Finite Group Databasebeta

← #13140368

Subgroups — #13140368

Order 1728.

496 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 85 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#11721838C211yesmaximal, characteristic, complemented
2C2×C3×#3673C211yesmaximal, characteristic, complemented
3#11721007C211yesmaximal, characteristic, complemented
4C2×(C3⋊D9)⋊C8C211yesmaximal, characteristic, complemented
5C9×#832C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic, complemented
7#11720262C211yesmaximal, characteristic, complemented
8#10591455—31nomaximal
9#10588455—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×C4×C3⋊D9C2211yescharacteristic
13C2×C3×D36C2211yescharacteristic
14C2×#6192—21no—
15#88551—21no—
16C4⋊(C3⋊Dic9)C2211yescharacteristic
17C3×#3673—22no—
18C9×C4⋊Dic3C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21(C2×D9)⋊Dic3—21no—
22(C3⋊D9)⋊C8—22no—
23#64147—31no—
24C2×C3×#213—31no—
25#63916—31no—
26C2×#3710—31no—
27C3×#832D311yescharacteristic
28#63575—31no—
29#63563—31no—
30#63500—31no—
31#63500—61no—
32C2×#3673D311yescharacteristic, complemented
33C2×C8×D9—31no—
34#63457—31no—
35C9×#102—31no—
36#63380—31no—
37#63366—31no—
38C23×C3×C9—21no—
39C22×C3⋊D9D411yescharacteristic
40C22×C3×D9—21no—
41C3×D4×C9—21no—
42C3×D4×C9—21no—
43C2×C3×C4×C9C2311yescharacteristic
44#6233—42no—
45C2×C3⋊Dic9—21no—
46C2×C3⋊Dic9D411yescharacteristic
47C4×C3⋊D9—21no—
48C4×C3⋊D9—21no—
49C2×C9×Dic3—21no—
50C3×D36—21no—
51C3×D36—21no—
52#6192—42no—
53C9×C3⋊C8—21no—
54C3×C9⋊C8D412yescomplemented
55#5157—91no—
56#4775—91no—
57C2×C32×D4C2×D311yescharacteristic
58C2×#638—61no—
59C2×C4×C3⋊D3—31no—
60C2×C3×D12—31no—
61C2×#626—61no—
62#3757—61no—
63C4⋊(C3⋊Dic3)—31no—
64C3×#213—62no—
65C3×C4⋊Dic3C2×D311yescharacteristic
66C2×C3×C3⋊C8—31no—
67C2×C3×C3⋊C8C2×D311yescharacteristic
68(C2×D3)⋊Dic3—61no—
69#3710—62no—
70C2×D4×C9—21no—
71C2×D4×C9C2×D311yescharacteristic
72C2×#611—61no—
73C2×#611—62no—
74C2×D36—61no—
75C2×D36C2×D311yescharacteristic
76C2×C4×D9—31no—
77C2×C4×D9—61no—
78C2×C8×C9—31no—
79C9×C4⋊C4—31no—
80#3676—61no—
81#3676—62no—
82#3673—22no—
83(C2×D9)⋊C4—61no—
84C4⋊Dic9—31no—
85C4⋊Dic9—61no—
86C2×C9⋊C8C2×D311yescharacteristic
87C8×D9—62no—
88C22×C3×C9—41no—
89C22×C3×C9—41no—
90C22×C3×C9—41no—
91C22×C3×C9C2×D411yescharacteristic
92C2×C3⋊D9—21no—
93C2×C3⋊D9—21no—
94C2×C3⋊D9D812yescomplemented
95C2×C3×D9—41no—
96C2×C3×D9—41no—
97C2×C3×D9—41no—
98C3×C4×C9C2×D411yescharacteristic
99C3×C4×C9#4111yescharacteristic
100C3⋊Dic9—21no—
101C3⋊Dic9—41no—
102C9×Dic3—41no—
103C2×C3×D8—91no—
104#938—91no—
105#938—91no—
106#938—91no—
107#938—181no—
108C2×#213—31no—
109C2×D3×C8—91no—
110C2×D3×C8—91no—
111#895—91no—
112C3×#102—31no—
113#832—91no—
114#832D911yescharacteristic, complemented
115#821—91no—
116#818—91no—
117#807—91no—
118C23×C32—21no—
119C22×C3⋊D3—31no—
120C22×C3×D3—61no—
121C32×D4—21no—
122C32×D4—21no—
123C2×C32×C4C22×D311yescharacteristic
124#638—122no—
125C2×C3⋊Dic3—31no—
126C2×C3⋊Dic3—61no—
127C4×C3⋊D3—61no—
128C4×C3⋊D3—61no—
129C2×C3×Dic3—21no—
130C3×D12—61no—
131C3×D12—61no—
132#626—122no—
133C23×C9—22no—
