Finite Group Databasebeta

← #13139500

Subgroups — #13139500

Order 1728.

530 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 97 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×C3×#3697C211yesmaximal, characteristic
2#11721813C211yesmaximal, characteristic, complemented
3C2×#88418C211yesmaximal, characteristic, complemented
4C2×#88415C211yesmaximal, characteristic, complemented
5C9×#830C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic, complemented
7#11720276C211yesmaximal, characteristic, complemented
8#10591173—31nomaximal
9#10588415—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×#611—21no—
13C3×#3697—22no—
14C3×#3697—22no—
15C2×C3×D36C2211yescharacteristic
16C2×C3×C4×D9—21no—
17C2×C3×Dic18C2211yescharacteristic
18#88551—21no—
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20#88534—21no—
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C9×C3⋊C8C2211yescharacteristic
23#88418—22no—
24#88415—22no—
25C2×C3×#235—31no—
26#64133—31no—
27C2×#3719—31no—
28C2×#3716—31no—
29C2×#3697D311yescharacteristic
30C3×#830D311yescharacteristic
31#63566—31no—
32#63563—31no—
33#63486—31no—
34#63486—61no—
35C2×#3664—31no—
36C2×Dic36—31no—
37C9×C22⋊C8—31no—
38#63380—31no—
39#63376—31no—
40C23×C3×C9D411yescharacteristic
41C22×C3×D9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9D411yescharacteristic
45C2×C3×C4×C9C2311yescharacteristic
46C2×C3⋊Dic9—21no—
47C2×C3⋊Dic9—21no—
48#6226—21no—
49#6226—21no—
50C3×#611—24no—
51C2×C3×Dic9—21no—
52C3×D36—21no—
53C3×D36D412yes—
54C3×C4×D9—24no—
55C3×Dic18—21no—
56C3×Dic18D412yes—
57C9×C3⋊C8—21no—
58C9×C3⋊C8—21no—
59#5239—91no—
60#4733—91no—
61C22×C32×C4C2×D311yescharacteristic
62C2×#634—31no—
63C2×C3×#67—61no—
64C3×#235—62no—
65C3×#235—62no—
66C2×C3×D12—31no—
67C2×C3×C4×D3—61no—
68C2×C3×Dic6—31no—
69#3757—61no—
70C4⋊(C3⋊Dic3)—31no—
71#3750—61no—
72C2×C3×C3⋊C8C2×D311yescharacteristic
73C2×C3×C3⋊C8C2×D311yescharacteristic
74#3719—62no—
75#3716—62no—
76C22×C4×C9—21no—
77C22×C4×C9C2×D311yescharacteristic
78C2×#611—21no—
79#3697—22no—
80#3697—22no—
81C2×D36C2×D311yescharacteristic
82C2×C4×D9—21no—
83C2×Dic18—31no—
84C2×Dic18—61no—
85C2×Dic18C2×D311yescharacteristic
86C2×C8×C9—31no—
87C2×C8×C9—31no—
88#3676—62no—
89#3676—61no—
90C4⋊Dic9—31no—
91C4⋊Dic9—61no—
92Dic9⋊C4—62no—
93Dic9⋊C4—61no—
94#3664—62no—
95Dic36—62no—
96C22×C3×C9—21no—
97C22×C3×C9—21no—
98C22×C3×C9—21no—
99C22×C3×C9C2×D411yescharacteristic
100C2×C3×D9—22no—
101C2×C3×D9—41no—
102C2×C3×D9—41no—
103C3×C4×C9—21no—
104C3×C4×C9C2×D411yescharacteristic
105C3×C4×C9C2×D411yescharacteristic
106C3⋊Dic9—41no—
107C3⋊Dic9—41no—
108C3×Dic9—22no—
109C2×C3×#41—91no—
110C2×#235—31no—
111C2×#216—91no—
112C2×#214—91no—
113#924—91no—
114#924—91no—
115#924—181no—
116C2×Dic12—91no—
117C2×#204—91no—
118C3×C22⋊C8—31no—
119#835—91no—
120#832—91no—
121#830D911yescharacteristic, complemented
122#821—91no—
123#816—91no—
124C23×C32D1211yescharacteristic
125C22×C3×D3—61no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4D1211yescharacteristic
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—61no—
