Finite Group Databasebeta

← #13137851

Subgroups — #13137851

Order 1728.

568 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 91 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×(C3⋊D9)⋊Q8C211yesmaximal, characteristic, complemented
2#11721751C211yesmaximal, characteristic
3C2×#88426C211yesmaximal, characteristic, complemented
4C2×#88416C211yesmaximal, characteristic, complemented
5C9×#810C211yesmaximal, characteristic, complemented
6C3×#63368C211yesmaximal, characteristic, complemented
7#11720282C211yesmaximal, characteristic, complemented
8#10590606—31nomaximal
9#10588486—31nomaximal
10C2×C4⋊(C3⋊D9)—21no—
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×C3⋊D9C2211yescharacteristic
13C2×C9×Dic6C2211yescharacteristic
14C2×C3×Dic18C2211yescharacteristic
15C2×(C3⋊D9)⋊C4—21no—
16C2×#6189—21no—
17(C3⋊D9)⋊Q8—22no—
18(C3⋊D9)⋊Q8—22no—
19C3×C9×C4⋊C4C2211yescharacteristic
20#88536—21no—
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23#88426—22no—
24#88416—22no—
25C2×#3797—31no—
26#64104—31no—
27C2×#3717—31no—
28C2×#3717—31no—
29C2×Q8×D9—31no—
30C3×#810—31no—
31C3×#810D311yescharacteristic
32#63557—31no—
33C2×#3675—31no—
34C2×#3664—31no—
35#63457—31no—
36#63457—61no—
37C9×#103—31no—
38(C2×D9)⋊C8—31no—
39#63368D311yescharacteristic, complemented
40C22×C3⋊D9—21no—
41C22×C3⋊D9D411yescharacteristic
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3⋊Dic9D411yescharacteristic
45C4⋊(C3⋊D9)—21no—
46C4⋊(C3⋊D9)—21no—
47C4⋊(C3⋊D9)—42no—
48C4×C3⋊D9—21no—
49C4×C3⋊D9—21no—
50C2×C9×Dic3—21no—
51C2×C3×Dic9—21no—
52C9×Dic6—21no—
53C9×Dic6D412yescomplemented
54C3×Dic18—21no—
55C3×Dic18D412yescomplemented
56(C3⋊D9)⋊C4—24no—
57#6189—24no—
58C9×C3⋊C8—21no—
59C3×C9⋊C8—21no—
60#4806—91no—
61#4806—91no—
62C2×C4⋊(C3⋊D3)—31no—
63C2×C4⋊(C3⋊D3)—61no—
64C2×C4×C3⋊D3—31no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6C2×D311yescharacteristic
67C2×#627—61no—
68C2×#624—61no—
69#3797—62no—
70#3797—62no—
71C32×C4⋊C4C2×D311yescharacteristic
72#3752—61no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8C2×D311yescharacteristic
75#3717—62no—
76#3717—62no—
77C2×Q8×C9—31no—
78Q8×D9—62no—
79Q8×D9—62no—
80C2×D36—31no—
81C2×D36—61no—
82C2×D36—61no—
83C2×D36—62no—
84C2×C4×D9—31no—
85C2×C4×D9—61no—
86C2×C4×D9—61no—
87C2×Dic18—61no—
88C2×Dic18C2×D311yescharacteristic
89C2×C8×C9—31no—
90C9×C4⋊C4—21no—
91C9×C4⋊C4C2×D311yescharacteristic
92#3675—62no—
93(C2×D9)⋊C4—61no—
94(C2×D9)⋊C4—62no—
95C2×C9⋊C8C2×D311yescharacteristic
96#3664—62no—
97C22×C3×C9C2×D411yescharacteristic
98C2×C3⋊D9—21no—
99C2×C3⋊D9—21no—
100C2×C3⋊D9—41no—
101C2×C3⋊D9—41no—
102C2×C3⋊D9—41no—
103C2×C3⋊D9SD1612yescomplemented
104C3×C4×C9—41no—
105C3×C4×C9C2×D411yescharacteristic
106C3×C4×C9C2×D411yescharacteristic
107C3⋊Dic9—21no—
108C9×Dic3—22no—
109C3×Dic9—22no—
110C2×D3×Q8—91no—
111C2×D3×Q8—91no—
112C2×#215—91no—
113C2×#215—91no—
114C2×#204—91no—
115C2×#204—91no—
116#895—91no—
117#895—91no—
118#895—181no—
119C3×#103—31no—
120C3×#103—91no—
121(C2×D3)⋊C8—91no—
122(C2×D3)⋊C8—91no—
123#810—31no—
124#810D911yescharacteristic, complemented
125C22×C3⋊D3—31no—
126C22×C3⋊D3—61no—
127C2×C32×C4—21no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—31no—
130C4⋊(C3⋊D3)—61no—
131C4⋊(C3⋊D3)—61no—
132C4⋊(C3⋊D3)—122no—
