Finite Group Databasebeta

← #13137811

Subgroups — #13137811

Order 1728.

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

normal subgroups only

This group has 466 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#11721749C211yesmaximal, characteristic, complemented
2#11720951C211yesmaximal, characteristic, complemented
3C2×#88426C211yesmaximal, characteristic, complemented
4C2×#88417C211yesmaximal, characteristic
5C9×#810C211yesmaximal, characteristic, complemented
6C3×#63369C211yesmaximal, characteristic, complemented
7#11720273C211yesmaximal, characteristic, complemented
8#10590540—31nomaximal
9#10588499—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C4×C3⋊D9—21no—
12C2×C3×D36C2211yescharacteristic
13C2×C9×Dic6C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15#88536—21no—
16C4×C3⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19(C2×D9)⋊Dic3—21no—
20(C2×D9)⋊Dic3—21no—
21#88426—22no—
22#88417—22no—
23#64102—31no—
24#63863—31no—
25C2×#3717—31no—
26C2×#3714—31no—
27C3×#810D311yescharacteristic
28C3×#809—31no—
29#63572—31no—
30#63509—31no—
31C2×#3675—31no—
32C2×D72—31no—
33#63455—31no—
34#63455—61no—
35C9×#103—31no—
36Dic9⋊C8—31no—
37#63369D311yescharacteristic, complemented
38C22×C3⋊D9—21no—
39C22×C3×D9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9D411yescharacteristic
43C2×C3⋊Dic9D411yescharacteristic
44C4⋊(C3⋊D9)—21no—
45C4⋊(C3⋊D9)D412yes—
46C4×C3⋊D9—42no—
47C2×C9×Dic3—21no—
48C3×D36—21no—
49C3×D36—21no—
50C9×Dic6—21no—
51C9×Dic6—21no—
52C9×C3⋊C8—21no—
53C3×C9⋊C8—21no—
54#4819—91no—
55#4787—91no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4×C3⋊D3—61no—
58C2×C3×D12—31no—
59C2×C3×Dic6C2×D311yescharacteristic
60C32×C4⋊C4C2×D311yescharacteristic
61#3752—61no—
62C4×C3⋊Dic3—31no—
63C2×C3×C3⋊C8—31no—
64C2×C3×C3⋊C8C2×D311yescharacteristic
65(C2×D3)⋊Dic3—61no—
66(C2×D3)⋊Dic3—61no—
67#3717—62no—
68#3714—62no—
69C2×Q8×C9—31no—
70C2×D36—31no—
71C2×D36—61no—
72C2×D36C2×D311yescharacteristic
73C2×C4×D9—61no—
74C2×C4×D9—62no—
75C2×C8×C9—31no—
76C9×C4⋊C4—21no—
77C9×C4⋊C4C2×D311yescharacteristic
78#3675—62no—
79(C2×D9)⋊C4—61no—
80(C2×D9)⋊C4—61no—
81(C2×D9)⋊C4—61no—
82(C2×D9)⋊C4—62no—
83C4×Dic9—31no—
84C4×Dic9—61no—
85C2×C9⋊C8C2×D311yescharacteristic
86D72—62no—
87C22×C3×C9C2×D411yescharacteristic
88C2×C3⋊D9—22no—
89C2×C3⋊D9—41no—
90C2×C3⋊D9—41no—
91C2×C3×D9—41no—
92C2×C3×D9—41no—
93C2×C3×D9—41no—
94C3×C4×C9—41no—
95C3×C4×C9C2×D411yescharacteristic
96C3×C4×C9#4111yescharacteristic
97C3⋊Dic9—21no—
98C3⋊Dic9—21no—
99C9×Dic3—41no—
100#947—91no—
101C2×#215—91no—
102#936—91no—
103C2×#213—91no—
104C2×D24—91no—
105C2×#204—91no—
106#893—91no—
107#893—91no—
108#893—181no—
109C3×#103—31no—
110C3×#102—91no—
111Dic3⋊C8—91no—
112Dic3⋊C8—91no—
113#810D911yescharacteristic, complemented
114#809—31no—
115C22×C3⋊D3—61no—
116C22×C3×D3—61no—
117C2×C32×C4—21no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—31no—
120C2×C3⋊Dic3—31no—
