—42no—
410C2×C3—42no—
411C2×C3#6344411yescharacteristic
412C2×C3#6347411yescharacteristic
413C2×C3#6347711yescharacteristic
414C2×C3#6393111yescharacteristic
415C2×C3#6393611yescharacteristic
416C2×C3#6398111yescharacteristic
417D3—362no—
418D3—361no—
419D3—361no—
420C22—22no—
421C22—41no—
422C22—41no—
423C22—541no—
424C22—541no—
425C22—541no—
426C22#8866811yescharacteristic
427C4—121no—
428C4—361no—
429C4—541no—
430C4—542no—
431C4C2×#619211yescharacteristic
432C4#8866811yescharacteristic
433C3—21no—
434C3#1058846311yescharacteristic, complemented
435C3#1059146611yescharacteristic
436C2—42no—
437C2—1081no—
438C2#1172117811yescharacteristic
439C2#1172118311yescharacteristic
440C2#1172127811yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×#88742C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#452
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3672C2semidirect
3. #11720936C2semidirect
4. C2×#88418C2semidirect
5. C9×#832C2semidirect
6. #11720282C2semidirect
7. #11720269C2semidirect
34. C2×#3672D3semidirect
121. #832D9semidirect
264. C3×C9#452semidirect
385. C9#5173semidirect
434. C3#10588463semidirect

Maximal subgroups

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

subgroupindex
C2×#887422
C2×C3×#36722
#117209362
C2×#884182
C9×#8322
#117202822
#117202692
#105914663
#105884633