131C2×C3⋊Dic3—61no—
132#634—61no—
133#634—61no—
134C3×#67—64no—
135C2×C3×Dic3—61no—
136C3×D12—32no—
137C3×D12—61no—
138C3×C4×D3—64no—
139C3×Dic6—32no—
140C3×Dic6—61no—
141C23×C9—21no—
142C23×C9#6711yescharacteristic
143C22×D9—21no—
144C3×C3⋊C8—21no—
145C3×C3⋊C8—21no—
146C2×C4×C9—21no—
147C2×C4×C9—21no—
148C2×C4×C9—21no—
149C2×C4×C9—21no—
150C2×C4×C9—41no—
151C2×C4×C9—41no—
152C2×C4×C9#6711yescharacteristic
153C2×C4×C9C22×D311yescharacteristic
154#611—24no—
155C2×Dic9—21no—
156C2×Dic9—62no—
157C2×Dic9—62no—
158C2×Dic9—61no—
159C2×Dic9—61no—
160D36—21no—
161D36#6712yes—
162C4×D9—24no—
163Dic18—21no—
164Dic18—61no—
165Dic18—61no—
166Dic18—121no—
167Dic18—121no—
168Dic18#6712yes—
169C8×C9—61no—
170C8×C9—61no—
171#452—271noSylow
172C2×C3×C9—41no—
173C2×C3×C9#12011yescharacteristic
174C2×C3×C9#13511yescharacteristic
175C2×C3×C9#13711yescharacteristic
176C3×D9—42no—
177C3×#41—182no—
178C3×#41—182no—
179C2×C3×Q8—91no—
180C2×C3×D4—91no—
181C2×C3×D4—181no—
182C22×C3×C4—21no—
183C22×C3×C4—181no—
184C22×C3×C4C2×D911yescharacteristic
185C22×C3×C4D3211yescharacteristic
186C2×#67—61no—
187#235—62no—
188#235—62no—
189C2×D12—31no—
190C2×C4×D3—61no—
191C2×Dic6—31no—
192C2×Dic6—91no—
193C2×Dic6—91no—
194C2×Dic6—181no—
195C2×C3×C8—31no—
196C2×C3×C8—31no—
197#217—182no—
198#217—181no—
199#217—181no—
200#216—182no—
201#214—182no—
202C4⋊Dic3—91no—
203C4⋊Dic3—91no—
204C4⋊Dic3—181no—
205Dic3⋊C4—182no—
206Dic3⋊C4—181no—
207Dic3⋊C4—181no—
208C2×C3⋊C8C2×D911yescharacteristic
209C2×C3⋊C8C2×D911yescharacteristic
210Dic12—182no—
211#204—182no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32C2×D1211yescharacteristic
216C2×C3×D3—62no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C32×C4—21no—
220C32×C4D3×D411yescharacteristic
221C32×C4D3×D411yescharacteristic
222C3⋊Dic3—121no—
223C3⋊Dic3—121no—
224C3×Dic3—62no—
225C22×C9—22no—
226C22×C9—21no—
227C22×C9—21no—
228C22×C9—21no—
229C22×C9—21no—
230C22×C9—41no—
231C22×C9—41no—
232C22×C9C2×#6711yescharacteristic
233C2×D9—22no—
234C2×D9—41no—
235C2×D9—41no—
236C4×C9—21no—
237C4×C9—21no—
238C4×C9—21no—
239C4×C9—41no—
240C4×C9C2×#6711yescharacteristic
241C4×C9C2×#6711yescharacteristic
242Dic9—22no—
243Dic9—122no—
244Dic9—122no—
245Dic9—121no—
246Dic9—121no—
247C2×#41—91no—
248C2×Q16—271no—
249C2×SD16—271no—
250C22⋊Q8—271no—
251#103—271no—
252#102—271no—
253C22⋊C8—31no—
254C3×C9#45211yescharacteristic, complemented, Sylow
255C23×C3—21no—
256C23×C3—181no—
257C23×C3D3611yescharacteristic
258C23×C3#62611yescharacteristic
259C22×D3—61no—
260C3×Q8—92no—
261C3×Q8—181no—
262C3×D4—92no—
263C3×D4—184no—
264C3×D4—181no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—184no—
274C2×C3×C4—181no—
275C2×C3×C4D3611yescharacteristic
276C2×C3×C4C22×D911yescharacteristic
277C2×C3×C4#62611yescharacteristic
278C2×C3×C4C2×D3211yescharacteristic
279#67—64no—
280C2×Dic3—61no—
281C2×Dic3—182no—
282C2×Dic3—182no—
283C2×Dic3—181no—
284C2×Dic3—181no—
285C2×Dic3—181no—
286C2×Dic3—181no—
287D12—32no—
288D12—61no—
289C4×D3—64no—
290Dic6—32no—
291Dic6—61no—
292Dic6—181no—
