Finite Group Databasebeta

← #13139791

Subgroups — #13139791

Order 1728.

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

normal subgroups only

This group has 506 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
2C9×#924C211yesmaximal, characteristic, complemented
3C2×#88427C211yesmaximal, characteristic, complemented
4C2×#88425C211yesmaximal, characteristic, complemented
5#11720571C211yesmaximal, characteristic, complemented
6#11720268C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic, complemented
8#10591170—31nomaximal
9#10588730—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×C3×#611—21no—
12C3×#3697—22no—
13C3×#3697—22no—
14C2×C3×D36C2211yescharacteristic
15C2×C3×C4×D9—21no—
16C2×C9×Dic6C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18C2×C3⋊(C9⋊C8)C2211yescharacteristic
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20C9×#217—21no—
21C9×C4⋊Dic3C2211yescharacteristic
22C9×Dic3⋊C4—21no—
23#88427—22no—
24#88425—22no—
25C2×C3×#235—31no—
26C3×#924D311yescharacteristic
27C2×#3718—31no—
28C2×#3715—31no—
29C2×#3697D311yescharacteristic
30#63657—31no—
31#63565—31no—
32#63562—31no—
33C9×C22⋊Q8—31no—
34C2×#3675—31no—
35C2×#3674—31no—
36#63389—31no—
37#63389—61no—
38#63369—31no—
39#63368—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
46C3×#611—24no—
47C2×C9×Dic3—21no—
48C2×C9×Dic3—21no—
49C2×C3×Dic9—21no—
50C3×D36—21no—
51C3×D36D412yes—
52C3×C4×D9—24no—
53C9×Dic6—21no—
54C9×Dic6—21no—
55C3×Dic18—21no—
56C3×Dic18D412yes—
57C3⋊(C9⋊C8)—21no—
58C3⋊(C9⋊C8)—21no—
59#5239—91no—
60#5047—91no—
61C22×C32×C4C2×D311yescharacteristic
62C2×C3×#67—61no—
63C3×#235—62no—
64C3×#235—62no—
65C2×C3×D12—31no—
66C2×C3×C4×D3—61no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—31no—
71C3×#217—21no—
72C3×C4⋊Dic3C2×D311yescharacteristic
73C3×Dic3⋊C4—21no—
74#3718—62no—
75#3715—62no—
76C2×Q8×C9—31no—
77C22×C4×C9—21no—
78C22×C4×C9C2×D311yescharacteristic
79C2×#611—21no—
80#3697—22no—
81#3697—22no—
82C2×D36C2×D311yescharacteristic
83C2×C4×D9—21no—
84C2×Dic18C2×D311yescharacteristic
85C9×C4⋊C4—31no—
86C9×C4⋊C4—61no—
87C9×C22⋊C4—61no—
88#3675—62no—
89#3674—62no—
90C2×C9⋊C8—31no—
91C2×C9⋊C8—31no—
92C2×C9⋊C8—61no—
93C2×C9⋊C8—61no—
94C22×C3×C9—21no—
95C22×C3×C9—21no—
96C22×C3×C9—21no—
97C22×C3×C9C2×D411yescharacteristic
98C2×C3×D9—22no—
99C2×C3×D9—41no—
100C2×C3×D9—41no—
101C3×C4×C9—21no—
102C3×C4×C9C2×D411yescharacteristic
103C3×C4×C9C2×D411yescharacteristic
104C9×Dic3—41no—
105C9×Dic3—41no—
106C3×Dic9—22no—
107C2×C3×#41—91no—
108C2×#235—31no—
109C3×C22⋊Q8—31no—
110C2×#216—91no—
111C2×#216—91no—
112C2×#215—91no—
113C2×#214—91no—
114#924D911yescharacteristic, complemented
115#835—91no—
116#832—91no—
117#830—91no—
118#830—91no—
119#830—181no—
120#810—91no—
121#809—91no—
122C23×C32#6711yescharacteristic
123C22×C3×D3—61no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4#6711yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C3×#67—64no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—61no—
