Finite Group Databasebeta

← #13140341

Subgroups — #13140341

Order 1728.

650 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 105 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×#88742C211yesmaximal, characteristic
2#11721000C211yesmaximal, characteristic, complemented
3C2×#88414C211yesmaximal, characteristic, complemented
4C9×#832C211yesmaximal, characteristic, complemented
5C3×#63392C211yesmaximal, characteristic, complemented
6#11720277C211yesmaximal, characteristic, complemented
7#11720269C211yesmaximal, characteristic, complemented
8#10591418—31nomaximal
9#10588451—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C22×C3⋊Dic9—21no—
13#88742—22no—
14#88742C414yescomplemented
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×#6226C2211yescharacteristic
17C2×(C3⋊D9)⋊C4—21no—
18C9×C4⋊Dic3C2211yescharacteristic
19C3×C4⋊Dic9C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21C2×C3×C9⋊C8C2211yescharacteristic
22#88442—21no—
23#88414—22no—
24C2×#3830—31no—
25#63900—31no—
26C2×#3712—31no—
27C2×#3699—31no—
28C2×#3699—61no—
29C3×#832—31no—
30C3×#832D311yescharacteristic
31#63567—31no—
32#63567—31no—
33C2×#3663—31no—
34C4⋊C4×D9—31no—
35C9×#102—31no—
36#63392D311yescharacteristic, complemented
37#63376—31no—
38#63368—31no—
39C23×C3×C9—21no—
40C22×C3⋊D9D411yescharacteristic
41C3×D4×C9—21no—
42C3×D4×C9C2×C412yescomplemented
43C2×C3×C4×C9C2311yescharacteristic
44#6233—24no—
45C2×C3⋊Dic9—21no—
46C2×C3⋊Dic9—24no—
47C2×C3⋊Dic9D411yescharacteristic
48C4×C3⋊D9C2×C412yes—
49C4×C3⋊D9D412yes—
50#6226—21no—
51#6226C2×C412yescomplemented
52C2×C9×Dic3—21no—
53C2×C3×Dic9—21no—
54(C3⋊D9)⋊C4—42no—
55C9×C3⋊C8—21no—
56C3×C9⋊C8—21no—
57#4771—91no—
58#4771—91no—
59C2×C32×D4C2×D311yescharacteristic
60C2×#638—61no—
61C22×C3⋊Dic3—61no—
62#3830—34no—
63#3830—62no—
64C2×C4×C3⋊D3—31no—
65C2×#634—31no—
66C2×#624—61no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3C2×D311yescharacteristic
69C2×C3×C3⋊C8—31no—
70C2×C3×C3⋊C8C2×D311yescharacteristic
71#3724—61no—
72#3712—62no—
73C2×D4×C9—21no—
74C2×D4×C9C2×D311yescharacteristic
75C2×#611—61no—
76C2×#611—62no—
77C22×Dic9—61no—
78C22×Dic9—62no—
79#3699—34no—
80#3699—62no—
81#3699—64no—
82#3699—64no—
83C2×C4×D9—31no—
84C2×C4×D9—61no—
85C2×C4×D9—61no—
86C2×Dic18—31no—
87C2×Dic18—61no—
88C2×C8×C9—31no—
89C9×C4⋊C4—31no—
90C4⋊Dic9C2×D311yescharacteristic
91Dic9⋊C4—61no—
92C2×C9⋊C8C2×D311yescharacteristic
93#3663—62no—
94C22×C3×C9—22no—
95C22×C3×C9—41no—
96C22×C3×C9—41no—
97C22×C3×C9C2×D411yescharacteristic
98C2×C3⋊D9—21no—
99C2×C3⋊D9—21no—
100C2×C3⋊D9C22⋊C412yes—
101C3×C4×C9C22×C411yescharacteristic
102C3×C4×C9C2×D411yescharacteristic
103C3⋊Dic9—22no—
104C3⋊Dic9C22⋊C412yes—
105C9×Dic3—41no—
106C3×Dic9—41no—
107C2×#237—91no—
108C2×#237—91no—
109C2×#237—181no—
110C2×ΓL(1,𝔽25)—91no—
111C2×ΓL(1,𝔽25)—91no—
112D3×C4⋊C4—91no—
113D3×C4⋊C4—91no—
114C3×#102—31no—
115C3×#102—91no—
116#832—31no—
117#832D911yescharacteristic, complemented
118#816—91no—
119#816—91no—
120#810—91no—
121#810—91no—
122C23×C32—21no—
123C22×C3⋊D3—31no—
124C32×D4—21no—
125C32×D4C4×D312yes—
126C2×C32×C4C22×D311yescharacteristic
127#638—64no—
128C2×C3⋊Dic3—31no—
129C2×C3⋊Dic3—64no—
130C2×C3⋊Dic3—61no—
131C4×C3⋊D3—32no—
132C4×C3⋊D3—32no—
133#634—32no—