134C23×C9—21no—
135C22×D9—21no—
136C22×D9—31no—
137C22×D9—61no—
138C3×C3⋊C8—21no—
139C3×C3⋊C8—32no—
140D4×C9—21no—
141D4×C9—21no—
142D4×C9—41no—
143D4×C9—41no—
144C2×C4×C9—21no—
145C2×C4×C9—61no—
146C2×C4×C9C22×D311yescharacteristic
147#611—122no—
148#611—124no—
149C2×Dic9—31no—
150C2×Dic9—61no—
151C2×Dic9—62no—
152C2×Dic9—61no—
153D36—21no—
154D36—21no—
155D36—122no—
156C4×D9—61no—
157C4×D9—61no—
158C4×D9—121no—
159C4×D9—121no—
160C8×C9—61no—
161C9⋊C8#6712yescomplemented
162#466—271noSylow
163C2×C3×C9—81no—
164C2×C3×C9#12111yescharacteristic
165C2×C3×C9C2×D811yescharacteristic
166C2×C3×C9#13511yescharacteristic
167C3⋊D9—22no—
168C3×D9—81no—
169C2×C3×D4—21no—
170C2×C3×D4—91no—
171C2×C3×D4C2×D911yescharacteristic
172C2×C3×D4D3211yescharacteristic
173C2×#67—181no—
174C2×#67—181no—
175C2×#67—181no—
176C2×#67—182no—
177C2×D12—31no—
178C2×D12—181no—
179C2×C4×D3—91no—
180C2×C4×D3—91no—
181C2×C4×D3—181no—
182C3×D8—182no—
183C2×C3×C8—31no—
184C2×C3×C8—91no—
185C3×C4⋊C4—31no—
186#217—181no—
187#217—181no—
188#217—181no—
189#217—182no—
190#213—62no—
191(C2×D3)⋊C4—181no—
192C4⋊Dic3—91no—
193C4⋊Dic3—91no—
194C4⋊Dic3—181no—
195C4⋊Dic3C2×D911yescharacteristic
196C2×C3⋊C8—31no—
197C2×C3⋊C8C2×D911yescharacteristic
198D3×C8—182no—
199D3×C8—182no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32—41no—
203C22×C32D3×D411yescharacteristic
204C2×C3⋊D3—32no—
205C2×C3⋊D3—61no—
206C2×C3⋊D3—61no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C32×C4#23511yescharacteristic
211C32×C4D3×D411yescharacteristic
212C3⋊Dic3—61no—
213C3⋊Dic3—121no—
214C3×Dic3—41no—
215C22×C9—21no—
216C22×C9—42no—
217C22×C9—42no—
218C22×C9—42no—
219C22×C9—41no—
220C22×C9—41no—
221C22×C9—41no—
222C22×C9D3×D411yescharacteristic
223C2×D9—32no—
224C2×D9—41no—
225C2×D9—41no—
226C2×D9—41no—
227C2×D9—62no—
228C2×D9—61no—
229C2×D9—61no—
230C2×D9—121no—
231C2×D9—121no—
232C4×C9—21no—
233C4×C9—21no—
234C4×C9—121no—
235C4×C9#23711yescharacteristic
236C4×C9C2×#6711yescharacteristic
237Dic9—61no—
238Dic9—121no—
239Dic9—122no—
240Dic9—121no—
241C2×D8—91no—
242C22×C8—271no—
243#121—271no—
244#121—271no—
245#107—271no—
246#102—31no—
247#102—271no—
248C3×C9#46611yescharacteristic, complemented, Sylow
249C23×C3—22no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—181no—
253C22×D3—61no—
254C22×D3—91no—
255C22×D3—91no—
256C22×D3—181no—
257C3×D4—21no—
258C3×D4—21no—
259C3×D4—21no—
260C3×D4—21no—
261C3×D4—41no—
262C3×D4—41no—
263C3×D4—181no—
264C3×D4—181no—
265C2×C3×C4—21no—
266C2×C3×C4—61no—
267C2×C3×C4C22×D911yescharacteristic
268C2×C3×C4C2×D3211yescharacteristic
269#67—362no—
270#67—362no—
271#67—362no—
272#67—364no—
273C2×Dic3—21no—
274C2×Dic3—91no—
275C2×Dic3—91no—
276C2×Dic3—181no—
277C2×Dic3—182no—
278C2×Dic3—181no—
279C2×Dic3—181no—
280D12—61no—
281D12—61no—
282D12—362no—
283C4×D3—181no—
284C4×D3—181no—
285C4×D3—181no—
286C4×D3—181no—
287C4×D3—361no—
288C4×D3—361no—
289C3×C8—61no—
290C3×C8—92no—
291C3⋊C8—21no—
292C3⋊C8—32no—
293C2×C32—81no—
294C2×C32#88411yescharacteristic
295C2×C32D3×D811yescharacteristic
296C2×C32#91611yescharacteristic
297C3⋊D3—62no—