293Dic6—181no—
294Dic6—181no—
295Dic6—181no—
296Dic6—361no—
297Dic6—361no—
298C3×C8—61no—
299C3×C8—61no—
300C3⋊C8—21no—
301C3⋊C8—21no—
302C2×C32—41no—
303C2×C32#88211yescharacteristic
304C2×C32#90411yescharacteristic
305C2×C32#90911yescharacteristic
306C3×D3—122no—
307C2×C9—21no—
308C2×C9—21no—
309C2×C9—21no—
310C2×C9—42no—
311C2×C9—41no—
312C2×C9#95011yescharacteristic
313C2×C9#95111yescharacteristic
314C2×C9#95311yescharacteristic
315D9—42no—
316#41—182no—
317#41—182no—
318C2×Q8—91no—
319C2×Q8—271no—
320C2×D4—91no—
321C2×D4—181no—
322C22×C4—181no—
323C22×C4D3×D911yescharacteristic
324Q16—542no—
325SD16—542no—
326C2×C8—31no—
327C2×C8—31no—
328C4⋊C4—271no—
329C4⋊C4—541no—
330C22⋊C4—541no—
331C22×C3—22no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—182no—
342C22×C3—361no—
343C22×C3—361no—
344C22×C3C2×D3611yescharacteristic
345C22×C3C2×#62611yescharacteristic
346C2×D3—62no—
347C2×D3—121no—
348C2×D3—121no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—41no—
354C3×C4—182no—
355C3×C4D4×D911yescharacteristic
356C3×C4D4×D911yescharacteristic
357C3×C4D3×#6711yescharacteristic
358C3×C4D3×#6711yescharacteristic
359Dic3—62no—
360Dic3—362no—
361Dic3—362no—
362Dic3—361no—
363Dic3—361no—
364Dic3—361no—
365Dic3—361no—
366C32#473311yescharacteristic
367C9—21no—
368C9#523911yescharacteristic, complemented
369C23—181no—
370C23#619211yescharacteristic
371Q8—92no—
372Q8—181no—
373Q8—541no—
374Q8—541no—
375D4—92no—
376D4—184no—
377D4—181no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—184no—
381C2×C4—181no—
382C2×C4—541no—
383C2×C4—541no—
384C2×C4#619211yescharacteristic
385C2×C4C2×D3×D911yescharacteristic
386C8—61no—
387C8—61no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—42no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—362no—
395C2×C3#6344411yescharacteristic
396C2×C3#6346711yescharacteristic
397C2×C3#6347111yescharacteristic
398C2×C3#6382911yescharacteristic
399C2×C3#6383111yescharacteristic
400C2×C3#6397711yescharacteristic
401D3—122no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—182no—
406C22—361no—
407C22—361no—
408C22C2×#619211yescharacteristic
409C4—21no—
410C4—182no—
411C4—1081no—
412C4—1081no—
413C4D9×#6711yescharacteristic
414C4D9×#6711yescharacteristic
415C3—21no—
416C3#1058841511yescharacteristic, complemented
417C3#1059117311yescharacteristic
418C2—41no—
419C2—362no—
420C2#1172086711yescharacteristic
421C2#1172086911yescharacteristic
422C2#1172127611yescharacteristic
subgroupG/H
centerC22C2×#6192
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×D12
prime radicalC2×C3×#3697C2
hypercenterC22×C4D3×D9
hypocenterC3×C9#452
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. #11721813C2semidirect
3. C2×#88418C2semidirect
4. C2×#88415C2semidirect
5. C9×#830C2semidirect
6. #11720283C2semidirect
7. #11720276C2semidirect
121. #830D9semidirect
254. C3×C9#452semidirect
368. C9#5239semidirect
416. C3#10588415semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36972
#117218132
C2×#884182
C2×#884152
C9×#8302
#117202832
#117202762
#105911733
#105884153