Finite Group Databasebeta

← #13140376

Subgroups — #13140376

Order 1728.

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

normal subgroups only

This group has 454 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
1#11721838C211yesmaximal, characteristic
2C2×C9×#214C211yesmaximal, characteristic, complemented
3#11720929C211yesmaximal, characteristic, complemented
4C2×#88414C211yesmaximal, characteristic, complemented
5C3×#63392C211yesmaximal, characteristic, complemented
6#11720298C211yesmaximal, characteristic, complemented
7#11720279C211yesmaximal, characteristic, complemented
8#10591464—31nomaximal
9#10588854—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×C4×C3⋊D9C2211yescharacteristic
13C2×C9×Dic6C2211yescharacteristic
14#88551—21no—
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16C9×#214—22no—
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88445—21no—
21Dic9⋊Dic3—21no—
22#88414—22no—
23#64147—31no—
24C2×C3×#214D311yescharacteristic
25#63871—31no—
26C2×#3712—31no—
27C3×#832—31no—
28#63575—31no—
29#63566—31no—
30C2×C9×SD16—31no—
31#63508—31no—
32#63500—31no—
33#63500—61no—
34C2×#3663—31no—
35#63395—31no—
36#63392D311yescharacteristic, complemented
37#63378—31no—
38C23×C3×C9—21no—
39C22×C3⋊D9D411yescharacteristic
40C3×D4×C9—21no—
41C3×D4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43#6233—42no—
44C2×C3⋊Dic9—21no—
45C2×C3⋊Dic9D411yescharacteristic
46C4×C3⋊D9—21no—
47C4×C3⋊D9—21no—
48C2×C9×Dic3—21no—
49C2×C3×Dic9—21no—
50C9×Dic6—21no—
51C9×Dic6—21no—
52C9×C3⋊C8D412yescomplemented
53C3×C9⋊C8—21no—
54#5174—91no—
55#4780—91no—
56C2×C32×D4C2×D311yescharacteristic
57C2×#638—61no—
58C2×C4×C3⋊D3—31no—
59C2×C3×Dic6C2×D311yescharacteristic
60#3757—61no—
61C4⋊(C3⋊Dic3)—31no—
62C3×#214—22no—
63C3×C4⋊Dic3—31no—
64C2×C3×C3⋊C8—31no—
65C2×C3×C3⋊C8C2×D311yescharacteristic
66Dic3⋊Dic3—61no—
67#3722—61no—
68#3712—62no—
69C2×Q8×C9—31no—
70C2×D4×C9—21no—
71C2×D4×C9C2×D311yescharacteristic
72C2×#611—61no—
73C2×#611—62no—
74C2×C4×D9—31no—
75C2×C4×D9—61no—
76C9×SD16—62no—
77C2×C8×C9—31no—
78#3676—61no—
79#3676—62no—
80(C2×D9)⋊C4—61no—
81C4⋊Dic9—31no—
82C4⋊Dic9—61no—
83C4⋊Dic9C2×D311yescharacteristic
84Dic9⋊C4—61no—
85C2×C9⋊C8C2×D311yescharacteristic
86#3663—62no—
87C22×C3×C9—41no—
88C22×C3×C9—41no—
89C22×C3×C9—41no—
90C22×C3×C9C2×D411yescharacteristic
91C2×C3⋊D9—21no—
92C2×C3⋊D9—21no—
93C2×C3⋊D9—21no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9#4111yescharacteristic
96C3⋊Dic9—21no—
97C3⋊Dic9—41no—
98C9×Dic3—41no—
99C3×Dic9—41no—
100C2×C3×SD16—31no—
101#946—91no—
102#938—91no—
103#938—91no—
104#938—181no—
105C2×#214D911yescharacteristic, complemented
106C2×ΓL(1,𝔽25)—91no—
107C2×ΓL(1,𝔽25)—91no—
108#897—91no—
109C3×#102—91no—
110#835—91no—
111#832—31no—
112#818—91no—
113#816—91no—
114#807—91no—
115C23×C32—21no—
116C22×C3⋊D3—31no—
117C32×D4—21no—
118C32×D4—21no—
119C2×C32×C4C22×D311yescharacteristic
120#638—122no—
121C2×C3⋊Dic3—31no—
122C2×C3⋊Dic3—61no—
123C4×C3⋊D3—61no—
