Finite Group Databasebeta

← #13164473

Subgroups — #13164473

Order 1728.

946 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 946 conjugacy classes of subgroups — too many to draw. Only its 107 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×C4×C22⋊C9C211yesmaximal, characteristic, complemented
2#11722396C212yesmaximal, complemented
3C4⋊(C3⋊((C22⋊C9)⋊C2))C211yesmaximal, characteristic, complemented
4C4×C3⋊((C22⋊C9)⋊C2)C212yesmaximal, complemented
5#11722377C211yesmaximal, characteristic, complemented
6#10596262—31nomaximal
7#10593087—33nomaximal
8C22×C3×C22⋊C9C2211yescharacteristic
9C2×C3⋊((C22⋊C9)⋊C2)C2212yes—
10#88736—41nomaximal
11C3×C4×C22⋊C9C2211yescharacteristic
12C3×C4×C22⋊C9C2211yescharacteristic
13C3⋊((C22⋊C9)⋊C4)C2212yes—
14C23×C32×C4D311yescharacteristic
15C2×#3828—31no—
16#64162—32no—
17#64139—31no—
18#64139—31no—
19#64138—32no—
20C4×#638—32no—
21C4×#638—32no—
22#64135—31no—
23#64133—31no—
24#64133—31no—
25C2×C4×C22⋊C9D313yescomplemented
26#63694—36no—
27C4⋊((C22⋊C9)⋊C2)—33no—
28C4×(C22⋊C9)⋊C2—36no—
29(C22⋊C9)⋊Q8—33no—
30C2×C3×C22⋊C9—21no—
31C2×C3×C22⋊C9C2311yescharacteristic
32C3⋊((C22⋊C9)⋊C2)—22no—
33C2×C3×C4×C9—41no—
34#6233—42no—
35C4⋊(C3⋊D9)—41no—
36C4×C3⋊D9—42no—
37#6226—41no—
38#5791—93no—
39#5791—91no—
40C24×C32C2×D311yescharacteristic
41C22×C32×C4—31no—
42C22×C32×C4—31no—
43C22×C32×C4—31no—
44C22×C32×C4—31no—
45C22×C32×C4C2×D311yescharacteristic
46C22×C32×C4C2×D311yescharacteristic
47C2×#638—32no—
48C2×#638—32no—
49C2×#638—32no—
50#3828—61no—
51#3828—61no—
52#3828—61no—
53#3828—61no—
54C2×C4⋊(C3⋊D3)—31no—
55C2×C4×C3⋊D3—32no—
56C2×#634—31no—
57C22×C22⋊C9C2×D313yes—
58C2×(C22⋊C9)⋊C2—36no—
59#3757—32no—
60#3757—32no—
61#3757—32no—
62#3752—32no—
63#3752—32no—
64C4⋊(C3⋊Dic3)—31no—
65C4⋊(C3⋊Dic3)—31no—
66#3750—32no—
67#3750—32no—
68C4×C3⋊Dic3—32no—
69#3697—123no—
70C4×C22⋊C9C2×D313yes—
71C4×C22⋊C9C2×D313yes—
72(C22⋊C9)⋊C4—36no—
73C22×C3×C9—41no—
74C2×C3⋊D9—42no—
75C3×C22⋊C9#4111yescharacteristic, complemented
76C3×C4×C9—41no—
77C3×C4×C9—41no—
78C3⋊Dic9—42no—
79C23×C3×C4D913yescomplemented
80C23×C3×C4C3⋊D311yescharacteristic
81C2×#235—93no—
82C2×#235—91no—
83#953—96no—
84#953—92no—
85#930—93no—
86#930—91no—
87#930—93no—
88#930—91no—
89#929—96no—
90#929—92no—
91C4×#67—96no—
92C4×#67—92no—
93C4×#67—96no—
94C4×#67—92no—
95#926—93no—
96#926—91no—
97#924—93no—
98#924—91no—
99#924—93no—
100#924—91no—
101C23×C32—21no—
102C23×C32—31no—
103C23×C32—31no—
104C23×C32—61no—
105C23×C32C22×D311yescharacteristic
106C22×C3⋊D3—32no—
107C2×C32×C4—31no—
108C2×C32×C4—31no—
109C2×C32×C4—31no—
110C2×C32×C4—31no—
111C2×C32×C4—31no—
112C2×C32×C4—31no—
113C2×C32×C4—31no—
114C2×C32×C4—61no—
115C2×C32×C4S411yescharacteristic
116#638—62no—
117#638—62no—
118#638—64no—
119#638—62no—
120#638—62no—
121C2×C3⋊Dic3—32no—
122C2×C3⋊Dic3—32no—
123C2×C3⋊Dic3—32no—
124C4⋊(C3⋊D3)—61no—
125C4⋊(C3⋊D3)—61no—
126C4×C3⋊D3—62no—
127C4×C3⋊D3—62no—
128#634—61no—
129#634—61no—
130C2×C22⋊C9—23no—
131C2×C22⋊C9C22×D313yes—
132(C22⋊C9)⋊C2—66no—
133C2×C4×C9—43no—
134#611—126no—
135D36—123no—
136C4×D9—126no—
137Dic18—123no—
138#525—271noSylow
139C2×C3×C9—41no—
140C2×C3×C9—81no—
141C3⋊D9—82no—
142C24×C3C2×D913yes—
