Finite Group Databasebeta

← #13140346

Subgroups — #13140346

Order 1728.

412 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#11721838C211yesmaximal, characteristic
2#11720922C211yesmaximal, characteristic, complemented
3C9×#832C211yesmaximal, characteristic, complemented
4C3×#63392C211yesmaximal, characteristic, complemented
5#11720282C211yesmaximal, characteristic, complemented
6#11720278C211yesmaximal, characteristic, complemented
7#11720262C211yesmaximal, characteristic, complemented
8#10591425—31nomaximal
9#10588459—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×C4×C3⋊D9C2211yescharacteristic
13#88551—21no—
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19#88445—21no—
20Dic3×Dic9—21no—
21#64147—31no—
22#63849—31no—
23C3×#832—31no—
24C3×#832D311yescharacteristic
25#63575—31no—
26#63573—31no—
27#63557—31no—
28#63500—31no—
29#63500—61no—
30#63454—31no—
31C9×#102—31no—
32#63392D311yescharacteristic, complemented
33(C2×D9)⋊C8—31no—
34#63377—31no—
35#63366—31no—
36C23×C3×C9—21no—
37C22×C3⋊D9D411yescharacteristic
38C3×D4×C9—21no—
39C3×D4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41#6233—42no—
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9D411yescharacteristic
44C4×C3⋊D9—21no—
45C4×C3⋊D9—21no—
46C2×C9×Dic3—21no—
47C2×C3×Dic9—21no—
48C9×C3⋊C8—21no—
49C3×C9⋊C8—21no—
50#4784—91no—
51#4779—91no—
52C2×C32×D4C2×D311yescharacteristic
53C2×#638—61no—
54C2×C4×C3⋊D3—31no—
55#3757—61no—
56C4⋊(C3⋊Dic3)—31no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3C2×D311yescharacteristic
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊C8C2×D311yescharacteristic
61#3722—61no—
62Dic32—61no—
63C2×D4×C9—21no—
64C2×D4×C9C2×D311yescharacteristic
65C2×#611—61no—
66C2×#611—62no—
67C2×C4×D9—31no—
68C2×C4×D9—61no—
69C2×C8×C9—31no—
70C9×C4⋊C4—31no—
71#3676—61no—
72#3676—62no—
73(C2×D9)⋊C4—61no—
74C4⋊Dic9—31no—
75C4⋊Dic9—61no—
76C4⋊Dic9C2×D311yescharacteristic
77C4×Dic9—61no—
78C2×C9⋊C8C2×D311yescharacteristic
79C22×C3×C9—41no—
80C22×C3×C9—41no—
81C22×C3×C9—41no—
82C22×C3×C9C2×D411yescharacteristic
83C2×C3⋊D9—21no—
84C2×C3⋊D9—21no—
85C2×C3⋊D9—21no—
86C3×C4×C9#4111yescharacteristic
87C3×C4×C9#4111yescharacteristic
88C3⋊Dic9—21no—
89C3⋊Dic9—41no—
90C9×Dic3—41no—
91C3×Dic9—41no—
92#938—91no—
93#938—91no—
94#938—181no—
95#892—91no—
96#892—91no—
97C3×#102—31no—
98C3×#102—91no—
99#832—31no—
100#832D911yescharacteristic, complemented
101(C2×D3)⋊C8—91no—
102(C2×D3)⋊C8—91no—
103#818—91no—
104#817—91no—
105#808—91no—
106#807—91no—
107C23×C32—21no—
108C22×C3⋊D3—31no—
109C32×D4—21no—
110C32×D4—21no—
111C2×C32×C4C22×D311yescharacteristic
112#638—122no—
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—61no—
115C4×C3⋊D3—61no—
116C4×C3⋊D3—61no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—61no—
119C23×C9—22no—
