Finite Group Databasebeta

← #13140456

Subgroups — #13140456

Order 1728.

514 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 93 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
1D4×C3⋊Dic9C211yesmaximal, characteristic, complemented
2C2×C9×#213C211yesmaximal, characteristic, complemented
3C2×C3×#3673C211yesmaximal, characteristic, complemented
4#11721043C211yesmaximal, characteristic, complemented
5#11720295C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic, complemented
7#11720273C211yesmaximal, characteristic, complemented
8#10591486—31nomaximal
9#10588830—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C22×C3⋊Dic9—21no—
12C2×C9×D12C2211yescharacteristic
13C2×C3×D36C2211yescharacteristic
14C2×#6194—21no—
15C2×#6194—21no—
16#88551—21no—
17C4⋊(C3⋊Dic9)C2211yescharacteristic
18C4×C3⋊Dic9C2211yescharacteristic
19C9×#213—22no—
20C3×#3673—22no—
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23D4×C3⋊Dic3—31no—
24C2×C3×#213—31no—
25C2×C3×#213D311yescharacteristic
26#63914—31no—
27#63572—31no—
28#63563—31no—
29#63563—31no—
30C2×C9×D8—31no—
31#63502—31no—
32D4×Dic9—31no—
33D4×Dic9—61no—
34C2×#3673D311yescharacteristic, complemented
35#63392—31no—
36#63380—31no—
37Dic9⋊C8—31no—
38C23×C3×C9—21no—
39C22×D3×C9—21no—
40C22×C3×D9—21no—
41C3×D4×C9—21no—
42C3×D4×C9D412yescomplemented
43C2×C3×C4×C9C2311yescharacteristic
44C2×C3⋊Dic9—21no—
45C2×C3⋊Dic9—42no—
46C2×C3⋊Dic9D411yescharacteristic
47C2×C3⋊Dic9D411yescharacteristic
48C9×D12—21no—
49C9×D12—21no—
50C3×D36—21no—
51C3×D36—21no—
52#6194—42no—
53#6194—42no—
54C9×C3⋊C8—21no—
55C3×C9⋊C8—21no—
56#5150—91no—
57#5150—91no—
58C2×C32×D4C2×D311yescharacteristic
59C22×C3⋊Dic3—61no—
60C2×C3×D12—31no—
61C2×C3×D12C2×D311yescharacteristic
62C2×#625—61no—
63C2×#625—61no—
64#3757—61no—
65C4⋊(C3⋊Dic3)—31no—
66C4×C3⋊Dic3—31no—
67C3×#213—22no—
68C3×#213—62no—
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8C2×D311yescharacteristic
71C2×D4×C9—21no—
72C2×D4×C9—31no—
73C2×D4×C9C2×D311yescharacteristic
74C2×#611—61no—
75C2×#611—61no—
76C22×Dic9—61no—
77C22×Dic9—62no—
78C2×D36C2×D311yescharacteristic
79C9×D8—62no—
80C2×C8×C9—31no—
81#3676—61no—
82#3676—62no—
83#3673—22no—
84C4⋊Dic9—31no—
85C4⋊Dic9—61no—
86C4×Dic9—31no—
87C4×Dic9—61no—
88C2×C9⋊C8C2×D311yescharacteristic
89C22×C3×C9—22no—
90C22×C3×C9—41no—
91C22×C3×C9—41no—
92C22×C3×C9C2×D411yescharacteristic
93C2×D3×C9—41no—
94C2×D3×C9—41no—
95C2×D3×C9—41no—
96C2×C3×D9—41no—
97C2×C3×D9—41no—
98C2×C3×D9—41no—
99C3×C4×C9C2×D411yescharacteristic
100C3×C4×C9#4111yescharacteristic
101C3⋊Dic9—21no—
102C3⋊Dic9—41no—
103C3⋊Dic9D812yescomplemented
104C2×C3×D8—31no—
105C2×C3×D8—91no—
106#940—91no—
107#940—91no—
108D4×Dic3—91no—
109D4×Dic3—91no—
110D4×Dic3—181no—
111C2×#213—31no—
112C2×#213D911yescharacteristic, complemented
113#832—91no—
114#832—91no—
115#821—91no—
116#821—91no—
117Dic3⋊C8—91no—
118Dic3⋊C8—91no—
119C23×C32—21no—
120C22×C3×D3—21no—
121C22×C3×D3—61no—
122C32×D4—21no—
123C32×D4#6712yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—31no—
126C2×C3⋊Dic3—31no—
127C2×C3⋊Dic3—61no—
128C2×C3⋊Dic3—122no—
129C3×D12—21no—
130C3×D12—21no—
