Finite Group Databasebeta

← #13137747

Subgroups — #13137747

Order 1728.

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

normal subgroups only

This group has 434 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, complemented
2#11721045C211yesmaximal, characteristic
3C9×#809C211yesmaximal, characteristic, complemented
4C3×#63366C211yesmaximal, characteristic, complemented
5#11720295C211yesmaximal, characteristic, complemented
6#11720282C211yesmaximal, characteristic, complemented
7#11720279C211yesmaximal, characteristic, complemented
8#10590510—31nomaximal
9#10588454—31nomaximal
10C2×C4×C3⋊D9—21no—
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×#6195—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16#88534—21no—
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20(C2×D3)⋊Dic9—21no—
21C4⋊C4×C3⋊D3—31no—
22#63916—31no—
23C3×#809D311yescharacteristic
24C3×#807—31no—
25#63575—31no—
26#63563—31no—
27#63557—31no—
28#63500—31no—
29C4⋊C4×D9—31no—
30C4⋊C4×D9—61no—
31C9×#102—31no—
32#63392—31no—
33(C2×D9)⋊C8—31no—
34#63378—31no—
35#63366D311yescharacteristic, complemented
36C22×C3⋊D9D411yescharacteristic
37C22×D3×C9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41C2×C3⋊Dic9D411yescharacteristic
42C4×C3⋊D9—21no—
43C4×C3⋊D9—21no—
44C4×C3⋊D9—42no—
45C2×C3×Dic9—21no—
46C9×D12—21no—
47C9×D12—21no—
48#6195—42no—
49C9×C3⋊C8—21no—
50C3×C9⋊C8—21no—
51#4882—91no—
52#4774—91no—
53C2×C4×C3⋊D3—31no—
54C2×C4×C3⋊D3—61no—
55C2×C3×D12C2×D311yescharacteristic
56C2×#626—61no—
57C32×C4⋊C4C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)—31no—
59#3750—61no—
60C3×C4⋊Dic3—31no—
61C2×C3×C3⋊C8—31no—
62C2×C3×C3⋊C8C2×D311yescharacteristic
63(C2×D3)⋊Dic3—61no—
64C2×D4×C9—31no—
65C2×#611—61no—
66C2×C4×D9—31no—
67C2×C4×D9—61no—
68C2×C4×D9—61no—
69C2×C4×D9—62no—
70C2×C8×C9—31no—
71C9×C4⋊C4—21no—
72C9×C4⋊C4C2×D311yescharacteristic
73#3676—61no—
74C4⋊Dic9—31no—
75C4⋊Dic9—61no—
76C4⋊Dic9C2×D311yescharacteristic
77Dic9⋊C4—61no—
78Dic9⋊C4—62no—
79C2×C9⋊C8C2×D311yescharacteristic
80C22×C3×C9C2×D411yescharacteristic
81C2×C3⋊D9—21no—
82C2×C3⋊D9—21no—
83C2×C3⋊D9D812yes—
84C2×D3×C9—41no—
85C2×D3×C9—41no—
86C2×D3×C9—41no—
87C3×C4×C9—41no—
88C3×C4×C9#4111yescharacteristic
89C3×C4×C9#4111yescharacteristic
90C3⋊Dic9—21no—
91C3⋊Dic9—41no—
92C3×Dic9—41no—
93#938—91no—
94#895—91no—
95D3×C4⋊C4—91no—
96D3×C4⋊C4—91no—
97D3×C4⋊C4—181no—
98C3×#107—91no—
99C3×#102—31no—
100#832—91no—
101#821—91no—
102(C2×D3)⋊C8—91no—
103(C2×D3)⋊C8—91no—
104#818—91no—
105#809D911yescharacteristic, complemented
106#807—31no—
107#807—91no—
108C22×C3⋊D3—31no—
109C22×C3×D3—21no—
110C2×C32×C4—21no—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—31no—
113C2×C3⋊Dic3—61no—
114C4×C3⋊D3—61no—
115C4×C3⋊D3—61no—
116C4×C3⋊D3—122no—
117C2×C3×Dic3—61no—