298C3×D3—241no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9—21no—
302C2×C9—82no—
303C2×C9—81no—
304C2×C9D3×D811yescharacteristic
305C2×C9#91211yescharacteristic
306C2×C9#93811yescharacteristic
307D9—62no—
308D9—81no—
309D9—122no—
310C2×D4—91no—
311C2×D4—541no—
312C2×D4—541no—
313C2×D4D3×D911yescharacteristic
314C22×C4—271no—
315D8—182no—
316C2×C8—31no—
317C2×C8—91no—
318C2×C8—542no—
319C4⋊C4—31no—
320C4⋊C4—271no—
321C22⋊C4—541no—
322C22⋊C4—541no—
323C22×C3—21no—
324C22×C3—42no—
325C22×C3—42no—
326C22×C3—42no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—361no—
334C22×C3—361no—
335C22×C3—361no—
336C22×C3D4×D911yescharacteristic
337C22×C3#381111yescharacteristic
338C2×D3—92no—
339C2×D3—92no—
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—121no—
343C2×D3—182no—
344C2×D3—181no—
345C2×D3—181no—
346C2×D3—181no—
347C2×D3—181no—
348C2×D3—361no—
349C2×D3—361no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—121no—
353C3×C4#369711yescharacteristic
354C3×C4D4×D911yescharacteristic
355C3×C4#380411yescharacteristic
356C3×C4D3×#6711yescharacteristic
357Dic3—41no—
358Dic3—181no—
359Dic3—181no—
360Dic3—361no—
361Dic3—362no—
362Dic3—361no—
363Dic3—361no—
364C32#477511yescharacteristic
365C9—21no—
366C9#515711yescharacteristic, complemented
367C23—21no—
368C23—181no—
369C23—271no—
370D4—21no—
371D4—21no—
372D4—181no—
373D4—181no—
374D4—1082no—
375D4—1082no—
376C2×C4—61no—
377C2×C4—271no—
378C2×C4—541no—
379C2×C4—541no—
380C2×C4—541no—
381C2×C4C2×D3×D911yescharacteristic
382C8—61no—
383C8—92no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—82no—
388C2×C3—81no—
389C2×C3—81no—
390C2×C3—721no—
391C2×C3#6344611yescharacteristic
392C2×C3D8×D911yescharacteristic
393C2×C3#6347811yescharacteristic
394C2×C3#6392611yescharacteristic
395C2×C3#6393611yescharacteristic
396C2×C3#6395811yescharacteristic
397D3—182no—
398D3—182no—
399D3—241no—
400D3—362no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—272no—
405C22—361no—
406C22—361no—
407C22—361no—
408C22—541no—
409C22—541no—
410C22#8866811yescharacteristic
411C4—121no—
412C4—541no—
413C4—1081no—
414C4#8865811yescharacteristic
415C4D9×#6711yescharacteristic
416C3—21no—
417C3#1058845511yescharacteristic, complemented
418C3#1059145511yescharacteristic
419C2—81no—
420C2—542no—
421C2—721no—
422C2#1172117411yescharacteristic
423C2#1172118611yescharacteristic
424C2#1172125611yescharacteristic
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#466
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. #11721838C2semidirect
2. C2×C3×#3673C2semidirect
3. #11721007C2semidirect
4. C2×(C3⋊D9)⋊C8C2semidirect
5. C9×#832C2semidirect
6. #11720283C2semidirect
7. #11720262C2semidirect
32. C2×#3673D3semidirect
54. C3×C9⋊C8D4semidirect
94. C2×C3⋊D9D8semidirect
114. #832D9semidirect
161. C9⋊C8#67semidirect
248. C3×C9#466semidirect
366. C9#5157semidirect
417. C3#10588455semidirect

Maximal subgroups

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

subgroupindex
#117218382
C2×C3×#36732
#117210072
C2×(C3⋊D9)⋊C82
C9×#8322
#117202832
#117202622
#105914553
#105884553