Finite Group Databasebeta

← #13140388

Subgroups — #13140388

Order 1728.

614 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#88742C211yesmaximal, characteristic
2#11720863C212yesmaximal, complemented
3C9×#833C211yesmaximal, characteristic, complemented
4C3×#63391C211yesmaximal, characteristic, complemented
5#11720290C212yesmaximal, complemented
6#10591504—31nomaximal
7#10588425—31nomaximal
8C2×C3×D4×C9C2211yescharacteristic
9C2×#6233C412yes—
10C22×C3⋊Dic9C412yes—
11#88742—22no—
12#88742—22no—
13C2×C4×C3⋊D9C2211yescharacteristic
14C2×#6226C2211yescharacteristic
15C9×#208C2212yes—
16C3×#3667C2212yes—
17#88414—22no—
18(C3⋊D9)⋊C8—22no—
19C2×#3830—31no—
20#63818—32no—
21C2×#3699—31no—
22C2×#3699—61no—
23C3×#833—31no—
24C3×#833D311yescharacteristic
25#63561—32no—
26M16×D9—32no—
27C9×#100—31no—
28#63391D311yescharacteristic, complemented
29#63382—32no—
30C23×C3×C9C2×C412yes—
31C22×C3⋊D9C2×C411yescharacteristic
32C3×D4×C9—22no—
33C2×C3×C4×C9C2311yescharacteristic
34#6233—24no—
35C2×C3⋊Dic9—22no—
36C2×C3⋊Dic9—22no—
37C2×C3⋊Dic9C2×C412yes—
38C2×C3⋊Dic9C2×C411yescharacteristic
39C4×C3⋊D9D412yes—
40C4×C3⋊D9D412yes—
41#6226—22no—
42C9×C3⋊C8—22no—
43C3×C9⋊C8—22no—
44#4745—91no—
45#4745—91no—
46C2×C32×D4C2×D311yescharacteristic
47C2×#638—32no—
48C22×C3⋊Dic3—32no—
49#3830—62no—
50#3830—62no—
51C2×C4×C3⋊D3—31no—
52C2×#634—31no—
53C3×#208—32no—
54C3×#208C2×D312yes—
55#3712—62no—
56#3710—62no—
57C2×D4×C9—21no—
58C2×D4×C9C2×D311yescharacteristic
59C2×#611—32no—
60C2×#611—62no—
61C22×Dic9—32no—
62C22×Dic9—62no—
63#3699—62no—
64#3699—62no—
65#3699—64no—
66#3699—64no—
67C2×C4×D9—31no—
68C2×C4×D9—61no—
69C2×Dic18—31no—
70C2×Dic18—61no—
71C9×M16—32no—
72#3667C2×D312yes—
73#3663—62no—
74C8×D9—62no—
75C22×C3×C9—22no—
76C22×C3×C9—42no—
77C22×C3×C9C22×C411yescharacteristic
78C2×C3⋊D9—41no—
79C2×C3⋊D9C22⋊C412yes—
80C3×C4×C9C2×D412yes—
81C3⋊Dic9—22no—
82C3⋊Dic9C22⋊C411yescharacteristic
83C3⋊Dic9C22⋊C411yescharacteristic
84C2×#237—91no—
85C2×#237—91no—
86C2×#237—181no—
87D3×M16—92no—
88D3×M16—92no—
89C3×#100—31no—
90C3×#100—91no—
91#833—31no—
92#833D911yescharacteristic, complemented
93#824—92no—
94#824—92no—
95C23×C32C4×D312yes—
96C22×C3⋊D3—31no—
97C32×D4—22no—
98C2×C32×C4C22×D311yescharacteristic
99#638—64no—
100C2×C3⋊Dic3—31no—
101C2×C3⋊Dic3—32no—
102C2×C3⋊Dic3—62no—
103C2×C3⋊Dic3—62no—
104C4×C3⋊D3—32no—
105C4×C3⋊D3—32no—
106#634—62no—
107C23×C9—22no—
108C23×C9C4×D312yes—
109C22×D9—31no—
110C22×D9—61no—
111C3×C3⋊C8—22no—
112C3×C3⋊C8—62no—
113D4×C9—24no—
114D4×C9—22no—
115C2×C4×C9—21no—
116C2×C4×C9C22×D311yescharacteristic
117#611—64no—
118#611—68no—
119C2×Dic9—31no—
120C2×Dic9—32no—
121C2×Dic9—62no—
122C2×Dic9—61no—
123C2×Dic9—62no—
124C2×Dic9—62no—
125C2×Dic9—64no—
126C2×Dic9—64no—
127C4×D9—32no—
128C4×D9—32no—
129C4×D9—62no—
130C4×D9—62no—
131Dic18—64no—
132Dic18—62no—
133C8×C9—62no—
134C9⋊C8—22no—
135#413—271noSylow
136C2×C3×C9—21no—
137C2×C3×C9—42no—
138C2×C3×C9C2×C22⋊C411yescharacteristic
139C3⋊D9—22no—
140C2×C3×D4—21no—
141C2×C3×D4C2×D911yescharacteristic
142C2×C3×D4D3211yescharacteristic
143C2×#67—92no—
144C2×#67—92no—
145C2×#67—182no—
146C22×Dic3—92no—
147C22×Dic3—92no—
148C22×Dic3—182no—
149#237—182no—
150#237—182no—
151#237—182no—
152#237—182no—
153#237—184no—
154#237—184no—
155C2×C4×D3—91no—
156C2×C4×D3—91no—
157C2×C4×D3—181no—
158C2×Dic6—91no—
159C2×Dic6—91no—
160C2×Dic6—181no—
161C3×M16—32no—
162C3×M16—92no—
163#208—32no—
164#208C2×D912yes—
165ΓL(1,𝔽25)—182no—
166ΓL(1,𝔽25)—182no—
167D3×C8—182no—
168D3×C8—182no—
169C22×C32—22no—
170C22×C32—42no—
171C22×C32C2×C4×D311yescharacteristic
172C2×C3⋊D3—32no—
