Finite Group Databasebeta

← #13137804

Subgroups — #13137804

Order 1728.

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

normal subgroups only

This group has 404 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
1C4⋊C4×C3⋊D9C211yesmaximal, characteristic
2#11720929C211yesmaximal, characteristic
3C9×#810C211yesmaximal, characteristic, complemented
4C3×#63367C211yesmaximal, characteristic, complemented
5#11720298C211yesmaximal, characteristic, complemented
6#11720282C211yesmaximal, characteristic
7#11720278C211yesmaximal, characteristic, complemented
8#10590529—31nomaximal
9#10588485—31nomaximal
10C2×C4×C3⋊D9—21no—
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15#88534—21no—
16C3×C4⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19#88445—21no—
20Dic9⋊Dic3—21no—
21C4⋊C4×C3⋊D3—31no—
22#63871—31no—
23C3×#810D311yescharacteristic
24C3×#808—31no—
25#63573—31no—
26#63566—31no—
27#63557—31no—
28#63508—31no—
29C4⋊C4×D9—31no—
30C4⋊C4×D9—61no—
31C9×#103—31no—
32#63395—31no—
33(C2×D9)⋊C8—31no—
34#63377—31no—
35#63367D311yescharacteristic, complemented
36C22×C3⋊D9D411yescharacteristic
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9D411yescharacteristic
41C4×C3⋊D9—21no—
42C4×C3⋊D9—21no—
43C4×C3⋊D9—42no—
44C2×C9×Dic3—21no—
45C2×C3×Dic9—21no—
46C9×Dic6—21no—
47C9×Dic6—21no—
48C9×C3⋊C8—21no—
49C3×C9⋊C8—21no—
50#4862—91no—
51#4805—91no—
52C2×C4×C3⋊D3—31no—
53C2×C4×C3⋊D3—61no—
54C2×C3×Dic6C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56C4⋊(C3⋊Dic3)—31no—
57#3750—61no—
58C3×C4⋊Dic3—31no—
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊C8C2×D311yescharacteristic
61Dic3⋊Dic3—61no—
62#3722—61no—
63C2×Q8×C9—31no—
64C2×C4×D9—31no—
65C2×C4×D9—61no—
66C2×C4×D9—61no—
67C2×C4×D9—62no—
68C2×C8×C9—31no—
69C9×C4⋊C4—21no—
70C9×C4⋊C4C2×D311yescharacteristic
71(C2×D9)⋊C4—61no—
72C4⋊Dic9—31no—
73C4⋊Dic9—61no—
74C4⋊Dic9C2×D311yescharacteristic
75Dic9⋊C4—61no—
76Dic9⋊C4—61no—
77Dic9⋊C4—62no—
78C2×C9⋊C8C2×D311yescharacteristic
79C22×C3×C9C2×D411yescharacteristic
80C2×C3⋊D9—21no—
81C2×C3⋊D9—21no—
82C2×C3⋊D9SD1612yes—
83C3×C4×C9—41no—
84C3×C4×C9#4111yescharacteristic
85C3×C4×C9#4111yescharacteristic
86C3⋊Dic9—21no—
87C3⋊Dic9—41no—
88C9×Dic3—41no—
89C3×Dic9—41no—
90#946—91no—
91#897—91no—
92D3×C4⋊C4—91no—
93D3×C4⋊C4—91no—
94D3×C4⋊C4—181no—
95C3×#106—91no—
96C3×#103—31no—
97#835—91no—
98(C2×D3)⋊C8—91no—
99(C2×D3)⋊C8—91no—
100#817—91no—
101#816—91no—
102#810D911yescharacteristic, complemented
103#808—31no—
104#808—91no—
105C22×C3⋊D3—31no—
106C2×C32×C4—21no—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—61no—
110C4×C3⋊D3—61no—
111C4×C3⋊D3—61no—