120C23×C9—21no—
121C22×D9—31no—
122C22×D9—61no—
123C3×C3⋊C8—21no—
124C3×C3⋊C8—61no—
125D4×C9—21no—
126D4×C9—21no—
127D4×C9—41no—
128D4×C9—41no—
129C2×C4×C9—21no—
130C2×C4×C9—61no—
131C2×C4×C9C22×D311yescharacteristic
132#611—124no—
133#611—122no—
134C2×Dic9—21no—
135C2×Dic9—31no—
136C2×Dic9—61no—
137C2×Dic9—62no—
138C2×Dic9—61no—
139C4×D9—61no—
140C4×D9—61no—
141C4×D9—121no—
142C4×D9—121no—
143C8×C9—61no—
144C9⋊C8—21no—
145#482—271noSylow
146C2×C3×C9—81no—
147C2×C3×C9#12311yescharacteristic
148C2×C3×C9#13511yescharacteristic
149C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
150C3⋊D9—41no—
151C2×C3×D4—21no—
152C2×C3×D4C2×D911yescharacteristic
153C2×C3×D4D3211yescharacteristic
154C2×#67—181no—
155C2×#67—181no—
156C2×#67—182no—
157C2×C4×D3—91no—
158C2×C4×D3—91no—
159C2×C4×D3—181no—
160C2×C3×C8—31no—
161C2×C3×C8—91no—
162C3×C4⋊C4—31no—
163C3×C4⋊C4—91no—
164#217—181no—
165#217—181no—
166#217—182no—
167(C2×D3)⋊C4—181no—
168(C2×D3)⋊C4—181no—
169C4⋊Dic3—31no—
170C4⋊Dic3—91no—
171C4⋊Dic3—91no—
172C4⋊Dic3—181no—
173C4⋊Dic3C2×D911yescharacteristic
174C4×Dic3—181no—
175C4×Dic3—181no—
176C2×C3⋊C8—31no—
177C2×C3⋊C8C2×D911yescharacteristic
178C22×C32—41no—
179C22×C32—41no—
180C22×C32—41no—
181C22×C32D3×D411yescharacteristic
182C2×C3⋊D3—61no—
183C2×C3⋊D3—61no—
184C2×C3⋊D3—61no—
185C32×C4#23511yescharacteristic
186C32×C4#23711yescharacteristic
187C3⋊Dic3—61no—
188C3⋊Dic3—121no—
189C3×Dic3—41no—
190C3×Dic3—121no—
191C22×C9—21no—
192C22×C9—42no—
193C22×C9—42no—
194C22×C9—42no—
195C22×C9—41no—
196C22×C9—41no—
197C22×C9—41no—
198C22×C9D3×D411yescharacteristic
199C2×D9—62no—
200C2×D9—61no—
201C2×D9—61no—
202C2×D9—61no—
203C2×D9—121no—
204C2×D9—121no—
205C4×C9—21no—
206C4×C9—21no—
207C4×C9—121no—
208C4×C9#23511yescharacteristic
209C4×C9#23711yescharacteristic
210Dic9—41no—
211Dic9—61no—
212Dic9—121no—
213Dic9—122no—
214Dic9—121no—
215#121—271no—
216#117—271no—
217#107—271no—
218#106—271no—
219#102—31no—
220#102—91no—
221C22⋊C8—271no—
222C3×C9#48211yescharacteristic, complemented, Sylow
223C23×C3—22no—
224C23×C3—21no—
225C23×C3—21no—
226C22×D3—91no—
227C22×D3—91no—
228C22×D3—181no—
229C3×D4—21no—
230C3×D4—21no—
231C3×D4—21no—
232C3×D4—21no—
233C3×D4—41no—
234C3×D4—41no—
235C2×C3×C4—21no—
236C2×C3×C4—61no—
237C2×C3×C4—181no—
238C2×C3×C4C22×D911yescharacteristic
239C2×C3×C4C2×D3211yescharacteristic
240#67—362no—
241#67—364no—
242#67—362no—
243C2×Dic3—21no—
244C2×Dic3—61no—
245C2×Dic3—91no—
246C2×Dic3—91no—
247C2×Dic3—181no—
248C2×Dic3—182no—
249C2×Dic3—181no—
250C2×Dic3—181no—
251C4×D3—181no—
252C4×D3—181no—
253C4×D3—181no—
254C4×D3—181no—
255C4×D3—361no—
256C4×D3—361no—
257C3×C8—61no—
258C3×C8—181no—
259C3⋊C8—21no—
260C3⋊C8—61no—
261C2×C32—81no—
262C2×C32#88311yescharacteristic
263C2×C32#91111yescharacteristic
264C2×C32#91611yescharacteristic
