Finite Group Databasebeta

← #13139505

Subgroups — #13139505

Order 1728.

454 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 95 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#11721819C211yesmaximal, characteristic
2C3×#63486C211yesmaximal, characteristic, complemented
3C2×C9×#208C211yesmaximal, characteristic, complemented
4C2×#88416C211yesmaximal, characteristic, complemented
5#11720284C211yesmaximal, characteristic, complemented
6#11720279C211yesmaximal, characteristic, complemented
7#11720276C211yesmaximal, characteristic, complemented
8#10591178—31nomaximal
9#10588816—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×C4⋊(C3⋊D9)C2211yescharacteristic
13C2×C3×Dic18C2211yescharacteristic
14#88536—21no—
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16C3×#3676—21no—
17C3×C4⋊Dic9C2211yescharacteristic
18C3×Dic9⋊C4—21no—
19C9×#208—22no—
20C2×C9×C3⋊C8C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22#88416—22no—
23#64139—31no—
24C3×#924—31no—
25C2×C3×#208D311yescharacteristic
26C2×#3717—31no—
27#63575—31no—
28#63566—31no—
29#63564—31no—
30C2×C9×M16—31no—
31#63492—31no—
32#63492—61no—
33#63486D311yescharacteristic, complemented
34C2×#3664—31no—
35#63380—31no—
36#63378—31no—
37#63376—31no—
38C23×C3×C9D411yescharacteristic
39C22×C3⋊D9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9D411yescharacteristic
43C2×C3×C4×C9C2311yescharacteristic
44#6233—42no—
45C2×C3⋊Dic9—21no—
46C4⋊(C3⋊D9)—21no—
47C4⋊(C3⋊D9)—21no—
48C2×C3×Dic9—21no—
49C2×C3×Dic9—21no—
50C3×Dic18—21no—
51C3×Dic18—21no—
52C9×C3⋊C8—21no—
53C9×C3⋊C8D412yescomplemented
54#5134—91no—
55#5049—91no—
56C22×C32×C4C2×D311yescharacteristic
57C2×#638—61no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C3×Dic6—31no—
60#3752—61no—
61C4⋊(C3⋊Dic3)—31no—
62C3×#217—61no—
63C3×C4⋊Dic3—31no—
64C3×Dic3⋊C4—61no—
65C3×#208—22no—
66C2×C3×C3⋊C8C2×D311yescharacteristic
67C2×C3×C3⋊C8C2×D311yescharacteristic
68#3717—62no—
69C22×C4×C9—21no—
70C22×C4×C9C2×D311yescharacteristic
71C2×#611—62no—
72C2×#611—61no—
73C2×D36—31no—
74C2×D36—61no—
75C2×Dic18C2×D311yescharacteristic
76C9×M16—62no—
77C2×C8×C9—31no—
78C2×C8×C9—31no—
79#3676—21no—
80(C2×D9)⋊C4—62no—
81(C2×D9)⋊C4—61no—
82C4⋊Dic9—31no—
83C4⋊Dic9—61no—
84C4⋊Dic9C2×D311yescharacteristic
85Dic9⋊C4—21no—
86#3664—62no—
87C22×C3×C9—21no—
88C22×C3×C9—21no—
89C22×C3×C9—21no—
90C22×C3×C9C2×D411yescharacteristic
91C2×C3⋊D9—41no—
92C2×C3⋊D9—41no—
93C2×C3⋊D9—41no—
94C3×C4×C9—21no—
95C3×C4×C9C2×D411yescharacteristic
96C3×C4×C9#4111yescharacteristic
97C3⋊Dic9—41no—
98C3×Dic9—41no—
99C3×Dic9—41no—
100C2×C3×M16—31no—
101C3×C22⋊Q8—91no—
102C2×#215—91no—
103#930—91no—
104#930—91no—
105#930—181no—
106#924—31no—
107C2×#208D911yescharacteristic, complemented
108C2×#204—91no—
109#835—91no—
110#821—91no—
111#818—91no—
112#816—91no—
113#809—91no—
114#807—91no—
115C23×C32D1211yescharacteristic
116C22×C3⋊D3—61no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4D1211yescharacteristic
120C2×C32×C4C22×D311yescharacteristic
121#638—122no—
122C2×C3⋊Dic3—61no—