132C3×D12—32no—
133C3×D12—61no—
134C3×C4×D3—64no—
135C3×Dic6—21no—
136C3×Dic6—21no—
137C3×Dic6—32no—
138C3×Dic6—61no—
139C23×C9—21no—
140C23×C9#6711yescharacteristic
141C22×D9—21no—
142C3⋊(C3⋊C8)—61no—
143C3⋊(C3⋊C8)—61no—
144Q8×C9—61no—
145Q8×C9—61no—
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—61no—
153C2×C4×C9—61no—
154C2×C4×C9#6711yescharacteristic
155C2×C4×C9C22×D311yescharacteristic
156#611—24no—
157C2×Dic9—21no—
158D36—21no—
159D36#6712yes—
160C4×D9—24no—
161Dic18—21no—
162Dic18#6712yes—
163C9⋊C8—61no—
164C9⋊C8—61no—
165C9⋊C8—121no—
166C9⋊C8—121no—
167#452—271noSylow
168C2×C3×C9—41no—
169C2×C3×C9#12011yescharacteristic
170C2×C3×C9#13511yescharacteristic
171C2×C3×C9#13711yescharacteristic
172C3×D9—42no—
173C3×#41—182no—
174C3×#41—182no—
175C2×C3×Q8—31no—
176C2×C3×Q8—91no—
177C2×C3×D4—91no—
178C2×C3×D4—181no—
179C22×C3×C4—21no—
180C22×C3×C4—181no—
181C22×C3×C4C2×D911yescharacteristic
182C22×C3×C4D3211yescharacteristic
183C2×#67—61no—
184#235—62no—
185#235—62no—
186C2×D12—31no—
187C2×C4×D3—61no—
188C2×Dic6—31no—
189C2×Dic6C2×D911yescharacteristic
190C3×C4⋊C4—31no—
191C3×C4⋊C4—61no—
192C3×C22⋊C4—61no—
193#217—21no—
194#216—182no—
195#216—182no—
196#215—182no—
197#214—182no—
198C4⋊Dic3C2×D911yescharacteristic
199Dic3⋊C4—21no—
200C2×C3⋊C8—91no—
201C2×C3⋊C8—91no—
202C2×C3⋊C8—91no—
203C2×C3⋊C8—91no—
204C2×C3⋊C8—181no—
205C2×C3⋊C8—181no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32C2×#6711yescharacteristic
210C2×C3×D3—62no—
211C2×C3×D3—121no—
212C2×C3×D3—121no—
213C32×C4—21no—
214C32×C4D3×D411yescharacteristic
215C32×C4D3×D411yescharacteristic
216C3×Dic3—41no—
217C3×Dic3—41no—
218C3×Dic3—62no—
219C22×C9—22no—
220C22×C9—21no—
221C22×C9—21no—
222C22×C9—21no—
223C22×C9—21no—
224C22×C9—41no—
225C22×C9—41no—
226C22×C9C2×#6711yescharacteristic
227C2×D9—22no—
228C2×D9—41no—
229C2×D9—41no—
230C4×C9—21no—
231C4×C9—21no—
232C4×C9—21no—
233C4×C9—41no—
234C4×C9—121no—
235C4×C9—121no—
236C4×C9C2×#6711yescharacteristic
237C4×C9C2×#6711yescharacteristic
238Dic9—22no—
239C2×#41—91no—
240C2×Q16—271no—
241C2×SD16—271no—
242C22⋊Q8—31no—
243#103—271no—
244#102—271no—
245C22⋊C8—271no—
246C3×C9#45211yescharacteristic, complemented, Sylow
247C23×C3—21no—
248C23×C3—181no—
249C23×C3#61111yescharacteristic
250C23×C3#62511yescharacteristic
251C22×D3—61no—
252C3×Q8—61no—
253C3×Q8—61no—
254C3×Q8—92no—
255C3×Q8—181no—
256C3×D4—92no—
257C3×D4—184no—
258C3×D4—181no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—184no—
270C2×C3×C4—181no—
271C2×C3×C4#61111yescharacteristic
272C2×C3×C4C22×D911yescharacteristic
273C2×C3×C4#62511yescharacteristic
274C2×C3×C4C2×D3211yescharacteristic
275#67—64no—
276C2×Dic3—21no—
277C2×Dic3—21no—
278C2×Dic3—61no—
279D12—32no—
280D12—61no—
281C4×D3—64no—
282Dic6—21no—
283Dic6—21no—
284Dic6—32no—