121C4⋊(C3⋊D3)—32no—
122C4⋊(C3⋊D3)—61no—
123C4×C3⋊D3—122no—
124C2×C3×Dic3—21no—
125C3×D12—61no—
126C3×D12—61no—
127C3×Dic6—21no—
128C3×Dic6—21no—
129C22×D9—21no—
130C22×D9—62no—
131C22×D9—61no—
132C3×C3⋊C8—21no—
133C3×C3⋊C8—61no—
134Q8×C9—61no—
135Q8×C9—61no—
136C2×C4×C9—22no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—61no—
140C2×C4×C9C22×D311yescharacteristic
141C2×Dic9—31no—
142C2×Dic9—31no—
143C2×Dic9—61no—
144C2×Dic9—61no—
145D36—21no—
146D36—21no—
147D36—32no—
148D36—62no—
149D36—62no—
150D36—61no—
151C4×D9—124no—
152C4×D9—122no—
153C8×C9—61no—
154C9⋊C8—21no—
155#463—271noSylow
156C2×C3×C9#12111yescharacteristic
157C2×C3×C9#13511yescharacteristic
158C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
159C3⋊D9—42no—
160C3×D9—81no—
161C2×C3×Q8—31no—
162C2×C3×D4—91no—
163C2×D12—31no—
164C2×D12—91no—
165C2×D12—91no—
166C2×D12—181no—
167C2×C4×D3—181no—
168C2×C4×D3—181no—
169C2×C4×D3—182no—
170C2×Dic6C2×D911yescharacteristic
171C2×C3×C8—31no—
172C2×C3×C8—91no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4C2×D911yescharacteristic
175C3×C4⋊C4D3211yescharacteristic
176#217—181no—
177#217—181no—
178#215—182no—
179#213—182no—
180(C2×D3)⋊C4—181no—
181(C2×D3)⋊C4—181no—
182(C2×D3)⋊C4—181no—
183(C2×D3)⋊C4—181no—
184(C2×D3)⋊C4—182no—
185C4×Dic3—91no—
186C4×Dic3—91no—
187C4×Dic3—181no—
188C2×C3⋊C8—31no—
189C2×C3⋊C8C2×D911yescharacteristic
190D24—182no—
191#204—182no—
192C22×C32D3×D411yescharacteristic
193C2×C3⋊D3—62no—
194C2×C3⋊D3—121no—
195C2×C3⋊D3—121no—
196C2×C3×D3—121no—
197C2×C3×D3—121no—
198C2×C3×D3—121no—
199C32×C4—41no—
200C32×C4#23511yescharacteristic
201C32×C4D3×D411yescharacteristic
202C3⋊Dic3—61no—
203C3⋊Dic3—61no—
204C3×Dic3—41no—
205C22×C9—21no—
206C22×C9D3×D411yescharacteristic
207C2×D9—41no—
208C2×D9—41no—
209C2×D9—41no—
210C2×D9—64no—
211C2×D9—62no—
212C2×D9—122no—
213C2×D9—122no—
214C2×D9—121no—
215C2×D9—121no—
216C4×C9—21no—
217C4×C9—21no—
218C4×C9—42no—
219C4×C9—41no—
220C4×C9—121no—
221C4×C9#23511yescharacteristic
222C4×C9D3×D411yescharacteristic
223Dic9—62no—
224Dic9—61no—
225Dic9—61no—
226Dic9—121no—
227C2×SD16—271no—
228C2×D8—271no—
229#124—271no—
230C4×D4—271no—
231C4⋊C8—271no—
232#103—31no—
233#102—91no—
234C3×C9#46311yescharacteristic, complemented, Sylow
235C23×C3—181no—
236C22×D3—61no—
237C22×D3—182no—
238C22×D3—181no—
239C22×D3—181no—
240C3×Q8—61no—
241C3×Q8—61no—
242C3×D4—181no—
243C3×D4—181no—
244C2×C3×C4—22no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—61no—
249C2×C3×C4C22×D911yescharacteristic
250C2×C3×C4C2×D3211yescharacteristic
251C2×Dic3—21no—
252C2×Dic3—91no—
253C2×Dic3—91no—
254C2×Dic3—91no—
255C2×Dic3—91no—
256C2×Dic3—181no—
257C2×Dic3—181no—
258D12—61no—
259D12—61no—
260D12—92no—
261D12—92no—
262D12—182no—
263D12—182no—
264D12—181no—
265D12—181no—
266C4×D3—362no—
267C4×D3—364no—
268C4×D3—362no—
269Dic6—21no—
270Dic6—21no—
271C3×C8—61no—
272C3×C8—181no—
273C3⋊C8—21no—
274C3⋊C8—61no—
