Finite Group Databasebeta

← #13140416

Subgroups — #13140416

Order 1728.

532 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C3×#63710C211yesmaximal, characteristic
2C9×#935C211yesmaximal, characteristic, complemented
3#11721259C212yesmaximal, complemented
4#11720577C211yesmaximal, characteristic, complemented
5#11720329C212yesmaximal, complemented
6#10591533—31nomaximal
7#10588733—31nomaximal
8C2×C3×D4×C9C2211yescharacteristic
9C2×C3×#611—21no—
10C2×C3×#611C2212yes—
11C22×C9×Dic3C2211yescharacteristic, complemented
12C3×#3699—22no—
13C3×D4×D9—22no—
14C3×#3697—21no—
15#88551C2212yescomplemented
16C9×#217—21no—
17C9×#217C2211yescharacteristic, complemented
18C9×Dic3⋊C4—21no—
19(C2×D9)⋊Dic3—22no—
20#88442—22no—
21C3×#1004—31no—
22C3×#935D311yescharacteristic
23#63961—32no—
24#63710D311yescharacteristic
25#63661—31no—
26#63580—32no—
27C9×#123—31no—
28#63491—32no—
29#63393—31no—
30#63393—61no—
31#63365—32no—
32C23×C3×C9D411yescharacteristic
33C23×C3×C9C2311yescharacteristic
34C22×C3×D9—21no—
35C22×C3×D9D412yes—
36C3×D4×C9—22no—
37C2×C3×C4×C9D411yescharacteristic
38C2×C3⋊Dic9—22no—
39C3×#611—22no—
40C3×#611—22no—
41C3×#611—22no—
42C2×C9×Dic3—22no—
43C2×C9×Dic3—21no—
44C2×C9×Dic3—21no—
45C2×C3×Dic9—21no—
46C2×C3×Dic9D412yes—
47C3×D36—21no—
48C3×C4×D9—22no—
49C3×Dic18—21no—
50#5243—91no—
51#5053—91no—
52C2×C32×D4C2×D311yescharacteristic
53C2×C3×#67—32no—
54C2×C3×#67—61no—
55C22×C3×Dic3C2×D311yescharacteristic
56C3×#237—62no—
57C3×D3×D4—62no—
58C3×#235—61no—
59#3757—32no—
60C3×#217—21no—
61C3×#217C2×D311yescharacteristic
62C3×Dic3⋊C4—21no—
63#3724—62no—
64(C2×D3)⋊Dic3—62no—
65C2×D4×C9—21no—
66C2×D4×C9C2×D311yescharacteristic
67C22×C4×C9—31no—
68C2×#611—21no—
69C2×#611C2×D312yes—
70#3699—22no—
71D4×D9—22no—
72#3697—21no—
73C9×C4⋊C4—61no—
74C9×C22⋊C4—31no—
75C9×C22⋊C4—61no—
76#3676—32no—
77#3676—62no—
78(C2×D9)⋊C4—62no—
79Dic9⋊C4—62no—
80C22×C3×C9—21no—
81C22×C3×C9—41no—
82C22×C3×C9—41no—
83C22×C3×C9C2×D412yes—
84C22×C3×C9C2×D411yescharacteristic
85C2×C3×D9—22no—
86C2×C3×D9—42no—
87C2×C3×D9—81no—
88C3×C4×C9—21no—
89C3⋊Dic9—42no—
90C9×Dic3—41no—
91C9×Dic3—41no—
92C3×Dic9—22no—
93C3×H(5,𝔽2)—91no—
94#1004—31no—
95C3×#123—31no—
96#935—92no—
97#935D911yescharacteristic, complemented
98#929—92no—
99#834—91no—
100#834—91no—
101#834—92no—
102#834—181no—
103#806—92no—
104C23×C32#6711yescharacteristic
105C23×C32C22×D311yescharacteristic
106C22×C3×D3—32no—
107C22×C3×D3—61no—
108C32×D4—22no—
109C2×C32×C4#6711yescharacteristic
110C2×C3⋊Dic3—62no—
111C3×#67—62no—
112C3×#67—62no—
113C3×#67—62no—
114C2×C3×Dic3—22no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—61no—
119C3×D12—61no—
120C3×C4×D3—62no—
121C3×Dic6—61no—
122C23×C9—21no—
123C23×C9—21no—