133C4×C3⋊D3—61no—
134C4×C3⋊D3—61no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—61no—
137C3×Dic6—21no—
138C3×Dic6—32no—
139C3×Dic6—61no—
140C3×Dic6D1212yes—
141#627—64no—
142#624—64no—
143C22×D9—31no—
144C22×D9—61no—
145C22×D9—62no—
146C22×D9—61no—
147C3×C3⋊C8—21no—
148C3×C3⋊C8—61no—
149Q8×C9—32no—
150Q8×C9—61no—
151C2×C4×C9—22no—
152C2×C4×C9—21no—
153C2×C4×C9—21no—
154C2×C4×C9—61no—
155C2×C4×C9C22×D311yescharacteristic
156C2×Dic9—21no—
157C2×Dic9—31no—
158C2×Dic9—61no—
159D36—61no—
160D36—61no—
161D36—121no—
162D36—121no—
163D36—122no—
164D36—124no—
165C4×D9—64no—
166C4×D9—61no—
167C4×D9—61no—
168C4×D9—121no—
169C4×D9—121no—
170Dic18—21no—
171Dic18—64no—
172Dic18D1212yescomplemented
173C8×C9—61no—
174C9⋊C8—21no—
175#448—271noSylow
176C2×C3×C9#12011yescharacteristic
177C2×C3×C9C2×SD1611yescharacteristic
178C2×C3×C9#13711yescharacteristic
179C3⋊D9—22no—
180C3⋊D9—81no—
181C2×C3×Q8—31no—
182C2×C3×Q8—91no—
183D3×Q8—182no—
184D3×Q8—182no—
185D3×Q8—182no—
186D3×Q8—182no—
187C2×D12—91no—
188C2×D12—91no—
189C2×D12—181no—
190C2×D12—181no—
191C2×D12—181no—
192C2×D12—182no—
193C2×C4×D3—91no—
194C2×C4×D3—91no—
195C2×C4×D3—181no—
196C2×C4×D3—181no—
197C2×C4×D3—181no—
198C2×Dic6—31no—
199C2×Dic6—181no—
200C2×Dic6—181no—
201C2×Dic6C2×D911yescharacteristic
202C2×C3×C8—31no—
203C2×C3×C8—91no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4C2×D911yescharacteristic
206C3×C4⋊C4D3211yescharacteristic
207#215—182no—
208#215—182no—
209(C2×D3)⋊C4—181no—
210(C2×D3)⋊C4—181no—
211(C2×D3)⋊C4—182no—
212C2×C3⋊C8—31no—
213C2×C3⋊C8C2×D911yescharacteristic
214#204—182no—
215#204—182no—
216C22×C32D3×D411yescharacteristic
217C2×C3⋊D3—32no—
218C2×C3⋊D3—61no—
219C2×C3⋊D3—61no—
220C2×C3⋊D3—121no—
221C2×C3⋊D3—121no—
222C2×C3⋊D3—121no—
223C32×C4—41no—
224C32×C4C2×D1211yescharacteristic
225C32×C4D3×D411yescharacteristic
226C3⋊Dic3—61no—
227C3×Dic3—22no—
228C3×Dic3—62no—
229C22×C9—21no—
230C22×C9D3×D411yescharacteristic
231C2×D9—32no—
232C2×D9—62no—
233C2×D9—61no—
234C2×D9—61no—
235C2×D9—121no—
236C2×D9—121no—
237C2×D9—122no—
238C2×D9—122no—
239C2×D9—122no—
240C2×D9—121no—
241C2×D9—121no—
242C2×D9—121no—
243C4×C9—21no—
244C4×C9—21no—
245C4×C9—42no—
246C4×C9—41no—
247C4×C9—62no—
248C4×C9C2×D1211yescharacteristic
249C4×C9D3×D411yescharacteristic
250Dic9—22no—
251Dic9—61no—
252Dic9—121no—
253C22×Q8—271no—
254C2×SD16—271no—
255C2×SD16—271no—
256#121—271no—
257#103—31no—
258#103—91no—
259C22⋊C8—271no—
260C3×C9#44811yescharacteristic, complemented, Sylow
261C22×D3—91no—
262C22×D3—91no—
263C22×D3—181no—
264C22×D3—182no—
265C22×D3—181no—
266C22×D3—181no—
267C3×Q8—32no—
268C3×Q8—61no—
269C3×Q8—92no—
270C3×Q8—181no—
271C2×C3×C4—22no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—61no—
276C2×C3×C4—181no—
277C2×C3×C4C22×D911yescharacteristic
278C2×C3×C4C2×D3211yescharacteristic
279C2×Dic3—21no—
280C2×Dic3—61no—
281C2×Dic3—91no—
282C2×Dic3—91no—
283C2×Dic3—181no—
284D12—181no—
285D12—181no—
286D12—181no—
287D12—181no—
288D12—361no—
289D12—361no—
290D12—362no—
291D12—362no—
292D12—364no—
293C4×D3—184no—
294C4×D3—184no—
295C4×D3—181no—
296C4×D3—181no—
297C4×D3—181no—
298C4×D3—181no—
299C4×D3—361no—
300C4×D3—361no—
301Dic6—21no—
302Dic6—32no—
303Dic6—61no—
304Dic6—184no—
305Dic6—184no—
306Dic6D3612yescomplemented