134#634—61no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—61no—
137#624—122no—
138C23×C9—22no—
139C23×C9—21no—
140C22×D9—31no—
141C22×D9—61no—
142C3×C3⋊C8—21no—
143C3×C3⋊C8—61no—
144D4×C9—22no—
145D4×C9—21no—
146D4×C9—22no—
147D4×C9C4×D312yescomplemented
148C2×C4×C9—21no—
149C2×C4×C9—61no—
150C2×C4×C9C22×D311yescharacteristic
151#611—64no—
152#611—68no—
153C2×Dic9—21no—
154C2×Dic9—31no—
155C2×Dic9—61no—
156C2×Dic9—64no—
157C2×Dic9—62no—
158C2×Dic9—68no—
159C2×Dic9—61no—
160C4×D9—32no—
161C4×D9—32no—
162C4×D9—62no—
163C4×D9—62no—
164C4×D9—122no—
165Dic18—32no—
166Dic18—62no—
167Dic18—62no—
168Dic18—61no—
169C8×C9—61no—
170C9⋊C8—21no—
171#417—271noSylow
172C2×C3×C9—42no—
173C2×C3×C9C2×C22⋊C411yescharacteristic
174C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
175C2×C3×C9#13711yescharacteristic
176C3⋊D9—22no—
177C2×C3×D4—21no—
178C2×C3×D4C2×D911yescharacteristic
179C2×C3×D4D3211yescharacteristic
180C2×#67—181no—
181C2×#67—181no—
182C2×#67—182no—
183C22×Dic3—181no—
184C22×Dic3—181no—
185C22×Dic3—182no—
186#237—94no—
187#237—94no—
188#237—184no—
189#237—182no—
190#237—182no—
191#237—184no—
192C2×C4×D3—91no—
193C2×C4×D3—91no—
194C2×C4×D3—181no—
195C2×C4×D3—181no—
196C2×C4×D3—181no—
197C2×Dic6—91no—
198C2×Dic6—91no—
199C2×Dic6—181no—
200C2×C3×C8—31no—
201C2×C3×C8—91no—
202C3×C4⋊C4—31no—
203C3×C4⋊C4—91no—
204C4⋊Dic3—31no—
205C4⋊Dic3C2×D911yescharacteristic
206Dic3⋊C4—181no—
207Dic3⋊C4—181no—
208C2×C3⋊C8—31no—
209C2×C3⋊C8C2×D911yescharacteristic
210ΓL(1,𝔽25)—182no—
211ΓL(1,𝔽25)—182no—
212C22×C32—22no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32D3×D411yescharacteristic
216C2×C3⋊D3—32no—
217C2×C3⋊D3—61no—
218C2×C3⋊D3—61no—
219C32×C4C2×C4×D311yescharacteristic
220C32×C4D3×D411yescharacteristic
221C3⋊Dic3—32no—
222C3⋊Dic3—62no—
223C3×Dic3—41no—
224C3×Dic3—121no—
225C22×C9—24no—
226C22×C9—22no—
227C22×C9—21no—
228C22×C9—42no—
229C22×C9—42no—
230C22×C9—41no—
231C22×C9—41no—
232C22×C9D3×D411yescharacteristic
233C2×D9—32no—
234C2×D9—62no—
235C2×D9—61no—
236C2×D9—61no—
237C2×D9—121no—
238C2×D9—121no—
239C4×C9—21no—
240C4×C9—21no—
241C4×C9—121no—
242C4×C9C2×C4×D311yescharacteristic
243C4×C9D3×D411yescharacteristic
244Dic9—32no—
245Dic9—41no—
246Dic9—62no—
247Dic9—64no—
248Dic9—62no—
249C2×#41—271no—
250C2×M16—271no—
251C2×C4⋊C4—271no—
252#103—271no—
253#103—271no—
254#102—31no—
255#102—91no—
256C3×C9#41711yescharacteristic, complemented, Sylow
257C23×C3—22no—
258C23×C3—21no—
259C23×C3—21no—
260C22×D3—91no—
261C22×D3—91no—
262C22×D3—181no—
263C3×D4—22no—
264C3×D4—21no—
265C3×D4—21no—
266C3×D4—22no—
267C3×D4C4×D912yescomplemented
268C3×D4#62412yes—
269C2×C3×C4—21no—
270C2×C3×C4—61no—
271C2×C3×C4—181no—
272C2×C3×C4C22×D911yescharacteristic
273C2×C3×C4C2×D3211yescharacteristic
274#67—184no—
275#67—184no—
276#67—188no—
277C2×Dic3—21no—
278C2×Dic3—61no—
279C2×Dic3—91no—
280C2×Dic3—91no—
281C2×Dic3—181no—
282C2×Dic3—184no—
283C2×Dic3—184no—
284C2×Dic3—182no—
285C2×Dic3—188no—
286C2×Dic3—181no—
287C2×Dic3—181no—
288C4×D3—92no—
289C4×D3—92no—
290C4×D3—92no—
291C4×D3—92no—
292C4×D3—182no—
293C4×D3—182no—
294C4×D3—362no—
295C4×D3—362no—
296Dic6—92no—
297Dic6—92no—
298Dic6—182no—
299Dic6—182no—
300Dic6—181no—
301Dic6—181no—
302C3×C8—61no—
303C3×C8—181no—
304C3⋊C8—21no—
305C3⋊C8—61no—
306C2×C32—42no—
