Finite Group Databasebeta

← #13140342

Subgroups — #13140342

Order 1728.

436 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 81 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#11721000C211yesmaximal, characteristic, complemented
3C9×#832C211yesmaximal, characteristic, complemented
4C3×#63392C211yesmaximal, characteristic, complemented
5#11720282C211yesmaximal, characteristic, complemented
6#11720278C211yesmaximal, characteristic, complemented
7#11720264C211yesmaximal, characteristic, complemented
8#10591419—31nomaximal
9#10588457—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×C4×C3⋊D9C2211yescharacteristic
13C2×(C3⋊D9)⋊C4—21no—
14#88551—21no—
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88442—21no—
21#64147—31no—
22#63900—31no—
23C3×#832—31no—
24C3×#832D311yescharacteristic
25#63573—31no—
26#63573—31no—
27#63557—31no—
28#63500—31no—
29#63500—61no—
30C4⋊C4×D9—31no—
31C9×#102—31no—
32#63392D311yescharacteristic, complemented
33(C2×D9)⋊C8—31no—
34#63377—31no—
35#63367—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—
48(C3⋊D9)⋊C4—42no—
49C9×C3⋊C8—21no—
50C3×C9⋊C8—21no—
51#4777—91no—
52#4777—91no—
53C2×C32×D4C2×D311yescharacteristic
54C2×#638—61no—
55C2×C4×C3⋊D3—31no—
56C2×#624—61no—
57#3757—61no—
58C4⋊(C3⋊Dic3)—31no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3C2×D311yescharacteristic
61C2×C3×C3⋊C8—31no—
62C2×C3×C3⋊C8C2×D311yescharacteristic
63#3724—61no—
64C2×D4×C9—21no—
65C2×D4×C9C2×D311yescharacteristic
66C2×#611—61no—
67C2×#611—62no—
68C2×C4×D9—31no—
69C2×C4×D9—61no—
70C2×C4×D9—61no—
71C2×C8×C9—31no—
72C9×C4⋊C4—31no—
73#3676—61no—
74#3676—62no—
75C4⋊Dic9—31no—
76C4⋊Dic9—61no—
77C4⋊Dic9C2×D311yescharacteristic
78Dic9⋊C4—61no—
79C2×C9⋊C8C2×D311yescharacteristic
80C22×C3×C9—41no—
81C22×C3×C9—41no—
82C22×C3×C9—41no—
83C22×C3×C9C2×D411yescharacteristic
84C2×C3⋊D9—21no—
85C2×C3⋊D9—21no—
86C2×C3⋊D9SD1612yes—
87C3×C4×C9#4111yescharacteristic
88C3×C4×C9#4111yescharacteristic
89C3⋊Dic9—21no—
90C3⋊Dic9—41no—
91C9×Dic3—41no—
92C3×Dic9—41no—
93#938—91no—
94#938—91no—
95#938—181no—
96D3×C4⋊C4—91no—
97D3×C4⋊C4—91no—
98C3×#102—31no—
99C3×#102—91no—
100#832—31no—
101#832D911yescharacteristic, complemented
102(C2×D3)⋊C8—91no—
103(C2×D3)⋊C8—91no—
104#817—91no—
105#817—91no—
106#808—91no—
107#808—91no—
108C23×C32—21no—
109C22×C3⋊D3—31no—
110C32×D4—21no—
111C32×D4—21no—
112C2×C32×C4C22×D311yescharacteristic
113#638—122no—
114C2×C3⋊Dic3—31no—
115C2×C3⋊Dic3—61no—
116C4×C3⋊D3—61no—
117C4×C3⋊D3—61no—
118C2×C3×Dic3—21no—
119C2×C3×Dic3—61no—
120#624—122no—
121C23×C9—22no—
122C23×C9—21no—
123C22×D9—31no—
124C22×D9—61no—
125C3×C3⋊C8—21no—
126C3×C3⋊C8—61no—
127D4×C9—21no—
128D4×C9—21no—
129D4×C9—41no—
130D4×C9—41no—
131C2×C4×C9—21no—
