Finite Group Databasebeta

← #13140334

Subgroups — #13140334

Order 1728.

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

normal subgroups only

This group has 438 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#11721840C211yesmaximal, characteristic
2#11720937C211yesmaximal, characteristic, complemented
3C9×#832C211yesmaximal, characteristic, complemented
4C3×#63392C211yesmaximal, characteristic, complemented
5#11720284C211yesmaximal, characteristic, complemented
6#11720275C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic, complemented
8#10591480—31nomaximal
9#10588444—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×#6233—21no—
13C2×C4⋊(C3⋊D9)C2211yescharacteristic
14C4×C3⋊Dic9C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20Dic9⋊Dic3—21no—
21#64149—31no—
22#63872—31no—
23C3×#832—31no—
24C3×#832D311yescharacteristic
25#63564—31no—
26#63564—31no—
27#63555—31no—
28#63502—31no—
29#63502—61no—
30#63452—31no—
31C9×#102—31no—
32#63392D311yescharacteristic, complemented
33#63380—31no—
34#63375—31no—
35#63369—31no—
36C23×C3×C9—21no—
37C22×C3⋊D9—21no—
38C3×D4×C9—21no—
39C3×D4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41#6233—42no—
42#6233—42no—
43C2×C3⋊Dic9D411yescharacteristic
44C2×C3⋊Dic9D411yescharacteristic
45C4⋊(C3⋊D9)—21no—
46C4⋊(C3⋊D9)—21no—
47C2×C9×Dic3—21no—
48C2×C3×Dic9—21no—
49C9×C3⋊C8—21no—
50C3×C9⋊C8—21no—
51#4764—91no—
52#4764—91no—
53C2×C32×D4C2×D311yescharacteristic
54C2×#638—61no—
55C2×#638—61no—
56C2×C4⋊(C3⋊D3)—31no—
57C4×C3⋊Dic3—31no—
58C3×C4⋊Dic3—31no—
59C3×C4⋊Dic3C2×D311yescharacteristic
60C2×C3×C3⋊C8—31no—
61C2×C3×C3⋊C8C2×D311yescharacteristic
62Dic3⋊Dic3—61no—
63Dic3⋊Dic3—61no—
64C2×D4×C9—21no—
65C2×D4×C9C2×D311yescharacteristic
66C2×#611—61no—
67C2×#611—61no—
68C2×#611—62no—
69C2×#611—62no—
70C2×D36—31no—
71C2×D36—61no—
72C2×C8×C9—31no—
73C9×C4⋊C4—31no—
74C4⋊Dic9—61no—
75C4⋊Dic9C2×D311yescharacteristic
76Dic9⋊C4—61no—
77C4×Dic9—31no—
78C4×Dic9—61no—
79C2×C9⋊C8C2×D311yescharacteristic
80C22×C3×C9—41no—
81C22×C3×C9—41no—
82C22×C3×C9—41no—
83C22×C3×C9C2×D411yescharacteristic
84C2×C3⋊D9—41no—
85C2×C3⋊D9—41no—
86C2×C3⋊D9—41no—
87C3×C4×C9#4111yescharacteristic
88C3×C4×C9#4111yescharacteristic
89C3⋊Dic9—21no—
90C3⋊Dic9—21no—
91C9×Dic3—41no—
92C3×Dic9—41no—
93#940—91no—
94#940—91no—
95#940—181no—
96#890—91no—
97#890—91no—
98C3×#102—31no—
99C3×#102—91no—
100#832—31no—
101#832D911yescharacteristic, complemented
102#821—91no—
103#821—91no—
104#815—91no—
105#815—91no—
106#809—91no—
107#809—91no—
108C23×C32—21no—
109C22×C3⋊D3—61no—
110C32×D4—21no—
111C32×D4—21no—
112C2×C32×C4C22×D311yescharacteristic
113#638—122no—
114#638—122no—
115C2×C3⋊Dic3—31no—
116C2×C3⋊Dic3—31no—
117C4⋊(C3⋊D3)—61no—
118C4⋊(C3⋊D3)—61no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—61no—