143C24×C3C2×C3⋊D311yescharacteristic
144C22×C3×C4—33no—
145C22×C3×C4—31no—
146C22×C3×C4—33no—
147C22×C3×C4—31no—
148C22×C3×C4—33no—
149C22×C3×C4—31no—
150C22×C3×C4—33no—
151C22×C3×C4—31no—
152C22×C3×C4C2×D913yes—
153C22×C3×C4C2×D913yes—
154C22×C3×C4C2×C3⋊D311yescharacteristic
155C22×C3×C4C2×C3⋊D311yescharacteristic
156C2×#67—96no—
157C2×#67—92no—
158C2×#67—92no—
159C2×#67—96no—
160C2×#67—96no—
161C2×#67—92no—
162#235—181no—
163#235—183no—
164#235—181no—
165#235—183no—
166#235—181no—
167#235—183no—
168#235—183no—
169#235—181no—
170C2×D12—91no—
171C2×D12—93no—
172C2×C4×D3—92no—
173C2×C4×D3—96no—
174C2×Dic6—91no—
175C2×Dic6—93no—
176#217—96no—
177#217—92no—
178#217—92no—
179#217—96no—
180#217—96no—
181#217—92no—
182(C2×D3)⋊C4—96no—
183(C2×D3)⋊C4—96no—
184(C2×D3)⋊C4—92no—
185(C2×D3)⋊C4—92no—
186C4⋊Dic3—91no—
187C4⋊Dic3—91no—
188C4⋊Dic3—93no—
189C4⋊Dic3—93no—
190Dic3⋊C4—96no—
191Dic3⋊C4—96no—
192Dic3⋊C4—92no—
193Dic3⋊C4—92no—
194C4×Dic3—92no—
195C4×Dic3—96no—
196C22×C32—31no—
197C22×C32—31no—
198C22×C32—32no—
199C22×C32—31no—
200C22×C32—61no—
201C22×C32—61no—
202C22×C32—61no—
203C22×C32#23511yescharacteristic
204C22×C32C2×S411yescharacteristic
205C2×C3⋊D3—62no—
206C2×C3⋊D3—62no—
207C2×C3⋊D3—62no—
208C32×C4—31no—
209C32×C4—31no—
210C32×C4C2×S411yescharacteristic
211C32×C4C2×S411yescharacteristic
212C3⋊Dic3—62no—
213C3⋊Dic3—62no—
214C3⋊Dic3—62no—
215C22×C9—43no—
216C2×D9—126no—
217C22⋊C9#23513yescomplemented
218C4×C9—43no—
219C4×C9—43no—
220Dic9—126no—
221C2×#41—271no—
222C23×C4C3⋊D911yescharacteristic, complemented
223#123—272no—
224C22⋊Q8—271no—
225C22⋊Q8—271no—
226#121—271no—
227#121—271no—
228#120—272no—
229C4×D4—272no—
230C4×D4—272no—
231#117—271no—
232C3×C9—41noSylow
233C23×C3—23no—
234C23×C3—21no—
235C23×C3—33no—
236C23×C3—31no—
237C23×C3—31no—
238C23×C3—33no—
239C23×C3—63no—
240C23×C3—61no—
241C23×C3C22×D913yes—
242C23×C3C22×C3⋊D311yescharacteristic
243C22×D3—92no—
244C22×D3—96no—
245C2×C3×C4—31no—
246C2×C3×C4—33no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—33no—
252C2×C3×C4—33no—
253C2×C3×C4—33no—
254C2×C3×C4—33no—
255C2×C3×C4—33no—
256C2×C3×C4—33no—
257C2×C3×C4—31no—
258C2×C3×C4—31no—
259C2×C3×C4—61no—
260C2×C3×C4—63no—
261C2×C3×C4(C22⋊C9)⋊C213yescomplemented
262C2×C3×C4C3⋊S411yescharacteristic
263#67—182no—
264#67—186no—
265#67—186no—
266#67—182no—
267#67—1812no—
268#67—184no—
269#67—182no—
270#67—186no—
271#67—186no—
272#67—182no—
273C2×Dic3—92no—
274C2×Dic3—92no—
275C2×Dic3—96no—
276C2×Dic3—96no—
277C2×Dic3—92no—
278C2×Dic3—96no—
279D12—181no—
280D12—183no—
281D12—181no—
282D12—183no—
283C4×D3—182no—
284C4×D3—186no—
285C4×D3—182no—
286C4×D3—186no—
287Dic6—181no—
288Dic6—183no—
289Dic6—181no—
290Dic6—183no—
291C2×C32—21no—
292C2×C32—31no—
293C2×C32—31no—
294C2×C32—61no—
295C2×C32C22×S411yescharacteristic
296C3⋊D3—122no—
297C2×C9—43no—
298C2×C9—83no—
299D9—246no—
300C24C2×C3⋊D911yescharacteristic
301#41—541no—
302#41—541no—
303#41—541no—
304#41—541no—
305C2×Q8—271no—
306C2×D4—271no—
307C2×D4—272no—
308C2×D4—272no—
309C2×D4—272no—
310C22×C4—31no—
311C22×C4—31no—
312C22×C4—31no—
313C22×C4—31no—
314C22×C4—272no—
315C22×C4C2×C3⋊D911yescharacteristic