124C4×C3⋊D3—61no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—61no—
127C3×Dic6—21no—
128C3×Dic6—21no—
129C23×C9—22no—
130C23×C9—21no—
131C22×D9—31no—
132C22×D9—61no—
133C3×C3⋊C8—61no—
134C3×C3⋊C8#6712yes—
135Q8×C9—61no—
136Q8×C9—61no—
137D4×C9—21no—
138D4×C9—21no—
139D4×C9—41no—
140D4×C9—41no—
141C2×C4×C9—21no—
142C2×C4×C9—61no—
143C2×C4×C9C22×D311yescharacteristic
144#611—124no—
145#611—122no—
146C2×Dic9—21no—
147C2×Dic9—31no—
148C2×Dic9—61no—
149C2×Dic9—62no—
150C2×Dic9—61no—
151C4×D9—61no—
152C4×D9—61no—
153C4×D9—121no—
154C4×D9—121no—
155C8×C9—32no—
156C9⋊C8—21no—
157#468—271noSylow
158C2×C3×C9—81no—
159C2×C3×C9#12111yescharacteristic
160C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
161C2×C3×C9#13711yescharacteristic
162C3⋊D9—41no—
163C2×C3×Q8—31no—
164C2×C3×D4—21no—
165C2×C3×D4C2×D911yescharacteristic
166C2×C3×D4D3211yescharacteristic
167C2×#67—181no—
168C2×#67—181no—
169C2×#67—182no—
170C2×C4×D3—91no—
171C2×C4×D3—91no—
172C2×C4×D3—181no—
173C2×Dic6C2×D911yescharacteristic
174C3×SD16—62no—
175C2×C3×C8—31no—
176C2×C3×C8—91no—
177C3×C4⋊C4—91no—
178#217—181no—
179#217—181no—
180#217—182no—
181#214—22no—
182(C2×D3)⋊C4—181no—
183(C2×D3)⋊C4—181no—
184C4⋊Dic3—31no—
185C4⋊Dic3—91no—
186C4⋊Dic3—91no—
187C4⋊Dic3—181no—
188C4⋊Dic3—181no—
189Dic3⋊C4—181no—
190C2×C3⋊C8—31no—
191C2×C3⋊C8C2×D911yescharacteristic
192ΓL(1,𝔽25)—182no—
193ΓL(1,𝔽25)—182no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32D3×D411yescharacteristic
198C2×C3⋊D3—61no—
199C2×C3⋊D3—61no—
200C2×C3⋊D3—61no—
201C32×C4#23711yescharacteristic
202C32×C4C2×#6711yescharacteristic
203C3⋊Dic3—61no—
204C3⋊Dic3—121no—
205C3×Dic3—41no—
206C3×Dic3—121no—
207C22×C9—21no—
208C22×C9—42no—
209C22×C9—42no—
210C22×C9—42no—
211C22×C9—41no—
212C22×C9—41no—
213C22×C9—41no—
214C22×C9D3×D411yescharacteristic
215C2×D9—62no—
216C2×D9—61no—
217C2×D9—61no—
218C2×D9—61no—
219C2×D9—121no—
220C2×D9—121no—
221C4×C9—21no—
222C4×C9—21no—
223C4×C9—121no—
224C4×C9#23511yescharacteristic
225C4×C9D3×D411yescharacteristic
226Dic9—41no—
227Dic9—61no—
228Dic9—121no—
229Dic9—122no—
230Dic9—121no—
231C2×SD16—31no—
232C2×M16—271no—
233C22⋊Q8—271no—
234#121—271no—
235#107—271no—
236#103—271no—
237#102—91no—
238C3×C9#46811yescharacteristic, complemented, Sylow
239C23×C3—22no—
240C23×C3—21no—
241C23×C3—21no—
242C22×D3—91no—
243C22×D3—91no—
244C22×D3—181no—
245C3×Q8—61no—
246C3×Q8—61no—
247C3×D4—21no—
248C3×D4—21no—
249C3×D4—21no—
250C3×D4—21no—
251C3×D4—41no—
252C3×D4—41no—
253C2×C3×C4—21no—
254C2×C3×C4—61no—
255C2×C3×C4—181no—
256C2×C3×C4C22×D911yescharacteristic
257C2×C3×C4C2×D3211yescharacteristic
258#67—362no—
259#67—364no—
260#67—362no—
261C2×Dic3—21no—
262C2×Dic3—61no—
263C2×Dic3—91no—
264C2×Dic3—91no—
265C2×Dic3—181no—
266C2×Dic3—182no—
267C2×Dic3—181no—
268C2×Dic3—181no—
269C4×D3—181no—
270C4×D3—181no—
271C4×D3—181no—
272C4×D3—181no—
273C4×D3—361no—
274C4×D3—361no—
275Dic6—21no—
276Dic6—21no—
277C3×C8—32no—
278C3×C8—181no—
279C3⋊C8—61no—
280C3⋊C8#61112yescomplemented
281C2×C32—81no—