121C23×C9—22no—
122C23×C9—21no—
123C22×D9—62no—
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—124no—
135#611—124no—
136#611—122no—
137#611—122no—
138C2×Dic9—21no—
139C2×Dic9—31no—
140C2×Dic9—31no—
141C2×Dic9—61no—
142C2×Dic9—61no—
143D36—61no—
144D36—61no—
145D36—121no—
146D36—121no—
147C8×C9—61no—
148C9⋊C8—21no—
149#490—271noSylow
150C2×C3×C9—81no—
151C2×C3×C9#12411yescharacteristic
152C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
153C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
154C3⋊D9—81no—
155C2×C3×D4—21no—
156C2×C3×D4C2×D911yescharacteristic
157C2×C3×D4D3211yescharacteristic
158C2×#67—181no—
159C2×#67—181no—
160C2×#67—181no—
161C2×#67—181no—
162C2×#67—182no—
163C2×#67—182no—
164C2×D12—91no—
165C2×D12—91no—
166C2×D12—181no—
167C2×C3×C8—31no—
168C2×C3×C8—91no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—91no—
171C4⋊Dic3—31no—
172C4⋊Dic3—181no—
173C4⋊Dic3—181no—
174C4⋊Dic3C2×D911yescharacteristic
175Dic3⋊C4—181no—
176Dic3⋊C4—181no—
177C4×Dic3—91no—
178C4×Dic3—91no—
179C4×Dic3—181no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8C2×D911yescharacteristic
182C22×C32—41no—
183C22×C32—41no—
184C22×C32—41no—
185C22×C32D3×D411yescharacteristic
186C2×C3⋊D3—121no—
187C2×C3⋊D3—121no—
188C2×C3⋊D3—121no—
189C32×C4#23511yescharacteristic
190C32×C4#23711yescharacteristic
191C3⋊Dic3—61no—
192C3⋊Dic3—61no—
193C3×Dic3—41no—
194C3×Dic3—121no—
195C22×C9—21no—
196C22×C9—42no—
197C22×C9—42no—
198C22×C9—42no—
199C22×C9—41no—
200C22×C9—41no—
201C22×C9—41no—
202C22×C9D3×D411yescharacteristic
203C2×D9—122no—
204C2×D9—122no—
205C2×D9—122no—
206C2×D9—121no—
207C2×D9—121no—
208C2×D9—121no—
209C4×C9—21no—
210C4×C9—21no—
211C4×C9—121no—
212C4×C9#23511yescharacteristic
213C4×C9#23711yescharacteristic
214Dic9—41no—
215Dic9—62no—
216Dic9—62no—
217Dic9—61no—
218Dic9—61no—
219#127—271no—
220#125—271no—
221#102—31no—
222#102—91no—
223#102—271no—
224#102—271no—
225#97—271no—
226C3×C9#49011yescharacteristic, complemented, Sylow
227C23×C3—22no—
228C23×C3—21no—
229C23×C3—21no—
230C22×D3—182no—
231C22×D3—181no—
232C22×D3—181no—
233C3×D4—21no—
234C3×D4—21no—
235C3×D4—21no—
236C3×D4—21no—
237C3×D4—41no—
238C3×D4—41no—
239C2×C3×C4—21no—
240C2×C3×C4—61no—
241C2×C3×C4—181no—
242C2×C3×C4C22×D911yescharacteristic
243C2×C3×C4C2×D3211yescharacteristic
244#67—362no—
245#67—362no—
246#67—364no—
247#67—364no—
248#67—362no—
249#67—362no—
250C2×Dic3—21no—
251C2×Dic3—61no—
252C2×Dic3—91no—
253C2×Dic3—91no—
254C2×Dic3—91no—
255C2×Dic3—91no—
256C2×Dic3—181no—
257C2×Dic3—181no—
258D12—181no—
259D12—181no—
260D12—181no—
261D12—181no—
262D12—361no—
263D12—361no—
264C3×C8—61no—
265C3×C8—181no—
266C3⋊C8—21no—
267C3⋊C8—61no—
268C2×C32—81no—
269C2×C32#88511yescharacteristic
270C2×C32#91111yescharacteristic
271C2×C32#91411yescharacteristic
272C3⋊D3—241no—