285Dic6—61no—
286C3⋊C8—181no—
287C3⋊C8—181no—
288C3⋊C8—181no—
289C3⋊C8—181no—
290C3⋊C8—361no—
291C3⋊C8—361no—
292C2×C32—41no—
293C2×C32#93711yescharacteristic
294C2×C32#94211yescharacteristic
295C2×C32#95011yescharacteristic
296C3×D3—122no—
297C2×C9—21no—
298C2×C9—21no—
299C2×C9—21no—
300C2×C9—42no—
301C2×C9—41no—
302C2×C9#95011yescharacteristic
303C2×C9#95111yescharacteristic
304C2×C9#95311yescharacteristic
305D9—42no—
306#41—182no—
307#41—182no—
308C2×Q8—31no—
309C2×Q8—91no—
310C2×D4—91no—
311C2×D4—181no—
312C22×C4—181no—
313C22×C4D3×D911yescharacteristic
314Q16—542no—
315SD16—542no—
316C2×C8—271no—
317C2×C8—271no—
318C4⋊C4—31no—
319C4⋊C4—61no—
320C22⋊C4—61no—
321C22×C3—22no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—182no—
332C22×C3—361no—
333C22×C3—361no—
334C22×C3C2×#61111yescharacteristic
335C22×C3C2×#62511yescharacteristic
336C2×D3—62no—
337C2×D3—121no—
338C2×D3—121no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—41no—
344C3×C4—121no—
345C3×C4—121no—
346C3×C4—182no—
347C3×C4D4×D911yescharacteristic
348C3×C4D4×D911yescharacteristic
349C3×C4D3×#6711yescharacteristic
350C3×C4D3×#6711yescharacteristic
351Dic3—41no—
352Dic3—41no—
353Dic3—62no—
354C32#504711yescharacteristic
355C9—21no—
356C9#523911yescharacteristic, complemented
357C23—181no—
358C23#619411yescharacteristic
359Q8—61no—
360Q8—61no—
361Q8—92no—
362Q8—181no—
363D4—92no—
364D4—184no—
365D4—181no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—184no—
371C2×C4—181no—
372C2×C4#619411yescharacteristic
373C2×C4C2×D3×D911yescharacteristic
374C8—541no—
375C8—541no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—42no—
380C2×C3—41no—
381C2×C3—41no—
382C2×C3—362no—
383C2×C3#6349911yescharacteristic
384C2×C3#6350511yescharacteristic
385C2×C3#6351311yescharacteristic
386C2×C3#6382211yescharacteristic
387C2×C3#6382711yescharacteristic
388C2×C3#6397611yescharacteristic
389D3—122no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—182no—
394C22—361no—
395C22—361no—
396C22C2×#619411yescharacteristic
397C4—21no—
398C4—121no—
399C4—121no—
400C4—182no—
401C4D9×#6711yescharacteristic
402C4D9×#6711yescharacteristic
403C3—21no—
404C3#1058873011yescharacteristic, complemented
405C3#1059117011yescharacteristic
406C2—41no—
407C2—362no—
408C2#1172088111yescharacteristic
409C2#1172089311yescharacteristic
410C2#1172128511yescharacteristic
subgroupG/H
centerC22C2×#6194
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×#67
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. C9×#924C2semidirect
3. C2×#88427C2semidirect
4. C2×#88425C2semidirect
5. #11720571C2semidirect
6. #11720268C2semidirect
7. #11720266C2semidirect
114. #924D9semidirect
246. C3×C9#452semidirect
356. C9#5239semidirect
404. C3#10588730semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36972
C9×#9242
C2×#884272
C2×#884252
#117205712
#117202682
#117202662
#105911703
#105887303