307C2×C32D3×C22⋊C411yescharacteristic
308C2×C32#91111yescharacteristic
309C2×C32#91511yescharacteristic
310C3⋊D3—62no—
311C2×C9—21no—
312C2×C9—21no—
313C2×C9—21no—
314C2×C9—44no—
315C2×C9—42no—
316C2×C9D3×C22⋊C411yescharacteristic
317C2×C9#91111yescharacteristic
318C2×C9#91511yescharacteristic
319D9—62no—
320D9—122no—
321#41—274no—
322#41—542no—
323C2×Q8—271no—
324C2×D4—541no—
325C2×D4D3×D911yescharacteristic
326C22×C4—271no—
327C22×C4—541no—
328C22×C4—541no—
329M16—542no—
330C2×C8—31no—
331C2×C8—91no—
332C4⋊C4—31no—
333C4⋊C4—91no—
334C4⋊C4—541no—
335C22×C3—24no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—21no—
339C22×C3—42no—
340C22×C3—42no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3D4×D911yescharacteristic
346C22×C3#381111yescharacteristic
347C2×D3—92no—
348C2×D3—92no—
349C2×D3—182no—
350C2×D3—181no—
351C2×D3—181no—
352C2×D3—181no—
353C2×D3—181no—
354C2×D3—361no—
355C2×D3—361no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—121no—
359C3×C4—361no—
360C3×C4C2×C4×D911yescharacteristic
361C3×C4D4×D911yescharacteristic
362C3×C4C2×#62411yescharacteristic
363C3×C4#381111yescharacteristic
364Dic3—41no—
365Dic3—92no—
366Dic3—92no—
367Dic3—121no—
368Dic3—182no—
369Dic3—184no—
370Dic3—182no—
371Dic3—182no—
372C32#477111yescharacteristic
373C9—21no—
374C9#477111yescharacteristic, complemented
375C23—21no—
376C23—271no—
377Q8—272no—
378Q8—541no—
379D4—21no—
380D4—544no—
381D4(C3⋊D9)⋊C412yescomplemented
382C2×C4—61no—
383C2×C4—181no—
384C2×C4—271no—
385C2×C4—272no—
386C2×C4—272no—
387C2×C4—544no—
388C2×C4—541no—
389C2×C4—1082no—
390C2×C4C2×D3×D911yescharacteristic
391C8—61no—
392C8—181no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—44no—
397C2×C3—42no—
398C2×C3—42no—
399C2×C3C22⋊C4×D911yescharacteristic
400C2×C3#6347311yescharacteristic
401C2×C3#6347711yescharacteristic
402C2×C3#6392711yescharacteristic
403C2×C3#6393111yescharacteristic
404C2×C3#6397411yescharacteristic
405D3—182no—
406D3—182no—
407D3—362no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—272no—
412C22—541no—
413C22—541no—
414C22#8866811yescharacteristic
415C4—121no—
416C4—272no—
417C4—361no—
418C4—542no—
419C4C2×(C3⋊D9)⋊C411yescharacteristic
420C4#8866811yescharacteristic
421C3—21no—
422C3#1058845111yescharacteristic, complemented
423C3#1059141811yescharacteristic
424C2—42no—
425C2—542no—
426C2#1172117511yescharacteristic
427C2#1172118311yescharacteristic
428C2#1172127411yescharacteristic
subgroupG/H
centerC22#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×#88742C2
cosocleC2×C3×C4×C9C23
hypercenterC2×D4D3×D9
hypocenterC3×C9#417
LCSG ▷ 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
2. #11721000C2semidirect
3. C2×#88414C2semidirect
4. C9×#832C2semidirect
5. C3×#63392C2semidirect
6. #11720277C2semidirect
7. #11720269C2semidirect
14. #88742C4semidirect
36. #63392D3semidirect
42. C3×D4×C9C2×C4semidirect
51. #6226C2×C4semidirect
117. #832D9semidirect
147. D4×C9C4×D3semidirect
256. C3×C9#417semidirect
267. C3×D4C4×D9semidirect
374. C9#4771semidirect
381. D4(C3⋊D9)⋊C4semidirect
422. C3#10588451semidirect

Maximal subgroups

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

subgroupindex
C2×#887422
#117210002
C2×#884142
C9×#8322
C3×#633922
#117202772
#117202692
#105914183
#105884513