273C2×C9—21no—
274C2×C9—21no—
275C2×C9—21no—
276C2×C9—82no—
277C2×C9—81no—
278C2×C9#88511yescharacteristic
279C2×C9#91111yescharacteristic
280C2×C9#91411yescharacteristic
281D9—242no—
282D9—241no—
283C2×D4—271no—
284C2×D4—541no—
285C2×D4—541no—
286C2×D4D3×D911yescharacteristic
287C2×C8—31no—
288C2×C8—91no—
289C4⋊C4—31no—
290C4⋊C4—91no—
291C4⋊C4—541no—
292C4⋊C4—541no—
293C42—271no—
294C22×C3—21no—
295C22×C3—42no—
296C22×C3—42no—
297C22×C3—42no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3D4×D911yescharacteristic
305C22×C3#381111yescharacteristic
306C2×D3—362no—
307C2×D3—362no—
308C2×D3—362no—
309C2×D3—361no—
310C2×D3—361no—
311C2×D3—361no—
312C2×D3—361no—
313C2×D3—361no—
314C2×D3—361no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—121no—
318C3×C4—361no—
319C3×C4#369711yescharacteristic
320C3×C4#369911yescharacteristic
321C3×C4#380411yescharacteristic
322C3×C4#380411yescharacteristic
323Dic3—41no—
324Dic3—121no—
325Dic3—182no—
326Dic3—182no—
327Dic3—181no—
328Dic3—181no—
329Dic3—181no—
330Dic3—181no—
331C32#476411yescharacteristic
332C9—21no—
333C9#476411yescharacteristic, complemented
334C23—21no—
335C23—541no—
336D4—21no—
337D4—21no—
338D4—541no—
339D4—541no—
340D4—1082no—
341D4—1082no—
342C2×C4—61no—
343C2×C4—181no—
344C2×C4—271no—
345C2×C4—271no—
346C2×C4C2×D3×D911yescharacteristic
347C8—61no—
348C8—181no—
349C2×C3—21no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—82no—
353C2×C3—81no—
354C2×C3—81no—
355C2×C3#6344711yescharacteristic
356C2×C3#6347311yescharacteristic
357C2×C3#6347611yescharacteristic
358C2×C3#6393711yescharacteristic
359C2×C3#6393711yescharacteristic
360C2×C3#6395311yescharacteristic
361D3—722no—
362D3—721no—
363D3—721no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—1081no—
368C22—1081no—
369C22—1081no—
370C22#8866811yescharacteristic
371C4—121no—
372C4—361no—
373C4—541no—
374C4—541no—
375C4#8865811yescharacteristic
376C4#8866211yescharacteristic
377C3—21no—
378C3#1058844411yescharacteristic, complemented
379C3#1059148011yescharacteristic
380C2—81no—
381C2—2161no—
382C2#1172117911yescharacteristic
383C2#1172118711yescharacteristic
384C2#1172124211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radical#11721840C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#490
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. #11720937C2semidirect
3. C9×#832C2semidirect
4. C3×#63392C2semidirect
5. #11720284C2semidirect
6. #11720275C2semidirect
7. #11720267C2semidirect
32. #63392D3semidirect
101. #832D9semidirect
226. C3×C9#490semidirect
333. C9#4764semidirect
378. C3#10588444semidirect

Maximal subgroups

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

subgroupindex
#117218402
#117209372
C9×#8322
C3×#633922
#117202842
#117202752
#117202672
#105914803
#105884443