132C2×C4×C9—61no—
133C2×C4×C9C22×D311yescharacteristic
134#611—122no—
135#611—124no—
136C2×Dic9—21no—
137C2×Dic9—31no—
138C2×Dic9—61no—
139C2×Dic9—62no—
140C2×Dic9—61no—
141C4×D9—61no—
142C4×D9—61no—
143C4×D9—121no—
144C4×D9—121no—
145C4×D9—122no—
146C8×C9—61no—
147C9⋊C8—21no—
148#481—271noSylow
149C2×C3×C9—81no—
150C2×C3×C9#12311yescharacteristic
151C2×C3×C9C2×SD1611yescharacteristic
152C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
153C3⋊D9—22no—
154C2×C3×D4—21no—
155C2×C3×D4C2×D911yescharacteristic
156C2×C3×D4D3211yescharacteristic
157C2×#67—181no—
158C2×#67—181no—
159C2×#67—182no—
160C2×C4×D3—91no—
161C2×C4×D3—91no—
162C2×C4×D3—181no—
163C2×C4×D3—181no—
164C2×C4×D3—181no—
165C2×C3×C8—31no—
166C2×C3×C8—91no—
167C3×C4⋊C4—31no—
168C3×C4⋊C4—91no—
169#217—181no—
170#217—181no—
171#217—182no—
172C4⋊Dic3—31no—
173C4⋊Dic3—91no—
174C4⋊Dic3—91no—
175C4⋊Dic3—181no—
176C4⋊Dic3C2×D911yescharacteristic
177Dic3⋊C4—181no—
178Dic3⋊C4—181no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8C2×D911yescharacteristic
181C22×C32—41no—
182C22×C32—41no—
183C22×C32—41no—
184C22×C32D3×D411yescharacteristic
185C2×C3⋊D3—32no—
186C2×C3⋊D3—61no—
187C2×C3⋊D3—61no—
188C32×C4#23511yescharacteristic
189C32×C4#23711yescharacteristic
190C3⋊Dic3—61no—
191C3⋊Dic3—121no—
192C3×Dic3—41no—
193C3×Dic3—121no—
194C22×C9—21no—
195C22×C9—42no—
196C22×C9—42no—
197C22×C9—42no—
198C22×C9—41no—
199C22×C9—41no—
200C22×C9—41no—
201C22×C9D3×D411yescharacteristic
202C2×D9—32no—
203C2×D9—62no—
204C2×D9—61no—
205C2×D9—61no—
206C2×D9—121no—
207C2×D9—121no—
208C4×C9—21no—
209C4×C9—21no—
210C4×C9—121no—
211C4×C9#23511yescharacteristic
212C4×C9#23711yescharacteristic
213Dic9—41no—
214Dic9—61no—
215Dic9—121no—
216Dic9—122no—
217Dic9—121no—
218#121—271no—
219C2×C4⋊C4—271no—
220#106—271no—
221#106—271no—
222#102—31no—
223#102—91no—
224C22⋊C8—271no—
225C3×C9#48111yescharacteristic, complemented, Sylow
226C23×C3—22no—
227C23×C3—21no—
228C23×C3—21no—
229C22×D3—91no—
230C22×D3—91no—
231C22×D3—181no—
232C3×D4—21no—
233C3×D4—21no—
234C3×D4—21no—
235C3×D4—21no—
236C3×D4—41no—
237C3×D4—41no—
238C2×C3×C4—21no—
239C2×C3×C4—61no—
240C2×C3×C4—181no—
241C2×C3×C4C22×D911yescharacteristic
242C2×C3×C4C2×D3211yescharacteristic
243#67—362no—
244#67—362no—
245#67—364no—
246C2×Dic3—21no—
247C2×Dic3—61no—
248C2×Dic3—91no—
249C2×Dic3—91no—
250C2×Dic3—181no—
251C2×Dic3—182no—
252C2×Dic3—181no—
253C2×Dic3—181no—
254C4×D3—181no—
255C4×D3—181no—
256C4×D3—181no—
257C4×D3—181no—
258C4×D3—361no—
259C4×D3—361no—
260C4×D3—362no—
261C4×D3—362no—
262C3×C8—61no—
263C3×C8—181no—
264C3⋊C8—21no—
265C3⋊C8—61no—
266C2×C32—81no—
267C2×C32#88311yescharacteristic
268C2×C32#91111yescharacteristic
269C2×C32D3×SD1611yescharacteristic