275C2×C32#88411yescharacteristic
276C2×C32#91411yescharacteristic
277C2×C32#91911yescharacteristic
278C3⋊D3—122no—
279C3×D3—241no—
280C2×C9—21no—
281C2×C9—21no—
282C2×C9—21no—
283C2×C9#88411yescharacteristic
284C2×C9#91111yescharacteristic
285C2×C9#91611yescharacteristic
286D9—81no—
287D9—124no—
288D9—122no—
289C2×Q8—31no—
290C2×D4—91no—
291C2×D4—271no—
292C22×C4—541no—
293SD16—542no—
294D8—542no—
295C2×C8—31no—
296C2×C8—91no—
297C4⋊C4D3×D911yescharacteristic
298C22⋊C4—541no—
299C22⋊C4—541no—
300C22⋊C4—541no—
301C42—271no—
302C22×C3—21no—
303C22×C3—361no—
304C22×C3—361no—
305C22×C3—361no—
306C22×C3D4×D911yescharacteristic
307C22×C3#381111yescharacteristic
308C2×D3—121no—
309C2×D3—121no—
310C2×D3—121no—
311C2×D3—184no—
312C2×D3—182no—
313C2×D3—182no—
314C2×D3—362no—
315C2×D3—362no—
316C2×D3—361no—
317C2×D3—361no—
318C2×D3—361no—
319C2×D3—361no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—42no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4—121no—
326C3×C4#369711yescharacteristic
327C3×C4D4×D911yescharacteristic
328C3×C4#379811yescharacteristic
329C3×C4#380211yescharacteristic
330Dic3—41no—
331Dic3—182no—
332Dic3—181no—
333Dic3—181no—
334Dic3—181no—
335Dic3—181no—
336Dic3—361no—
337C32#481911yescharacteristic
338C9—21no—
339C9#478711yescharacteristic, complemented
340C23—181no—
341C23—541no—
342Q8—61no—
343Q8—61no—
344D4—181no—
345D4—181no—
346D4—272no—
347D4—541no—
348C2×C4—21no—
349C2×C4—61no—
350C2×C4—271no—
351C2×C4—271no—
352C2×C4—1082no—
353C2×C4C2×D3×D911yescharacteristic
354C8—61no—
355C8—181no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—721no—
360C2×C3#6344611yescharacteristic
361C2×C3#6347611yescharacteristic
362C2×C3#6348111yescharacteristic
363C2×C3#6391211yescharacteristic
364C2×C3#6393711yescharacteristic
365C2×C3#6395011yescharacteristic
366D3—241no—
367D3—364no—
368D3—362no—
369D3—362no—
370C22—361no—
371C22—361no—
372C22—361no—
373C22—542no—
374C22—1081no—
375C22—1081no—
376C22#8866811yescharacteristic
377C4—41no—
378C4—121no—
379C4—541no—
380C4—541no—
381C4#8864011yescharacteristic
382C4#8864611yescharacteristic
383C3—21no—
384C3#1058849911yescharacteristic, complemented
385C3#1059054011yescharacteristic
386C2—721no—
387C2—1082no—
388C2#1172104111yescharacteristic
389C2#1172118711yescharacteristic
390C2#1172122411yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×#88417C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#463
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. #11721749C2semidirect
2. #11720951C2semidirect
3. C2×#88426C2semidirect
5. C9×#810C2semidirect
6. C3×#63369C2semidirect
7. #11720273C2semidirect
37. #63369D3semidirect
113. #810D9semidirect
234. C3×C9#463semidirect
339. C9#4787semidirect
384. C3#10588499semidirect

Maximal subgroups

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

subgroupindex
#117217492
#117209512
C2×#884262
C2×#884172
C9×#8102
C3×#633692
#117202732
#105905403
#105884993