Finite Group Databasebeta

← #13137860

Subgroups — #13137860

Order 1728.

580 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 89 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, complemented
2#11721751C211yesmaximal, characteristic, complemented
3C2×#88426C211yesmaximal, characteristic, complemented
4C2×#88417C211yesmaximal, characteristic, complemented
5C9×#810C211yesmaximal, characteristic, complemented
6C3×#63369C211yesmaximal, characteristic, complemented
7#11720282C211yesmaximal, characteristic, complemented
8#10590617—31nomaximal
9#10588490—31nomaximal
10C2×C4⋊(C3⋊D9)—21no—
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×C3⋊D9C2211yescharacteristic
13C2×C3×D36C2211yescharacteristic
14C2×C9×Dic6C2211yescharacteristic
15C2×#6192—21no—
16C2×Dic3×D9—21no—
17#88634—22no—
18#88634—22no—
19C3×C9×C4⋊C4C2211yescharacteristic
20#88536—21no—
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23#88426—22no—
24#88417—22no—
25C2×#3796—31no—
26#64104—31no—
27C2×#3717—31no—
28C2×#3714—31no—
29C2×#3701—31no—
30C3×#810D311yescharacteristic
31C3×#809—31no—
32#63557—31no—
33C2×#3675—31no—
34C2×D72—31no—
35#63457—31no—
36#63457—61no—
37C9×#103—31no—
38(C2×D9)⋊C8—31no—
39#63369D311yescharacteristic, complemented
40C22×C3⋊D9—21no—
41C22×C3⋊D9D411yescharacteristic
42C22×C3×D9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3⋊Dic9D411yescharacteristic
46C4⋊(C3⋊D9)—21no—
47C4⋊(C3⋊D9)—21no—
48C4⋊(C3⋊D9)—42no—
49C4×C3⋊D9—21no—
50C4×C3⋊D9—21no—
51C2×C9×Dic3—21no—
52C3×D36—21no—
53C3×D36D412yescomplemented
54C9×Dic6—21no—
55C9×Dic6D412yescomplemented
56#6192—24no—
57Dic3×D9—24no—
58C9×C3⋊C8—21no—
59C3×C9⋊C8—21no—
60#4810—91no—
61#4781—91no—
62C2×C4⋊(C3⋊D3)—31no—
63C2×C4⋊(C3⋊D3)—61no—
64C2×C4×C3⋊D3—31no—
65C2×C3×D12—31no—
66C2×C3×Dic6C2×D311yescharacteristic
67C2×#626—61no—
68C2×D3×Dic3—61no—
69#3796—62no—
70#3796—62no—
71C32×C4⋊C4C2×D311yescharacteristic
72#3752—61no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8C2×D311yescharacteristic
75#3717—62no—
76#3714—62no—
77C2×Q8×C9—31no—
78#3701—62no—
79#3701—62no—
80C2×D36—31no—
81C2×D36—61no—
82C2×D36—61no—
83C2×D36—61no—
84C2×D36—62no—
85C2×D36C2×D311yescharacteristic
86C2×C4×D9—31no—
87C2×C4×D9—61no—
88C2×C4×D9—61no—
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
96D72—62no—
97C22×C3×C9C2×D411yescharacteristic
98C2×C3⋊D9—21no—
99C2×C3⋊D9—21no—
100C2×C3⋊D9—21no—
101C2×C3⋊D9—41no—
102C2×C3⋊D9—41no—
103C2×C3⋊D9—41no—
104C2×C3×D9—22no—
105C2×C3×D9—41no—
106C2×C3×D9—41no—
107C3×C4×C9—41no—
108C3×C4×C9C2×D411yescharacteristic
109C3×C4×C9C2×D411yescharacteristic
110C3⋊Dic9—21no—
111C9×Dic3—22no—
112C2×#239—91no—
113C2×#237—91no—
114C2×#215—91no—
115C2×#213—91no—
116C2×D24—91no—
117C2×#204—91no—
118#895—91no—
119#895—91no—
120#895—181no—
121C3×#103—31no—
122C3×#102—91no—
123(C2×D3)⋊C8—91no—
124(C2×D3)⋊C8—91no—
125#810D911yescharacteristic, complemented
126#809—31no—
127C22×C3⋊D3—31no—
128C22×C3⋊D3—61no—
129C22×C3×D3—61no—
130C2×C32×C4—21no—
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—31no—
133C4⋊(C3⋊D3)—61no—
134C4⋊(C3⋊D3)—61no—
135C4⋊(C3⋊D3)—122no—
136C4×C3⋊D3—61no—
137C4×C3⋊D3—61no—
138C2×C3×Dic3—21no—
139C3×D12—32no—
140C3×D12—61no—
141C3×Dic6—21no—
142C3×Dic6D1212yes—
143#626—64no—
144D3×Dic3—64no—