112C4×C3⋊D3—122no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—61no—
115C3×Dic6—21no—
116C3×Dic6—21no—
117C22×D9—31no—
118C22×D9—61no—
119C3×C3⋊C8—21no—
120C3×C3⋊C8—61no—
121Q8×C9—61no—
122Q8×C9—61no—
123C2×C4×C9—22no—
124C2×C4×C9—21no—
125C2×C4×C9—21no—
126C2×C4×C9—61no—
127C2×C4×C9C22×D311yescharacteristic
128C2×Dic9—21no—
129C2×Dic9—31no—
130C2×Dic9—61no—
131C2×Dic9—62no—
132C2×Dic9—61no—
133C4×D9—62no—
134C4×D9—62no—
135C4×D9—61no—
136C4×D9—61no—
137C4×D9—122no—
138C4×D9—124no—
139C8×C9—61no—
140C9⋊C8—21no—
141#483—271noSylow
142C2×C3×C9#12311yescharacteristic
143C2×C3×C9C2×SD1611yescharacteristic
144C2×C3×C9#13711yescharacteristic
145C3⋊D9—22no—
146C2×C3×Q8—31no—
147C2×C4×D3—91no—
148C2×C4×D3—91no—
149C2×C4×D3—181no—
150C2×C4×D3—181no—
151C2×C4×D3—181no—
152C2×C4×D3—182no—
153C2×Dic6C2×D911yescharacteristic
154C2×C3×C8—31no—
155C2×C3×C8—91no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—91no—
158C3×C4⋊C4C2×D911yescharacteristic
159C3×C4⋊C4D3211yescharacteristic
160(C2×D3)⋊C4—181no—
161(C2×D3)⋊C4—181no—
162C4⋊Dic3—31no—
163C4⋊Dic3—91no—
164C4⋊Dic3—91no—
165C4⋊Dic3—181no—
166C4⋊Dic3—181no—
167Dic3⋊C4—181no—
168Dic3⋊C4—181no—
169Dic3⋊C4—181no—
170Dic3⋊C4—182no—
171C2×C3⋊C8—31no—
172C2×C3⋊C8C2×D911yescharacteristic
173C22×C32D3×D411yescharacteristic
174C2×C3⋊D3—32no—
175C2×C3⋊D3—61no—
176C2×C3⋊D3—61no—
177C32×C4—41no—
178C32×C4#23511yescharacteristic
179C32×C4#23711yescharacteristic
180C3⋊Dic3—61no—
181C3⋊Dic3—121no—
182C3×Dic3—41no—
183C3×Dic3—121no—
184C22×C9—21no—
185C22×C9D3×D411yescharacteristic
186C2×D9—32no—
187C2×D9—62no—
188C2×D9—62no—
189C2×D9—62no—
190C2×D9—61no—
191C2×D9—61no—
192C4×C9—21no—
193C4×C9—21no—
194C4×C9—42no—
195C4×C9—41no—
196C4×C9—121no—
197C4×C9#23511yescharacteristic
198C4×C9#23911yescharacteristic
199Dic9—41no—
200Dic9—62no—
201Dic9—61no—
202Dic9—122no—
203Dic9—121no—
204C22⋊Q8—271no—
205C2×C4⋊C4—271no—
206#106—91no—
207#106—271no—
208#103—31no—
209#103—271no—
210C22⋊C8—271no—
211C3×C9#48311yescharacteristic, complemented, Sylow
212C22×D3—91no—
213C22×D3—91no—
214C22×D3—181no—
215C3×Q8—61no—
216C3×Q8—61no—
217C2×C3×C4—22no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—61no—
222C2×C3×C4—181no—
223C2×C3×C4C22×D911yescharacteristic
224C2×C3×C4C2×D3211yescharacteristic
225C2×Dic3—21no—
226C2×Dic3—61no—
227C2×Dic3—91no—
228C2×Dic3—91no—
229C2×Dic3—181no—
230C2×Dic3—182no—
231C2×Dic3—181no—
232C2×Dic3—181no—
233C4×D3—182no—
234C4×D3—182no—
235C4×D3—181no—
236C4×D3—181no—
237C4×D3—181no—
238C4×D3—181no—
239C4×D3—362no—
240C4×D3—362no—
241C4×D3—364no—
242Dic6—21no—
243Dic6—21no—
244C3×C8—61no—
245C3×C8—181no—
246C3⋊C8—21no—
247C3⋊C8—61no—