118C3×D12—21no—
119C3×D12—21no—
120#626—122no—
121C23×C9—61no—
122C22×D9—31no—
123C22×D9—61no—
124C3×C3⋊C8—21no—
125C3×C3⋊C8—61no—
126D4×C9—61no—
127D4×C9—61no—
128C2×C4×C9—22no—
129C2×C4×C9—21no—
130C2×C4×C9—21no—
131C2×C4×C9C22×D311yescharacteristic
132#611—122no—
133C2×Dic9—21no—
134C2×Dic9—31no—
135C2×Dic9—61no—
136C2×Dic9—62no—
137C2×Dic9—61no—
138C4×D9—62no—
139C4×D9—62no—
140C4×D9—61no—
141C4×D9—61no—
142C4×D9—122no—
143C4×D9—124no—
144C8×C9—61no—
145C9⋊C8—21no—
146#480—271noSylow
147C2×C3×C9#12311yescharacteristic
148C2×C3×C9C2×D811yescharacteristic
149C2×C3×C9#13711yescharacteristic
150C3⋊D9—22no—
151D3×C9—81no—
152C2×C3×D4—31no—
153C2×#67—181no—
154C2×D12—181no—
155C2×D12C2×D911yescharacteristic
156C2×C4×D3—91no—
157C2×C4×D3—91no—
158C2×C4×D3—181no—
159C2×C4×D3—181no—
160C2×C4×D3—181no—
161C2×C4×D3—182no—
162C2×C3×C8—31no—
163C2×C3×C8—91no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—91no—
166C3×C4⋊C4C2×D911yescharacteristic
167C3×C4⋊C4D3211yescharacteristic
168#217—181no—
169(C2×D3)⋊C4—181no—
170C4⋊Dic3—31no—
171C4⋊Dic3—91no—
172C4⋊Dic3—91no—
173C4⋊Dic3—181no—
174Dic3⋊C4—181no—
175Dic3⋊C4—181no—
176Dic3⋊C4—182no—
177C2×C3⋊C8—31no—
178C2×C3⋊C8C2×D911yescharacteristic
179C22×C32D3×D411yescharacteristic
180C2×C3⋊D3—32no—
181C2×C3⋊D3—61no—
182C2×C3⋊D3—61no—
183C2×C3×D3—41no—
184C2×C3×D3—41no—
185C2×C3×D3—41no—
186C32×C4—41no—
187C32×C4#23511yescharacteristic
188C32×C4#23711yescharacteristic
189C3⋊Dic3—61no—
190C3⋊Dic3—121no—
191C3×Dic3—121no—
192C22×C9—21no—
193C22×C9—121no—
194C22×C9—121no—
195C22×C9—121no—
196C22×C9D3×D411yescharacteristic
197C2×D9—32no—
198C2×D9—62no—
199C2×D9—62no—
200C2×D9—62no—
201C2×D9—61no—
202C2×D9—61no—
203C4×C9—21no—
204C4×C9—21no—
205C4×C9—42no—
206C4×C9—41no—
207C4×C9#23511yescharacteristic
208C4×C9#23911yescharacteristic
209Dic9—41no—
210Dic9—62no—
211Dic9—61no—
212Dic9—122no—
213Dic9—121no—
214#121—271no—
215C2×C4⋊C4—271no—
216#107—91no—
217#107—271no—
218#102—31no—
219#102—271no—
220C22⋊C8—271no—
221C3×C9#48011yescharacteristic, complemented, Sylow
222C23×C3—61no—
223C22×D3—21no—
224C22×D3—91no—
225C22×D3—91no—
226C22×D3—181no—
227C3×D4—61no—
228C3×D4—61no—
229C2×C3×C4—22no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—181no—
234C2×C3×C4C22×D911yescharacteristic
235C2×C3×C4C2×D3211yescharacteristic
236#67—362no—
237C2×Dic3—61no—
238C2×Dic3—91no—
239C2×Dic3—91no—
240C2×Dic3—181no—
241C2×Dic3—182no—
242C2×Dic3—181no—
243C2×Dic3—181no—
244D12—21no—
245D12—21no—
246D12—362no—
247C4×D3—182no—
248C4×D3—182no—
249C4×D3—181no—
250C4×D3—181no—
251C4×D3—181no—
252C4×D3—181no—
253C4×D3—362no—
254C4×D3—362no—
255C4×D3—364no—
256C3×C8—61no—
257C3×C8—181no—
258C3⋊C8—21no—
259C3⋊C8—61no—
260C2×C32#88311yescharacteristic
261C2×C32D3×D811yescharacteristic
262C2×C32#91511yescharacteristic