307C3×C8—61no—
308C3×C8—181no—
309C3⋊C8—21no—
310C3⋊C8—61no—
311C2×C32#88211yescharacteristic
312C2×C32D3×SD1611yescharacteristic
313C2×C32#91811yescharacteristic
314C3⋊D3—62no—
315C3⋊D3—241no—
316C2×C9—21no—
317C2×C9—21no—
318C2×C9—21no—
319C2×C9#88211yescharacteristic
320C2×C9D3×SD1611yescharacteristic
321C2×C9#91811yescharacteristic
322D9—62no—
323D9—122no—
324D9—242no—
325D9—241no—
326C2×Q8—31no—
327C2×Q8—91no—
328C2×Q8—541no—
329C2×Q8—542no—
330C2×Q8—542no—
331C2×D4—271no—
332C2×D4—541no—
333C22×C4—271no—
334C22×C4—541no—
335SD16—542no—
336SD16—542no—
337C2×C8—31no—
338C2×C8—91no—
339C4⋊C4D3×D911yescharacteristic
340C22⋊C4—541no—
341C22×C3—21no—
342C22×C3D4×D911yescharacteristic
343C22×C3#381111yescharacteristic
344C2×D3—92no—
345C2×D3—92no—
346C2×D3—182no—
347C2×D3—181no—
348C2×D3—181no—
349C2×D3—181no—
350C2×D3—181no—
351C2×D3—361no—
352C2×D3—361no—
353C2×D3—362no—
354C2×D3—362no—
355C2×D3—362no—
356C2×D3—361no—
357C2×D3—361no—
358C2×D3—361no—
359C2×D3—361no—
360C2×D3—361no—
361C2×D3—361no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—42no—
365C3×C4—41no—
366C3×C4—41no—
367C3×C4—62no—
368C3×C4—182no—
369C3×C4C2×D3611yescharacteristic
370C3×C4D4×D911yescharacteristic
371C3×C4D3×D1211yescharacteristic
372C3×C4D3×D1211yescharacteristic
373Dic3—22no—
374Dic3—62no—
375Dic3—181no—
376Dic3—181no—
377Dic3—361no—
378C32#480611yescharacteristic
379C9—21no—
380C9#480611yescharacteristic, complemented
381C23—271no—
382C23—541no—
383Q8—32no—
384Q8—61no—
385Q8—92no—
386Q8—181no—
387Q8—544no—
388D4—541no—
389D4—541no—
390D4—1082no—
391C2×C4—21no—
392C2×C4—61no—
393C2×C4—181no—
394C2×C4—271no—
395C2×C4—544no—
396C2×C4—541no—
397C2×C4—541no—
398C2×C4C2×D3×D911yescharacteristic
399C8—61no—
400C8—181no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3#6344411yescharacteristic
405C2×C3SD16×D911yescharacteristic
406C2×C3#6348011yescharacteristic
407C2×C3#6392311yescharacteristic
408C2×C3#6393011yescharacteristic
409C2×C3#6394511yescharacteristic
410D3—182no—
411D3—182no—
412D3—362no—
413D3—722no—
414D3—721no—
415D3—721no—
416C22—272no—
417C22—541no—
418C22—541no—
419C22—1081no—
420C22—1081no—
421C22—1081no—
422C22#8866811yescharacteristic
423C4—41no—
424C4—62no—
425C4—182no—
426C4—541no—
427C4D3×D3611yescharacteristic
428C4D9×D1211yescharacteristic
429C3—21no—
430C3#1058848611yescharacteristic, complemented
431C3#1059060611yescharacteristic
432C2—542no—
433C2—2161no—
434C2#1172105111yescharacteristic
435C2#1172118211yescharacteristic
436C2#1172122111yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radical#11721751C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#448
LCSG ▷ C2×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
1. C2×(C3⋊D9)⋊Q8C2semidirect
3. C2×#88426C2semidirect
4. C2×#88416C2semidirect
5. C9×#810C2semidirect
6. C3×#63368C2semidirect
7. #11720282C2semidirect
39. #63368D3semidirect
53. C9×Dic6D4semidirect
55. C3×Dic18D4semidirect
103. C2×C3⋊D9SD16semidirect
124. #810D9semidirect
172. Dic18D12semidirect
260. C3×C9#448semidirect
306. Dic6D36semidirect
380. C9#4806semidirect
430. C3#10588486semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊D9)⋊Q82
#117217512
C2×#884262
C2×#884162
C9×#8102
C3×#633682
#117202822
#105906063
#105884863