316C22×C4C2×C3⋊D911yescharacteristic
317C4⋊C4—271no—
318C4⋊C4—271no—
319C4⋊C4—272no—
320C4⋊C4—272no—
321C22⋊C4—272no—
322C22⋊C4—272no—
323C22⋊C4—272no—
324C22⋊C4—272no—
325C22⋊C4—272no—
326C42—272no—
327C22×C3—33no—
328C22×C3—31no—
329C22×C3—31no—
330C22×C3—33no—
331C22×C3—36no—
332C22×C3—32no—
333C22×C3—31no—
334C22×C3—33no—
335C22×C3—63no—
336C22×C3—61no—
337C22×C3—63no—
338C22×C3—61no—
339C22×C3—63no—
340C22×C3—61no—
341C22×C3#369713yescomplemented
342C22×C3C2×(C22⋊C9)⋊C213yes—
343C22×C3#382811yescharacteristic
344C22×C3C2×C3⋊S411yescharacteristic
345C2×D3—186no—
346C2×D3—186no—
347C2×D3—182no—
348C2×D3—182no—
349C2×D3—182no—
350C2×D3—186no—
351C3×C4—31no—
352C3×C4—31no—
353C3×C4—33no—
354C3×C4—33no—
355C3×C4C2×(C22⋊C9)⋊C213yes—
356C3×C4C2×(C22⋊C9)⋊C213yes—
357C3×C4C2×C3⋊S411yescharacteristic
358C3×C4C2×C3⋊S411yescharacteristic
359Dic3—186no—
360Dic3—186no—
361Dic3—182no—
362Dic3—182no—
363Dic3—182no—
364Dic3—186no—
365C32#591211yescharacteristic
366C9—43no—
367C23—21no—
368C23—31no—
369C23—31no—
370C23—61no—
371C23—272no—
372C23C22×C3⋊D911yescharacteristic
373Q8—541no—
374Q8—541no—
375D4—541no—
376D4—542no—
377D4—542no—
378D4—544no—
379D4—542no—
380D4—542no—
381D4—541no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—31no—
385C2×C4—31no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—31no—
389C2×C4—61no—
390C2×C4—272no—
391C2×C4—272no—
392C2×C4—272no—
393C2×C4—542no—
394C2×C4—542no—
395C2×C4C3⋊((C22⋊C9)⋊C2)11yescharacteristic, complemented
396C2×C3—23no—
397C2×C3—21no—
398C2×C3—33no—
399C2×C3—33no—
400C2×C3—31no—
401C2×C3—31no—
402C2×C3—63no—
403C2×C3—61no—
404C2×C3C22×(C22⋊C9)⋊C213yes—
405C2×C3C22×C3⋊S411yescharacteristic
406D3—366no—
407D3—362no—
408C22—31no—
409C22—31no—
410C22—32no—
411C22—31no—
412C22—61no—
413C22—61no—
414C22—61no—
415C22—542no—
416C22—542no—
417C22—542no—
418C22#8873611yescharacteristic, complemented
419C22C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
420C4—31no—
421C4—31no—
422C4—542no—
423C4—542no—
424C4—542no—
425C4C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
426C4C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
427C3#1059308713yescomplemented
428C3#1059657511yescharacteristic
429C2—21no—
430C2—31no—
431C2—31no—
432C2—61no—
433C2—1082no—
434C2C22×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
subgroupG/H
centerC4C2×C3⋊((C22⋊C9)⋊C2)
commutator, cosocleC2×C3×C22⋊C9C23
FrattiniC2×C3C22×C3⋊S4
FittingC23×C32×C4D3
socleC23×C32C22×D3
hypercenterC2×C4C3⋊((C22⋊C9)⋊C2)
hypocenterC3×C22⋊C9#41
LCSG ▷ C2×C3×C22⋊C9 ▷ C3×C22⋊C9
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C22⋊C9C2semidirect
2. #11722396C2semidirect
3. C4⋊(C3⋊((C22⋊C9)⋊C2))C2semidirect
4. C4×C3⋊((C22⋊C9)⋊C2)C2semidirect
5. #11722377C2semidirect
25. C2×C4×C22⋊C9D3semidirect
75. C3×C22⋊C9#41semidirect
79. C23×C3×C4D9semidirect
217. C22⋊C9#235semidirect
222. C23×C4C3⋊D9semidirect
261. C2×C3×C4(C22⋊C9)⋊C2semidirect
341. C22×C3#3697semidirect
395. C2×C4C3⋊((C22⋊C9)⋊C2)semidirect
418. C22#88736semidirect
427. C3#10593087semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊C92
#117223962
C4⋊(C3⋊((C22⋊C9)⋊C2))2
C4×C3⋊((C22⋊C9)⋊C2)2
#117223772
#105962623
#105930873
#887364