123C4⋊(C3⋊D3)—61no—
124C4⋊(C3⋊D3)—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C3×Dic6—61no—
128C3×Dic6—61no—
129C23×C9—21no—
130C23×C9#6711yescharacteristic
131C22×D9—62no—
132C22×D9—61no—
133C3×C3⋊C8—21no—
134C3×C3⋊C8#6712yes—
135C2×C4×C9—21no—
136C2×C4×C9—21no—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—41no—
140C2×C4×C9—41no—
141C2×C4×C9#6711yescharacteristic
142C2×C4×C9C22×D311yescharacteristic
143#611—124no—
144#611—122no—
145C2×Dic9—21no—
146C2×Dic9—21no—
147C2×Dic9—62no—
148C2×Dic9—61no—
149D36—61no—
150D36—61no—
151D36—121no—
152D36—121no—
153Dic18—21no—
154Dic18—21no—
155C8×C9—32no—
156C8×C9—61no—
157#468—271noSylow
158C2×C3×C9—41no—
159C2×C3×C9#12111yescharacteristic
160C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
161C2×C3×C9#13711yescharacteristic
162C3⋊D9—81no—
163C2×C3×Q8—91no—
164C22×C3×C4—21no—
165C22×C3×C4C2×D911yescharacteristic
166C22×C3×C4D3211yescharacteristic
167C2×#67—182no—
168C2×#67—181no—
169C2×#67—181no—
170C2×D12—91no—
171C2×D12—91no—
172C2×D12—181no—
173C2×Dic6—31no—
174C3×M16—62no—
175C2×C3×C8—31no—
176C2×C3×C8—31no—
177C3×C4⋊C4—91no—
178C3×C4⋊C4—181no—
179C3×C22⋊C4—181no—
180#217—61no—
181#215—182no—
182(C2×D3)⋊C4—182no—
183(C2×D3)⋊C4—181no—
184(C2×D3)⋊C4—181no—
185C4⋊Dic3—31no—
186C4⋊Dic3—91no—
187C4⋊Dic3—91no—
188C4⋊Dic3—181no—
189Dic3⋊C4—61no—
190#208—22no—
191C2×C3⋊C8C2×D911yescharacteristic
192C2×C3⋊C8C2×D911yescharacteristic
193#204—182no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32C2×D1211yescharacteristic
198C2×C3⋊D3—121no—
199C2×C3⋊D3—121no—
200C2×C3⋊D3—121no—
201C32×C4—21no—
202C32×C4#23511yescharacteristic
203C32×C4C2×#6711yescharacteristic
204C3⋊Dic3—121no—
205C3×Dic3—121no—
206C3×Dic3—121no—
207C22×C9—22no—
208C22×C9—21no—
209C22×C9—21no—
210C22×C9—21no—
211C22×C9—21no—
212C22×C9—41no—
213C22×C9—41no—
214C22×C9C2×#6711yescharacteristic
215C2×D9—122no—
216C2×D9—122no—
217C2×D9—122no—
218C2×D9—121no—
219C2×D9—121no—
220C2×D9—121no—
221C4×C9—21no—
222C4×C9—21no—
223C4×C9—21no—
224C4×C9—41no—
225C4×C9D3×D411yescharacteristic
226C4×C9#23711yescharacteristic
227Dic9—41no—
228Dic9—41no—
229Dic9—122no—
230Dic9—121no—
231C2×SD16—271no—
232C2×M16—31no—
233C22⋊Q8—91no—
234#121—271no—
235#107—271no—
236#103—271no—
237#102—271no—
238C3×C9#46811yescharacteristic, complemented, Sylow
239C23×C3—21no—
240C23×C3D3611yescharacteristic
241C23×C3#62611yescharacteristic
242C22×D3—182no—
243C22×D3—181no—
244C22×D3—181no—
245C3×Q8—181no—
246C3×Q8—181no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—181no—
256C2×C3×C4—181no—
257C2×C3×C4D3611yescharacteristic
258C2×C3×C4C22×D911yescharacteristic
259C2×C3×C4#62611yescharacteristic
260C2×C3×C4C2×D3211yescharacteristic
261#67—362no—
262#67—364no—
263#67—362no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—182no—
267C2×Dic3—181no—
268C2×Dic3—181no—
269D12—181no—
270D12—181no—
271D12—181no—
272D12—181no—
273D12—361no—
274D12—361no—
275Dic6—61no—
276Dic6—61no—
277C3×C8—32no—
278C3×C8—61no—
279C3⋊C8—21no—
280C3⋊C8#61112yescomplemented