131C3×D12—61no—
132C3×D12—61no—
133#625—122no—
134#625—122no—
135C23×C9—22no—
136C23×C9—21no—
137C23×C9—61no—
138C22×D9—21no—
139C3×C3⋊C8—21no—
140C3×C3⋊C8—61no—
141D4×C9—22no—
142D4×C9—21no—
143D4×C9—22no—
144D4×C9—61no—
145D4×C9—61no—
146D4×C9#6712yescomplemented
147C2×C4×C9—21no—
148C2×C4×C9C22×D311yescharacteristic
149#611—122no—
150#611—122no—
151C2×Dic9—31no—
152C2×Dic9—31no—
153C2×Dic9—61no—
154C2×Dic9—61no—
155C2×Dic9—62no—
156C2×Dic9—61no—
157C2×Dic9—122no—
158C2×Dic9—124no—
159D36—21no—
160D36—21no—
161C8×C9—61no—
162C9⋊C8—21no—
163#459—271noSylow
164C2×C3×C9—42no—
165C2×C3×C9#12111yescharacteristic
166C2×C3×C9C2×D811yescharacteristic
167C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
168D3×C9—81no—
169C3×D9—81no—
170C2×C3×D4—21no—
171C2×C3×D4—31no—
172C2×C3×D4—91no—
173C2×C3×D4C2×D911yescharacteristic
174C2×C3×D4D3211yescharacteristic
175C2×#67—181no—
176C2×#67—181no—
177C2×#67—181no—
178C2×#67—181no—
179C22×Dic3—181no—
180C22×Dic3—181no—
181C22×Dic3—182no—
182C2×D12—31no—
183C2×D12C2×D911yescharacteristic
184C3×D8—62no—
185C3×D8—182no—
186C2×C3×C8—31no—
187C2×C3×C8—91no—
188#217—181no—
189#217—181no—
190#217—182no—
191#213—22no—
192#213—62no—
193C4⋊Dic3—91no—
194C4⋊Dic3—91no—
195C4⋊Dic3—181no—
196C4×Dic3—91no—
197C4×Dic3—91no—
198C4×Dic3—181no—
199C2×C3⋊C8—31no—
200C2×C3⋊C8C2×D911yescharacteristic
201C22×C32—22no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32D3×D411yescharacteristic
205C2×C3×D3—41no—
206C2×C3×D3—41no—
207C2×C3×D3—41no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C2×C3×D3—121no—
211C32×C4#23711yescharacteristic
212C32×C4C2×#6711yescharacteristic
213C3⋊Dic3—32no—
214C3⋊Dic3—61no—
215C3⋊Dic3—121no—
216C22×C9—24no—
217C22×C9—22no—
218C22×C9—21no—
219C22×C9—42no—
220C22×C9—42no—
221C22×C9—41no—
222C22×C9—41no—
223C22×C9—121no—
224C22×C9—121no—
225C22×C9—121no—
226C22×C9D3×D411yescharacteristic
227C2×D9—41no—
228C2×D9—41no—
229C2×D9—41no—
230C4×C9—21no—
231C4×C9—21no—
232C4×C9#23711yescharacteristic
233C4×C9C2×#6711yescharacteristic
234Dic9—32no—
235Dic9—62no—
236Dic9—61no—
237Dic9—121no—
238Dic9—122no—
239Dic9—121no—
240C2×D8—31no—
241C2×D8—91no—
242#127—271no—
243C4×D4—271no—
244C4⋊C8—271no—
245#102—271no—
246#102—271no—
247C3×C9#45911yescharacteristic, complemented, Sylow
248C23×C3—22no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—61no—
252C23×C3—181no—
253C22×D3—21no—
254C22×D3—61no—
255C3×D4—22no—
256C3×D4—21no—
257C3×D4—21no—
258C3×D4—22no—
259C3×D4—61no—
260C3×D4—61no—
261C3×D4—181no—
262C3×D4—181no—
263C3×D4#61112yescomplemented
264C3×D4#62512yes—
265C2×C3×C4—21no—
266C2×C3×C4C22×D911yescharacteristic
267C2×C3×C4C2×D3211yescharacteristic
268#67—362no—
269#67—362no—
270#67—362no—
271#67—362no—
272C2×Dic3—91no—
273C2×Dic3—91no—
274C2×Dic3—91no—
275C2×Dic3—91no—
276C2×Dic3—181no—
277C2×Dic3—181no—
278C2×Dic3—182no—
279C2×Dic3—181no—
280C2×Dic3—181no—
281C2×Dic3—362no—
282C2×Dic3—362no—
283C2×Dic3—364no—
284D12—21no—
285D12—21no—
286D12—61no—
287D12—61no—
288C3×C8—61no—
289C3×C8—181no—
290C3⋊C8—21no—
291C3⋊C8—61no—
292C2×C32—42no—
293C2×C32D3×D811yescharacteristic
294C2×C32#91111yescharacteristic