173C2×C3⋊D3—121no—
174C32×C4D3×D412yes—
175C3⋊Dic3—31no—
176C3⋊Dic3—31no—
177C3⋊Dic3—62no—
178C22×C9—24no—
179C22×C9—22no—
180C22×C9—21no—
181C22×C9—42no—
182C22×C9—44no—
183C22×C9C2×C4×D311yescharacteristic
184C2×D9—32no—
185C2×D9—62no—
186C2×D9—121no—
187C2×D9—122no—
188C4×C9—22no—
189C4×C9D3×D412yes—
190Dic9—31no—
191Dic9—31no—
192Dic9—61no—
193Dic9—61no—
194Dic9—64no—
195Dic9—62no—
196C2×#41—271no—
197C2×M16—272no—
198#101—272no—
199#100—31no—
200#100—91no—
201C3×C9#41311yescharacteristic, complemented, Sylow
202C23×C3—22no—
203C23×C3C4×D912yes—
204C23×C3#62412yes—
205C22×D3—91no—
206C22×D3—91no—
207C22×D3—181no—
208C3×D4—24no—
209C3×D4—22no—
210C3×D4—22no—
211C2×C3×C4—21no—
212C2×C3×C4C22×D911yescharacteristic
213C2×C3×C4C2×D3211yescharacteristic
214#67—184no—
215#67—184no—
216#67—188no—
217C2×Dic3—91no—
218C2×Dic3—91no—
219C2×Dic3—92no—
220C2×Dic3—92no—
221C2×Dic3—182no—
222C2×Dic3—181no—
223C2×Dic3—182no—
224C2×Dic3—182no—
225C2×Dic3—182no—
226C2×Dic3—182no—
227C2×Dic3—184no—
228C2×Dic3—184no—
229C4×D3—92no—
230C4×D3—92no—
231C4×D3—92no—
232C4×D3—92no—
233C4×D3—182no—
234C4×D3—182no—
235Dic6—184no—
236Dic6—182no—
237Dic6—182no—
238C3×C8—62no—
239C3×C8—182no—
240C3⋊C8—22no—
241C3⋊C8—62no—
242C2×C32—21no—
243C2×C32—42no—
244C2×C32D3×C22⋊C411yescharacteristic
245C3⋊D3—62no—
246C2×C9—22no—
247C2×C9—21no—
248C2×C9—21no—
249C2×C9—44no—
250C2×C9—42no—
251C2×C9D3×C22⋊C411yescharacteristic
252D9—62no—
253D9—122no—
254#41—542no—
255#41—542no—
256C2×Q8—271no—
257C2×D4—272no—
258C2×D4D3×D911yescharacteristic
259C22×C4—271no—
260C22×C4—272no—
261M16—32no—
262M16—92no—
263M16—542no—
264C2×C8—542no—
265C22×C3—24no—
266C22×C3—22no—
267C22×C3—22no—
268C22×C3—21no—
269C22×C3—42no—
270C22×C3—42no—
271C22×C3—44no—
272C22×C3C2×C4×D911yescharacteristic
273C22×C3C2×#62411yescharacteristic
274C2×D3—92no—
275C2×D3—92no—
276C2×D3—182no—
277C2×D3—361no—
278C2×D3—361no—
279C2×D3—362no—
280C3×C4—22no—
281C3×C4D4×D912yes—
282C3×C4#381112yes—
283Dic3—91no—
284Dic3—91no—
285Dic3—91no—
286Dic3—91no—
287Dic3—181no—
288Dic3—181no—
289Dic3—184no—
290Dic3—182no—
291Dic3—182no—
292C32#474511yescharacteristic
293C9—21no—
294C9#474511yescharacteristic, complemented
295C23—271no—
296C23(C3⋊D9)⋊C412yes—
297Q8—542no—
298D4—22no—
299D4—544no—
300C2×C4—271no—
301C2×C4—272no—
302C2×C4—272no—
303C2×C4—272no—
304C2×C4—542no—
305C2×C4—542no—
306C2×C4C2×D3×D911yescharacteristic
307C8—62no—
308C8—182no—
309C2×C3—22no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—44no—
314C2×C3—42no—
315C2×C3—42no—
316C2×C3C22⋊C4×D911yescharacteristic
317C2×C3#6397411yescharacteristic
318D3—182no—
319D3—182no—
320D3—362no—
321C22—22no—
322C22—42no—
323C22—272no—
324C22—1081no—
325C22C2×(C3⋊D9)⋊C411yescharacteristic
326C4—271no—
327C4—271no—
328C4—542no—
329C4#8866812yes—
330C3—21no—
331C3#1058842511yescharacteristic, complemented
332C3#1059150411yescharacteristic
333C2—21no—
334C2—42no—
335C2—542no—
336C2#1172127411yescharacteristic
subgroupG/H
centerC2#11721274
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC2×C32D3×C22⋊C4
prime radicalC2×#88742C2
cosocleC2×C3×C4×C9C23
hypercenterC2×D4D3×D9
hypocenterC3×C9#413
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11720863C2semidirect
3. C9×#833C2semidirect
4. C3×#63391C2semidirect
5. #11720290C2semidirect
28. #63391D3semidirect
92. #833D9semidirect
201. C3×C9#413semidirect
294. C9#4745semidirect
331. C3#10588425semidirect

Maximal subgroups

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

subgroupindex
C2×#887422
#117208632
C9×#8332
C3×#633912
#117202902
#105915043
#105884253