248C2×C32#88311yescharacteristic
249C2×C32D3×SD1611yescharacteristic
250C2×C32#91811yescharacteristic
251C3⋊D3—62no—
252C2×C9—21no—
253C2×C9—21no—
254C2×C9—21no—
255C2×C9#89411yescharacteristic
256C2×C9D3×SD1611yescharacteristic
257C2×C9#91511yescharacteristic
258D9—64no—
259D9—62no—
260C2×Q8—31no—
261C22×C4—271no—
262C22×C4—541no—
263C2×C8—31no—
264C2×C8—91no—
265C4⋊C4—91no—
266C4⋊C4—271no—
267C4⋊C4—541no—
268C4⋊C4—541no—
269C4⋊C4D3×D911yescharacteristic
270C22⋊C4—541no—
271C22×C3—21no—
272C22×C3D4×D911yescharacteristic
273C22×C3#381111yescharacteristic
274C2×D3—92no—
275C2×D3—92no—
276C2×D3—182no—
277C2×D3—182no—
278C2×D3—182no—
279C2×D3—181no—
280C2×D3—181no—
281C2×D3—181no—
282C2×D3—181no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—42no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4—121no—
289C3×C4—361no—
290C3×C4#369711yescharacteristic
291C3×C4#369911yescharacteristic
292C3×C4#379611yescharacteristic
293C3×C4#379811yescharacteristic
294Dic3—41no—
295Dic3—121no—
296Dic3—182no—
297Dic3—181no—
298Dic3—181no—
299Dic3—362no—
300Dic3—361no—
301Dic3—361no—
302C32#480511yescharacteristic
303C9—21no—
304C9#486211yescharacteristic, complemented
305C23—271no—
306Q8—61no—
307Q8—61no—
308C2×C4—21no—
309C2×C4—61no—
310C2×C4—181no—
311C2×C4—271no—
312C2×C4—541no—
313C2×C4—541no—
314C2×C4—541no—
315C2×C4—1082no—
316C2×C4C2×D3×D911yescharacteristic
317C8—61no—
318C8—181no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3#6344511yescharacteristic
323C2×C3SD16×D911yescharacteristic
324C2×C3#6348011yescharacteristic
325C2×C3#6385311yescharacteristic
326C2×C3#6393011yescharacteristic
327C2×C3#6395111yescharacteristic
328D3—184no—
329D3—182no—
330D3—182no—
331C22—272no—
332C22—541no—
333C22—541no—
334C22#8866811yescharacteristic
335C4—41no—
336C4—121no—
337C4—361no—
338C4—541no—
339C4—1081no—
340C4#8863911yescharacteristic
341C4#8864011yescharacteristic
342C3—21no—
343C3#1058848511yescharacteristic, complemented
344C3#1059052911yescharacteristic
345C2—542no—
346C2#1172095711yescharacteristic
347C2#1172118211yescharacteristic
348C2#1172122511yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC22×C3⋊D9D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#483
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
3. C9×#810C2semidirect
4. C3×#63367C2semidirect
5. #11720298C2semidirect
7. #11720278C2semidirect
35. #63367D3semidirect
102. #810D9semidirect
211. C3×C9#483semidirect
304. C9#4862semidirect
343. C3#10588485semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×C3⋊D92
#117209292
C9×#8102
C3×#633672
#117202982
#117202822
#117202782
#105905293
#105884853