145C22×D9—21no—
146C22×D9—31no—
147C22×D9—61no—
148C22×D9—62no—
149C22×D9—61no—
150C3×C3⋊C8—21no—
151C3×C3⋊C8—61no—
152Q8×C9—32no—
153Q8×C9—61no—
154C2×C4×C9—22no—
155C2×C4×C9—21no—
156C2×C4×C9—21no—
157C2×C4×C9—61no—
158C2×C4×C9C22×D311yescharacteristic
159C2×Dic9—31no—
160C2×Dic9—61no—
161D36—21no—
162D36—64no—
163D36—61no—
164D36—61no—
165D36—121no—
166D36—121no—
167D36—124no—
168D36—122no—
169D36D1212yescomplemented
170C4×D9—64no—
171C4×D9—61no—
172C4×D9—61no—
173C4×D9—121no—
174C4×D9—121no—
175C8×C9—61no—
176C9⋊C8—21no—
177#449—271noSylow
178C2×C3×C9#12011yescharacteristic
179C2×C3×C9#13511yescharacteristic
180C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
181C3⋊D9—41no—
182C3⋊D9—81no—
183C3×D9—42no—
184C2×C3×Q8—31no—
185C2×C3×D4—91no—
186C2×#67—181no—
187C22×Dic3—181no—
188#239—182no—
189#239—182no—
190#237—182no—
191#237—182no—
192C2×D12—31no—
193C2×D12—91no—
194C2×D12—91no—
195C2×D12—181no—
196C2×D12—181no—
197C2×D12—181no—
198C2×D12—181no—
199C2×D12—182no—
200C2×C4×D3—91no—
201C2×C4×D3—91no—
202C2×C4×D3—181no—
203C2×C4×D3—181no—
204C2×Dic6C2×D911yescharacteristic
205C2×C3×C8—31no—
206C2×C3×C8—91no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4C2×D911yescharacteristic
209C3×C4⋊C4D3211yescharacteristic
210#215—182no—
211#213—182no—
212(C2×D3)⋊C4—181no—
213(C2×D3)⋊C4—181no—
214(C2×D3)⋊C4—182no—
215C2×C3⋊C8—31no—
216C2×C3⋊C8C2×D911yescharacteristic
217D24—182no—
218#204—182no—
219C22×C32D3×D411yescharacteristic
220C2×C3⋊D3—61no—
221C2×C3⋊D3—61no—
222C2×C3⋊D3—61no—
223C2×C3⋊D3—121no—
224C2×C3⋊D3—121no—
225C2×C3⋊D3—121no—
226C2×C3×D3—62no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C32×C4—41no—
230C32×C4C2×D1211yescharacteristic
231C32×C4D3×D411yescharacteristic
232C3⋊Dic3—61no—
233C3×Dic3—22no—
234C22×C9—21no—
235C22×C9D3×D411yescharacteristic
236C2×D9—22no—
237C2×D9—41no—
238C2×D9—41no—
239C2×D9—62no—
240C2×D9—61no—
241C2×D9—61no—
242C2×D9—61no—
243C2×D9—121no—
244C2×D9—121no—
245C2×D9—122no—
246C2×D9—122no—
247C2×D9—122no—
248C2×D9—121no—
249C2×D9—121no—
250C2×D9—121no—
251C4×C9—21no—
252C4×C9—21no—
253C4×C9—42no—
254C4×C9—41no—
255C4×C9—62no—
256C4×C9C2×D1211yescharacteristic
257C4×C9D3×D411yescharacteristic
258Dic9—61no—
259Dic9—121no—
260C2×#41—271no—
261C2×SD16—271no—
262C2×D8—271no—
263#121—271no—
264#103—31no—
265#102—91no—
266C22⋊C8—271no—
267C3×C9#44911yescharacteristic, complemented, Sylow
268C23×C3—181no—
269C22×D3—61no—
270C22×D3—91no—
271C22×D3—91no—
272C22×D3—181no—
273C22×D3—182no—
274C22×D3—181no—
275C22×D3—181no—
276C3×Q8—32no—
277C3×Q8—61no—
278C3×D4—92no—
279C3×D4—181no—
280C2×C3×C4—22no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—61no—
285C2×C3×C4C22×D911yescharacteristic
286C2×C3×C4C2×D3211yescharacteristic
287#67—184no—
288C2×Dic3—21no—
289C2×Dic3—91no—
290C2×Dic3—91no—
291C2×Dic3—181no—
292C2×Dic3—184no—
293D12—32no—
294D12—61no—
295D12—184no—
296D12—181no—
297D12—181no—
298D12—181no—
299D12—181no—
300D12—361no—
301D12—361no—
302D12—362no—
303D12—364no—
304D12—362no—
305C4×D3—184no—
306C4×D3—181no—
307C4×D3—181no—
308C4×D3—181no—
309C4×D3—181no—
310C4×D3—361no—
311C4×D3—361no—
312Dic6—21no—
313Dic6D3612yescomplemented
314C3×C8—61no—
315C3×C8—181no—
316C3⋊C8—21no—
317C3⋊C8—61no—
318C2×C32#88211yescharacteristic