263C3⋊D3—62no—
264C3×D3—81no—
265C2×C9—21no—
266C2×C9—21no—
267C2×C9—21no—
268C2×C9—241no—
269C2×C9#89411yescharacteristic
270C2×C9D3×D811yescharacteristic
271C2×C9#91811yescharacteristic
272D9—64no—
273D9—62no—
274C2×D4—31no—
275C2×D4—541no—
276C22×C4—271no—
277C22×C4—541no—
278C2×C8—31no—
279C2×C8—91no—
280C4⋊C4—91no—
281C4⋊C4—271no—
282C4⋊C4—541no—
283C4⋊C4D3×D911yescharacteristic
284C22⋊C4—541no—
285C22×C3—21no—
286C22×C3—121no—
287C22×C3—121no—
288C22×C3—121no—
289C22×C3D4×D911yescharacteristic
290C22×C3#381111yescharacteristic
291C2×D3—41no—
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—92no—
295C2×D3—92no—
296C2×D3—182no—
297C2×D3—182no—
298C2×D3—182no—
299C2×D3—181no—
300C2×D3—181no—
301C2×D3—181no—
302C2×D3—181no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—42no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4—361no—
309C3×C4#369711yescharacteristic
310C3×C4#369911yescharacteristic
311C3×C4#379611yescharacteristic
312C3×C4#379811yescharacteristic
313Dic3—121no—
314Dic3—182no—
315Dic3—181no—
316Dic3—181no—
317Dic3—362no—
318Dic3—361no—
319Dic3—361no—
320C32#477411yescharacteristic
321C9—21no—
322C9#488211yescharacteristic, complemented
323C23—61no—
324C23—271no—
325D4—61no—
326D4—61no—
327D4—1082no—
328C2×C4—21no—
329C2×C4—181no—
330C2×C4—271no—
331C2×C4—541no—
332C2×C4—541no—
333C2×C4—541no—
334C2×C4—1082no—
335C2×C4C2×D3×D911yescharacteristic
336C8—61no—
337C8—181no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—241no—
342C2×C3#6344511yescharacteristic
343C2×C3D8×D911yescharacteristic
344C2×C3#6347711yescharacteristic
345C2×C3#6385311yescharacteristic
346C2×C3#6392611yescharacteristic
347C2×C3#6395111yescharacteristic
348D3—81no—
349D3—184no—
350D3—182no—
351D3—182no—
352C22—121no—
353C22—121no—
354C22—121no—
355C22—272no—
356C22—541no—
357C22—541no—
358C22#8866811yescharacteristic
359C4—41no—
360C4—361no—
361C4—541no—
362C4—1081no—
363C4#8863911yescharacteristic
364C4#8864011yescharacteristic
365C3—21no—
366C3#1058845411yescharacteristic, complemented
367C3#1059051011yescharacteristic
368C2—241no—
369C2—542no—
370C2#1172095711yescharacteristic
371C2#1172117411yescharacteristic
372C2#1172121711yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radical#11721045C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#480
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. C4⋊C4×C3⋊D9C2semidirect
3. C9×#809C2semidirect
4. C3×#63366C2semidirect
5. #11720295C2semidirect
6. #11720282C2semidirect
7. #11720279C2semidirect
35. #63366D3semidirect
105. #809D9semidirect
221. C3×C9#480semidirect
322. C9#4882semidirect
366. C3#10588454semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×C3⋊D92
#117210452
C9×#8092
C3×#633662
#117202952
#117202822
#117202792
#105905103
#105884543