124C23×C9#6711yescharacteristic
125C23×C9C22×D311yescharacteristic
126C22×D9—21no—
127C22×D9#6712yes—
128D4×C9—22no—
129D4×C9—42no—
130C2×C4×C9—21no—
131C2×C4×C9—62no—
132C2×C4×C9—61no—
133C2×C4×C9—61no—
134C2×C4×C9#6711yescharacteristic
135#611—22no—
136#611—22no—
137#611—22no—
138C2×Dic9—21no—
139C2×Dic9—62no—
140C2×Dic9—122no—
141C2×Dic9#6712yes—
142D36—21no—
143C4×D9—22no—
144Dic18—21no—
145#458—271noSylow
146C2×C3×C9—21no—
147C2×C3×C9—22no—
148C2×C3×C9—41no—
149C2×C3×C9#12011yescharacteristic
150C3×D9—42no—
151C3×#41—182no—
152C3×#41—181no—
153C2×C3×D4—21no—
154C2×C3×D4—92no—
155C2×C3×D4—182no—
156C2×C3×D4—181no—
157C2×C3×D4C2×D911yescharacteristic
158C2×C3×D4D3211yescharacteristic
159C22×C3×C4—31no—
160C2×#67—32no—
161C2×#67—61no—
162C22×Dic3C2×D911yescharacteristic, complemented
163#237—62no—
164D3×D4—62no—
165#235—61no—
166C3×C4⋊C4—61no—
167C3×C22⋊C4—31no—
168C3×C22⋊C4—61no—
169#217—21no—
170#217—92no—
171#217—92no—
172#217—182no—
173#217—182no—
174#217C2×D911yescharacteristic, complemented
175(C2×D3)⋊C4—182no—
176Dic3⋊C4—21no—
177Dic3⋊C4—182no—
178Dic3⋊C4—182no—
179C22×C32—21no—
180C22×C32—41no—
181C22×C32—41no—
182C22×C32D3×D412yes—
183C22×C32C2×#6711yescharacteristic
184C2×C3×D3—62no—
185C2×C3×D3—122no—
186C2×C3×D3—241no—
187C32×C4—21no—
188C3⋊Dic3—122no—
189C3×Dic3—41no—
190C3×Dic3—41no—
191C3×Dic3—62no—
192C22×C9—21no—
193C22×C9—21no—
194C22×C9—22no—
195C22×C9—41no—
196C22×C9—41no—
197C22×C9—41no—
198C22×C9—81no—
199C22×C9—81no—
200C22×C9C2×#6711yescharacteristic
201C22×C9C2×#6712yes—
202C2×D9—22no—
203C2×D9—42no—
204C2×D9—81no—
205C4×C9—21no—
206C4×C9—41no—
207C4×C9—121no—
208C4×C9—121no—
209Dic9—22no—
210Dic9—122no—
211Dic9—242no—
212H(5,𝔽2)—91no—
213#123—31no—
214#123—272no—
215#99—272no—
216#99—271no—
217C3×C9#45811yescharacteristic, complemented, Sylow
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—92no—
221C23×C3—181no—
222C23×C3#61111yescharacteristic
223C23×C3C22×D911yescharacteristic
224C23×C3#62511yescharacteristic
225C23×C3C2×D3211yescharacteristic
226C22×D3—32no—
227C22×D3—61no—
228C3×Q8—181no—
229C3×D4—22no—
230C3×D4—22no—
231C3×D4—42no—
232C3×D4—181no—
233C3×D4—182no—
234C3×D4—182no—
235C3×D4—182no—
236C2×C3×C4—21no—
237C2×C3×C4—61no—
238C2×C3×C4—61no—
239C2×C3×C4—62no—
240C2×C3×C4—92no—
241C2×C3×C4—182no—
242C2×C3×C4—181no—
243C2×C3×C4#61111yescharacteristic
244C2×C3×C4#62511yescharacteristic
245#67—62no—
246#67—62no—
247#67—62no—
248C2×Dic3—22no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—32no—
252C2×Dic3—61no—
253C2×Dic3—182no—
254C2×Dic3—182no—
255C2×Dic3—362no—
256D12—61no—
257C4×D3—62no—
258Dic6—61no—
259C2×C32—21no—
260C2×C32—22no—
261C2×C32—41no—
262C2×C32#93711yescharacteristic
263C3×D3—122no—
264C2×C9—22no—