265C3⋊D3—121no—
266C2×C9—21no—
267C2×C9—21no—
268C2×C9—21no—
269C2×C9—82no—
270C2×C9—81no—
271C2×C9#88311yescharacteristic
272C2×C9#91211yescharacteristic
273C2×C9#91411yescharacteristic
274D9—122no—
275D9—121no—
276C2×D4—541no—
277C2×D4D3×D911yescharacteristic
278C22×C4—271no—
279C2×C8—31no—
280C2×C8—91no—
281C4⋊C4—31no—
282C4⋊C4—91no—
283C4⋊C4—271no—
284C22⋊C4—541no—
285C22⋊C4—541no—
286C42—541no—
287C22×C3—21no—
288C22×C3—42no—
289C22×C3—42no—
290C22×C3—42no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3D4×D911yescharacteristic
298C22×C3#381111yescharacteristic
299C2×D3—182no—
300C2×D3—181no—
301C2×D3—181no—
302C2×D3—181no—
303C2×D3—181no—
304C2×D3—181no—
305C2×D3—181no—
306C2×D3—361no—
307C2×D3—361no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—121no—
311C3×C4—361no—
312C3×C4#369711yescharacteristic
313C3×C4#369911yescharacteristic
314C3×C4#380411yescharacteristic
315C3×C4#380411yescharacteristic
316Dic3—41no—
317Dic3—121no—
318Dic3—181no—
319Dic3—181no—
320Dic3—361no—
321Dic3—362no—
322Dic3—361no—
323Dic3—361no—
324C32#477911yescharacteristic
325C9—21no—
326C9#478411yescharacteristic, complemented
327C23—21no—
328C23—271no—
329D4—21no—
330D4—21no—
331D4—1082no—
332C2×C4—61no—
333C2×C4—181no—
334C2×C4—271no—
335C2×C4—541no—
336C2×C4—541no—
337C2×C4—541no—
338C2×C4C2×D3×D911yescharacteristic
339C8—61no—
340C8—181no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—82no—
345C2×C3—81no—
346C2×C3—81no—
347C2×C3#6344511yescharacteristic
348C2×C3#6347311yescharacteristic
349C2×C3#6347811yescharacteristic
350C2×C3#6393611yescharacteristic
351C2×C3#6393711yescharacteristic
352C2×C3#6397511yescharacteristic
353D3—362no—
354D3—361no—
355D3—361no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—541no—
360C22—541no—
361C22—541no—
362C22#8866811yescharacteristic
363C4—121no—
364C4—361no—
365C4—541no—
366C4—1081no—
367C4#8865811yescharacteristic
368C4#8866211yescharacteristic
369C3—21no—
370C3#1058845911yescharacteristic, complemented
371C3#1059142511yescharacteristic
372C2—81no—
373C2—1081no—
374C2#1172117911yescharacteristic
375C2#1172118611yescharacteristic
376C2#1172127511yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radical#11721838C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#482
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. #11720922C2semidirect
3. C9×#832C2semidirect
4. C3×#63392C2semidirect
5. #11720282C2semidirect
6. #11720278C2semidirect
7. #11720262C2semidirect
32. #63392D3semidirect
100. #832D9semidirect
222. C3×C9#482semidirect
326. C9#4784semidirect
370. C3#10588459semidirect

Maximal subgroups

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

subgroupindex
#117218382
#117209222
C9×#8322
C3×#633922
#117202822
#117202782
#117202622
#105914253
#105884593