281C2×C32—41no—
282C2×C32#90811yescharacteristic
283C2×C32#90911yescharacteristic
284C2×C32#93011yescharacteristic
285C3⋊D3—241no—
286C2×C9—21no—
287C2×C9—21no—
288C2×C9—21no—
289C2×C9—42no—
290C2×C9—41no—
291C2×C9#93911yescharacteristic
292C2×C9#94211yescharacteristic
293C2×C9#94911yescharacteristic
294D9—242no—
295D9—241no—
296C2×Q8—91no—
297C2×D4—271no—
298C2×D4—541no—
299C22×C4D3×D911yescharacteristic
300SD16—542no—
301M16—62no—
302C2×C8—31no—
303C2×C8—31no—
304C4⋊C4—91no—
305C4⋊C4—181no—
306C4⋊C4—271no—
307C22⋊C4—181no—
308C22⋊C4—541no—
309C22×C3—22no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3C2×D3611yescharacteristic
320C22×C3C2×#62611yescharacteristic
321C2×D3—362no—
322C2×D3—362no—
323C2×D3—362no—
324C2×D3—361no—
325C2×D3—361no—
326C2×D3—361no—
327C2×D3—361no—
328C2×D3—361no—
329C2×D3—361no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—41no—
335C3×C4—361no—
336C3×C4—361no—
337C3×C4#369711yescharacteristic
338C3×C4C2×#61111yescharacteristic
339C3×C4#380411yescharacteristic
340C3×C4D3×#6711yescharacteristic
341Dic3—121no—
342Dic3—121no—
343Dic3—362no—
344Dic3—361no—
345Dic3—361no—
346C32#513411yescharacteristic
347C9—21no—
348C9#504911yescharacteristic, complemented
349C23—541no—
350C23#619211yescharacteristic
351Q8—181no—
352Q8—181no—
353D4—541no—
354D4—541no—
355D4—1082no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4—541no—
361C2×C4#619211yescharacteristic
362C2×C4C2×D3×D911yescharacteristic
363C8—32no—
364C8—61no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—42no—
369C2×C3—41no—
370C2×C3—41no—
371C2×C3#6347011yescharacteristic
372C2×C3#6347111yescharacteristic
373C2×C3#6349211yescharacteristic
374C2×C3#6382511yescharacteristic
375C2×C3#6383311yescharacteristic
376C2×C3#6397811yescharacteristic
377D3—722no—
378D3—721no—
379D3—721no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—1081no—
384C22—1081no—
385C22—1081no—
386C22C2×#619211yescharacteristic
387C4—21no—
388C4—361no—
389C4—361no—
390C4—1081no—
391C4#8865811yescharacteristic
392C4D3×#61111yescharacteristic
393C3—21no—
394C3#1058881611yescharacteristic, complemented
395C3#1059117811yescharacteristic
396C2—41no—
397C2—2161no—
398C2#1172087111yescharacteristic
399C2#1172087311yescharacteristic
400C2#1172127711yescharacteristic
subgroupG/H
centerC22C2×#6192
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×D12
prime radical#11721819C2
hypercenterC22×C4D3×D9
hypocenterC3×C9#468
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C3×#63486C2semidirect
3. C2×C9×#208C2semidirect
4. C2×#88416C2semidirect
5. #11720284C2semidirect
6. #11720279C2semidirect
7. #11720276C2semidirect
33. #63486D3semidirect
53. C9×C3⋊C8D4semidirect
107. C2×#208D9semidirect
238. C3×C9#468semidirect
280. C3⋊C8#611semidirect
348. C9#5049semidirect
394. C3#10588816semidirect

Maximal subgroups

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

subgroupindex
#117218192
C3×#634862
C2×C9×#2082
C2×#884162
#117202842
#117202792
#117202762
#105911783
#105888163