265C2×C9—21no—
266C2×C9—21no—
267C2×C9—42no—
268C2×C9—41no—
269C2×C9—41no—
270C2×C9—81no—
271C2×C9#95311yescharacteristic
272D9—42no—
273#41—182no—
274#41—181no—
275C2×D4—92no—
276C2×D4—182no—
277C2×D4—181no—
278C2×D4D3×D911yescharacteristic
279C22×C4—31no—
280C4⋊C4—61no—
281C4⋊C4—542no—
282C22⋊C4—31no—
283C22⋊C4—61no—
284C22⋊C4—272no—
285C22⋊C4—542no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—22no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—81no—
296C22×C3—81no—
297C22×C3—182no—
298C22×C3—362no—
299C22×C3—721no—
300C22×C3D4×D912yes—
301C22×C3C2×#61111yescharacteristic
302C22×C3C2×#62511yescharacteristic
303C22×C3D3×#6712yes—
304C2×D3—62no—
305C2×D3—122no—
306C2×D3—241no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—41no—
310C3×C4—121no—
311C3×C4—121no—
312C3×C4—182no—
313Dic3—41no—
314Dic3—41no—
315Dic3—62no—
316Dic3—362no—
317Dic3—362no—
318Dic3—722no—
319C32#505311yescharacteristic
320C9—21no—
321C9#524311yescharacteristic, complemented
322C23—92no—
323C23—181no—
324C23#619411yescharacteristic
325C23C2×D3×D911yescharacteristic
326Q8—181no—
327D4—22no—
328D4—181no—
329D4—182no—
330D4—182no—
331D4—182no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—62no—
335C2×C4—92no—
336C2×C4—182no—
337C2×C4—181no—
338C2×C4—542no—
339C2×C4#619411yescharacteristic
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—22no—
344C2×C3—21no—
345C2×C3—42no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—41no—
349C2×C3—81no—
350C2×C3—362no—
351C2×C3#6349911yescharacteristic
352C2×C3#6397611yescharacteristic
353D3—122no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—182no—
358C22—362no—
359C22—721no—
360C22C2×#619411yescharacteristic
361C22D9×#6712yes—
362C4—21no—
363C4—121no—
364C4—121no—
365C4—182no—
366C4—1082no—
367C3—21no—
368C3#1058873311yescharacteristic, complemented
369C3#1059153311yescharacteristic
370C2—21no—
371C2—22no—
372C2—41no—
373C2—362no—
374C2#1172128511yescharacteristic
subgroupG/H
centerC2#11721285
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C2×D32
FittingC2×C3×D4×C9C22
socleC2×C32#937
prime radicalC3×#63710C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#458
LCSG ▷ C23×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C9×#935C2semidirect
3. #11721259C2semidirect
4. #11720577C2semidirect
5. #11720329C2semidirect
11. C22×C9×Dic3C22semidirect
15. #88551C22semidirect
17. C9×#217C22semidirect
97. #935D9semidirect
162. C22×Dic3C2×D9semidirect
174. #217C2×D9semidirect
217. C3×C9#458semidirect
321. C9#5243semidirect
368. C3#10588733semidirect

Maximal subgroups

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

subgroupindex
C3×#637102
C9×#9352
#117212592
#117205772
#117203292
#105915333
#105887333