295C2×C32#93911yescharacteristic
296C3×D3—81no—
297C3×D3—241no—
298C2×C9—21no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9—44no—
302C2×C9—42no—
303C2×C9—241no—
304C2×C9D3×D811yescharacteristic
305C2×C9#91111yescharacteristic
306C2×C9#93911yescharacteristic
307D9—81no—
308C2×D4—31no—
309C2×D4—91no—
310C2×D4—541no—
311C2×D4—541no—
312C2×D4D3×D911yescharacteristic
313C22×C4—541no—
314D8—62no—
315D8—182no—
316C2×C8—31no—
317C2×C8—91no—
318C4⋊C4—271no—
319C22⋊C4—541no—
320C42—271no—
321C22×C3—24no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—21no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—121no—
332C22×C3—121no—
333C22×C3—121no—
334C22×C3—361no—
335C22×C3—361no—
336C22×C3—361no—
337C22×C3D4×D911yescharacteristic
338C22×C3#381111yescharacteristic
339C2×D3—41no—
340C2×D3—41no—
341C2×D3—41no—
342C2×D3—121no—
343C2×D3—121no—
344C2×D3—121no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4#369911yescharacteristic
348C3×C4C2×#61111yescharacteristic
349C3×C4#380511yescharacteristic
350C3×C4C2×#62511yescharacteristic
351Dic3—92no—
352Dic3—92no—
353Dic3—182no—
354Dic3—181no—
355Dic3—181no—
356Dic3—361no—
357Dic3—362no—
358Dic3—361no—
359Dic3—361no—
360C32#515011yescharacteristic
361C9—21no—
362C9#515011yescharacteristic, complemented
363C23—21no—
364C23—61no—
365C23—181no—
366D4—21no—
367D4—61no—
368D4—61no—
369D4—181no—
370D4—181no—
371D4—1082no—
372D4—1082no—
373D4#619412yescomplemented
374C2×C4—271no—
375C2×C4—271no—
376C2×C4—541no—
377C2×C4—1082no—
378C2×C4C2×D3×D911yescharacteristic
379C8—61no—
380C8—181no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3—44no—
385C2×C3—42no—
386C2×C3—42no—
387C2×C3—241no—
388C2×C3—721no—
389C2×C3D8×D911yescharacteristic
390C2×C3#6347311yescharacteristic
391C2×C3#6350111yescharacteristic
392C2×C3#6392611yescharacteristic
393C2×C3#6392711yescharacteristic
394C2×C3#6398011yescharacteristic
395D3—81no—
396D3—241no—
397C22—22no—
398C22—41no—
399C22—41no—
400C22—121no—
401C22—121no—
402C22—121no—
403C22—361no—
404C22—361no—
405C22—361no—
406C22#8866811yescharacteristic
407C4—272no—
408C4—541no—
409C4—1081no—
410C4#8866311yescharacteristic
411C4C2×#619411yescharacteristic
412C3—21no—
413C3#1058883011yescharacteristic, complemented
414C3#1059148611yescharacteristic
415C2—42no—
416C2—241no—
417C2—721no—
418C2#1172117411yescharacteristic
419C2#1172117511yescharacteristic
420C2#1172129011yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
hypercenterC2×D4D3×D9
hypocenterC3×C9#459
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
1. D4×C3⋊Dic9C2semidirect
2. C2×C9×#213C2semidirect
3. C2×C3×#3673C2semidirect
4. #11721043C2semidirect
5. #11720295C2semidirect
6. #11720283C2semidirect
7. #11720273C2semidirect
34. C2×#3673D3semidirect
42. C3×D4×C9D4semidirect
103. C3⋊Dic9D8semidirect
112. C2×#213D9semidirect
146. D4×C9#67semidirect
247. C3×C9#459semidirect
263. C3×D4#611semidirect
362. C9#5150semidirect
373. D4#6194semidirect
413. C3#10588830semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊Dic92
C2×C9×#2132
C2×C3×#36732
#117210432
#117202952
#117202832
#117202732
#105914863
#105888303