282C2×C32#91411yescharacteristic
283C2×C32#91511yescharacteristic
284C2×C32#93811yescharacteristic
285C3⋊D3—121no—
286C2×C9—21no—
287C2×C9—21no—
288C2×C9—21no—
289C2×C9—82no—
290C2×C9—81no—
291C2×C9#88411yescharacteristic
292C2×C9#91111yescharacteristic
293C2×C9#91511yescharacteristic
294D9—122no—
295D9—121no—
296C2×Q8—31no—
297C2×D4—541no—
298C2×D4D3×D911yescharacteristic
299C22×C4—271no—
300SD16—62no—
301M16—542no—
302C2×C8—31no—
303C2×C8—91no—
304C4⋊C4—91no—
305C4⋊C4—271no—
306C4⋊C4—541no—
307C22⋊C4—541no—
308C22⋊C4—541no—
309C22×C3—21no—
310C22×C3—42no—
311C22×C3—42no—
312C22×C3—42no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3D4×D911yescharacteristic
320C22×C3#381111yescharacteristic
321C2×D3—182no—
322C2×D3—181no—
323C2×D3—181no—
324C2×D3—181no—
325C2×D3—181no—
326C2×D3—181no—
327C2×D3—181no—
328C2×D3—361no—
329C2×D3—361no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—121no—
333C3×C4—361no—
334C3×C4#369911yescharacteristic
335C3×C4C2×#61111yescharacteristic
336C3×C4#380411yescharacteristic
337C3×C4D3×#6711yescharacteristic
338Dic3—41no—
339Dic3—121no—
340Dic3—181no—
341Dic3—181no—
342Dic3—361no—
343Dic3—362no—
344Dic3—361no—
345Dic3—361no—
346C32#517411yescharacteristic
347C9—21no—
348C9#478011yescharacteristic, complemented
349C23—21no—
350C23—271no—
351Q8—61no—
352Q8—61no—
353D4—21no—
354D4—21no—
355D4—1082no—
356C2×C4—61no—
357C2×C4—181no—
358C2×C4—271no—
359C2×C4—541no—
360C2×C4—541no—
361C2×C4—541no—
362C2×C4C2×D3×D911yescharacteristic
363C8—32no—
364C8—181no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—82no—
369C2×C3—81no—
370C2×C3—81no—
371C2×C3#6347611yescharacteristic
372C2×C3#6347711yescharacteristic
373C2×C3#6350011yescharacteristic
374C2×C3#6393111yescharacteristic
375C2×C3#6393711yescharacteristic
376C2×C3#6395811yescharacteristic
377D3—362no—
378D3—361no—
379D3—361no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—541no—
384C22—541no—
385C22—541no—
386C22#8866811yescharacteristic
387C4—121no—
388C4—361no—
389C4—541no—
390C4—1081no—
391C4#8866211yescharacteristic
392C4D3×#61111yescharacteristic
393C3—21no—
394C3#1058885411yescharacteristic, complemented
395C3#1059146411yescharacteristic
396C2—81no—
397C2—1081no—
398C2#1172118311yescharacteristic
399C2#1172118711yescharacteristic
400C2#1172126211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radical#11721838C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#468
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C9×#214C2semidirect
3. #11720929C2semidirect
4. C2×#88414C2semidirect
5. C3×#63392C2semidirect
6. #11720298C2semidirect
7. #11720279C2semidirect
36. #63392D3semidirect
52. C9×C3⋊C8D4semidirect
105. C2×#214D9semidirect
238. C3×C9#468semidirect
280. C3⋊C8#611semidirect
348. C9#4780semidirect
394. C3#10588854semidirect

Maximal subgroups

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

subgroupindex
#117218382
C2×C9×#2142
#117209292
C2×#884142
C3×#633922
#117202982
#117202792
#105914643
#105888543