319C2×C32#91411yescharacteristic
320C2×C32#91911yescharacteristic
321C3⋊D3—121no—
322C3⋊D3—241no—
323C3×D3—122no—
324C2×C9—21no—
325C2×C9—21no—
326C2×C9—21no—
327C2×C9#88211yescharacteristic
328C2×C9#91111yescharacteristic
329C2×C9#91611yescharacteristic
330D9—42no—
331D9—122no—
332D9—121no—
333D9—242no—
334D9—241no—
335#41—542no—
336#41—542no—
337C2×Q8—31no—
338C2×D4—91no—
339C2×D4—271no—
340C2×D4—541no—
341C2×D4—541no—
342C22×C4—271no—
343C22×C4—541no—
344SD16—542no—
345D8—542no—
346C2×C8—31no—
347C2×C8—91no—
348C4⋊C4D3×D911yescharacteristic
349C22⋊C4—541no—
350C22×C3—21no—
351C22×C3—182no—
352C22×C3—361no—
353C22×C3—361no—
354C22×C3D4×D911yescharacteristic
355C22×C3#381111yescharacteristic
356C2×D3—62no—
357C2×D3—121no—
358C2×D3—121no—
359C2×D3—182no—
360C2×D3—181no—
361C2×D3—181no—
362C2×D3—181no—
363C2×D3—181no—
364C2×D3—181no—
365C2×D3—181no—
366C2×D3—361no—
367C2×D3—361no—
368C2×D3—362no—
369C2×D3—362no—
370C2×D3—362no—
371C2×D3—361no—
372C2×D3—361no—
373C2×D3—361no—
374C2×D3—361no—
375C2×D3—361no—
376C2×D3—361no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—42no—
380C3×C4—41no—
381C3×C4—41no—
382C3×C4—62no—
383C3×C4C2×D3611yescharacteristic
384C3×C4D4×D911yescharacteristic
385C3×C4D3×D1211yescharacteristic
386C3×C4D3×D1211yescharacteristic
387Dic3—22no—
388Dic3—181no—
389Dic3—181no—
390Dic3—361no—
391C32#481011yescharacteristic
392C9—21no—
393C9#478111yescharacteristic, complemented
394C23—181no—
395C23—271no—
396C23—541no—
397Q8—32no—
398Q8—61no—
399D4—92no—
400D4—181no—
401D4—544no—
402D4—541no—
403D4—541no—
404D4—1082no—
405C2×C4—21no—
406C2×C4—61no—
407C2×C4—271no—
408C2×C4—544no—
409C2×C4—541no—
410C2×C4—541no—
411C2×C4C2×D3×D911yescharacteristic
412C8—61no—
413C8—181no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—21no—
417C2×C3—362no—
418C2×C3#6344411yescharacteristic
419C2×C3#6347611yescharacteristic
420C2×C3#6348111yescharacteristic
421C2×C3#6392311yescharacteristic
422C2×C3#6393711yescharacteristic
423C2×C3#6395011yescharacteristic
424D3—122no—
425D3—362no—
426D3—361no—
427D3—361no—
428D3—722no—
429D3—721no—
430D3—721no—
431C22—182no—
432C22—361no—
433C22—361no—
434C22—541no—
435C22—541no—
436C22—541no—
437C22—1081no—
438C22—1081no—
439C22—1081no—
440C22#8866811yescharacteristic
441C4—41no—
442C4—62no—
443C4—541no—
444C4D3×D3611yescharacteristic
445C4D9×D1211yescharacteristic
446C3—21no—
447C3#1058849011yescharacteristic, complemented
448C3#1059061711yescharacteristic
449C2—362no—
450C2—1081no—
451C2—2161no—
452C2#1172105111yescharacteristic
453C2#1172118711yescharacteristic
454C2#1172122411yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#449
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×#88634C2semidirect
2. #11721751C2semidirect
3. C2×#88426C2semidirect
4. C2×#88417C2semidirect
5. C9×#810C2semidirect
6. C3×#63369C2semidirect
7. #11720282C2semidirect
39. #63369D3semidirect
53. C3×D36D4semidirect
55. C9×Dic6D4semidirect
125. #810D9semidirect
169. D36D12semidirect
267. C3×C9#449semidirect
313. Dic6D36semidirect
393. C9#4781semidirect
447. C3#10588490semidirect

Maximal subgroups

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

subgroupindex
C2×#886342
#117217512
C2×#884262
C2×#884172
C9×#8102
C3×#633692
#117202822
#105906173
#105884903