270C3⋊D3—62no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9—21no—
274C2×C9—82no—
275C2×C9—81no—
276C2×C9#88311yescharacteristic
277C2×C9#91111yescharacteristic
278C2×C9D3×SD1611yescharacteristic
279D9—62no—
280D9—122no—
281C2×D4—541no—
282C2×D4D3×D911yescharacteristic
283C22×C4—271no—
284C22×C4—541no—
285C2×C8—31no—
286C2×C8—91no—
287C4⋊C4—31no—
288C4⋊C4—91no—
289C4⋊C4—271no—
290C4⋊C4—541no—
291C22⋊C4—541no—
292C22×C3—21no—
293C22×C3—42no—
294C22×C3—42no—
295C22×C3—42no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3D4×D911yescharacteristic
303C22×C3#381111yescharacteristic
304C2×D3—92no—
305C2×D3—92no—
306C2×D3—182no—
307C2×D3—181no—
308C2×D3—181no—
309C2×D3—181no—
310C2×D3—181no—
311C2×D3—361no—
312C2×D3—361no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—121no—
316C3×C4—361no—
317C3×C4#369711yescharacteristic
318C3×C4#369911yescharacteristic
319C3×C4#380411yescharacteristic
320C3×C4#380411yescharacteristic
321Dic3—41no—
322Dic3—121no—
323Dic3—181no—
324Dic3—181no—
325Dic3—361no—
326Dic3—362no—
327Dic3—361no—
328Dic3—361no—
329C32#477711yescharacteristic
330C9—21no—
331C9#477711yescharacteristic, complemented
332C23—21no—
333C23—271no—
334D4—21no—
335D4—21no—
336D4—1082no—
337C2×C4—61no—
338C2×C4—181no—
339C2×C4—271no—
340C2×C4—541no—
341C2×C4—541no—
342C2×C4—541no—
343C2×C4—1082no—
344C2×C4C2×D3×D911yescharacteristic
345C8—61no—
346C8—181no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—82no—
351C2×C3—81no—
352C2×C3—81no—
353C2×C3#6344511yescharacteristic
354C2×C3#6347311yescharacteristic
355C2×C3SD16×D911yescharacteristic
356C2×C3#6392711yescharacteristic
357C2×C3#6393011yescharacteristic
358C2×C3#6397511yescharacteristic
359D3—182no—
360D3—182no—
361D3—362no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—272no—
366C22—541no—
367C22—541no—
368C22#8866811yescharacteristic
369C4—121no—
370C4—361no—
371C4—541no—
372C4—1081no—
373C4#8865811yescharacteristic
374C4#8866211yescharacteristic
375C3—21no—
376C3#1058845711yescharacteristic, complemented
377C3#1059141911yescharacteristic
378C2—81no—
379C2—542no—
380C2#1172117511yescharacteristic
381C2#1172118211yescharacteristic
382C2#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#481
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. #11721000C2semidirect
3. C9×#832C2semidirect
4. C3×#63392C2semidirect
5. #11720282C2semidirect
6. #11720278C2semidirect
7. #11720264C2semidirect
32. #63392D3semidirect
101. #832D9semidirect
225. C3×C9#481semidirect
331. C9#4777semidirect
376. C3#10588457semidirect

Maximal subgroups

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

subgroupindex
#117218382
#117210002
C9×#8322
C3×#633922
#117202822
#117202782
#117202642
#105914193
#105884573