Finite Group Databasebeta

← #13140507

Subgroups — #13140507

Order 1728.

568 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 91 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⋊D9)⋊Q8C211yesmaximal, characteristic, complemented
2#11721838C211yesmaximal, characteristic
3C2×C9×#214C211yesmaximal, characteristic, complemented
4C2×C3×#3672C211yesmaximal, characteristic, complemented
5#11720298C211yesmaximal, characteristic, complemented
6#11720282C211yesmaximal, characteristic, complemented
7#11720276C211yesmaximal, characteristic, complemented
8#10591518—31nomaximal
9#10588850—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—21no—
12C2×C4×C3⋊D9C2211yescharacteristic
13C2×C9×Dic6C2211yescharacteristic
14C2×C3×Dic18C2211yescharacteristic
15C2×(C3⋊D9)⋊C4—21no—
16C2×#6189—21no—
17(C3⋊D9)⋊Q8—22no—
18(C3⋊D9)⋊Q8—22no—
19#88551—21no—
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21C9×#214—22no—
22C3×#3672—22no—
23C2×C9×C3⋊C8C2211yescharacteristic
24C2×C3×C9⋊C8C2211yescharacteristic
25C2×#3797—31no—
26#64147—31no—
27C2×C3×#214—31no—
28C2×C3×#214D311yescharacteristic
29C2×Q8×D9—31no—
30#63566—31no—
31#63566—31no—
32#63557—31no—
33C2×C9×SD16—31no—
34#63500—31no—
35#63500—61no—
36C2×#3672D311yescharacteristic, complemented
37#63395—31no—
38(C2×D9)⋊C8—31no—
39#63376—31no—
40C23×C3×C9—21no—
41C22×C3⋊D9D411yescharacteristic
42C3×D4×C9—21no—
43C3×D4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45#6233—42no—
46C2×C3⋊Dic9—21no—
47C2×C3⋊Dic9D411yescharacteristic
48C4×C3⋊D9—21no—
49C4×C3⋊D9—21no—
50C2×C9×Dic3—21no—
51C2×C3×Dic9—21no—
52C9×Dic6—21no—
53C9×Dic6D412yescomplemented
54C3×Dic18—21no—
55C3×Dic18D412yescomplemented
56(C3⋊D9)⋊C4—24no—
57#6189—24no—
58C9×C3⋊C8—21no—
59C3×C9⋊C8—21no—
60#5170—91no—
61#5170—91no—
62C2×C32×D4C2×D311yescharacteristic
63C2×#638—61no—
64C2×C4×C3⋊D3—31no—
65C2×C3×Dic6—31no—
66C2×C3×Dic6C2×D311yescharacteristic
67C2×#627—61no—
68C2×#624—61no—
69#3797—62no—
70#3797—62no—
71#3757—61no—
72C4⋊(C3⋊Dic3)—31no—
73C3×#214—22no—
74C3×#214—62no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8C2×D311yescharacteristic
77C2×Q8×C9—31no—
78C2×D4×C9—21no—
79C2×D4×C9C2×D311yescharacteristic
80C2×#611—61no—
81C2×#611—62no—
82Q8×D9—62no—
83Q8×D9—62no—
84C2×C4×D9—31no—
85C2×C4×D9—61no—
86C2×C4×D9—61no—
87C2×Dic18—61no—
88C2×Dic18C2×D311yescharacteristic
89C9×SD16—62no—
90C2×C8×C9—31no—
91#3676—61no—
92#3676—62no—
93#3672—22no—
94C4⋊Dic9—31no—
95C4⋊Dic9—61no—
96C2×C9⋊C8C2×D311yescharacteristic
97C22×C3×C9—41no—
98C22×C3×C9—41no—
99C22×C3×C9—41no—
100C22×C3×C9C2×D411yescharacteristic
101C2×C3⋊D9—21no—
102C2×C3⋊D9—21no—
103C2×C3⋊D9SD1612yescomplemented
104C3×C4×C9C2×D411yescharacteristic
105C3×C4×C9C2×D411yescharacteristic
106C3⋊Dic9—21no—
107C3⋊Dic9—41no—
108C9×Dic3—22no—
109C3×Dic9—22no—
110C2×D3×Q8—91no—
111C2×D3×Q8—91no—
112C2×C3×SD16—31no—
113C2×C3×SD16—91no—
114#938—91no—
115#938—91no—
116#938—181no—
117C2×#214—31no—
118C2×#214D911yescharacteristic, complemented
119#835—91no—
120#835—91no—
121(C2×D3)⋊C8—91no—
122(C2×D3)⋊C8—91no—
123#816—91no—
124#816—91no—
125C23×C32—21no—
126C22×C3⋊D3—31no—
127C32×D4—21no—
128C32×D4—21no—
129C2×C32×C4C22×D311yescharacteristic
130#638—122no—
131C2×C3⋊Dic3—31no—
132C2×C3⋊Dic3—61no—
133C4×C3⋊D3—61no—
134C4×C3⋊D3—61no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—61no—
137C3×Dic6—21no—
138C3×Dic6—32no—
139C3×Dic6—61no—
140C3×Dic6#6712yes—
141#627—64no—
142#624—64no—
143C23×C9—22no—
144C23×C9—21no—
145C22×D9—31no—
146C22×D9—61no—
147C3×C3⋊C8—21no—
148C3×C3⋊C8—61no—
149Q8×C9—32no—
150Q8×C9—61no—
151D4×C9—21no—
152D4×C9—21no—
153D4×C9—41no—
154D4×C9—41no—
155C2×C4×C9—21no—
156C2×C4×C9—61no—
157C2×C4×C9C22×D311yescharacteristic
158#611—122no—
159#611—124no—
160C2×Dic9—21no—
161C2×Dic9—31no—
162C2×Dic9—61no—
163C2×Dic9—62no—
164C2×Dic9—61no—
165C4×D9—64no—
166C4×D9—61no—
167C4×D9—61no—
168C4×D9—121no—
169C4×D9—121no—
170Dic18—21no—
171Dic18—64no—
172Dic18#6712yescomplemented
173C8×C9—61no—
174C9⋊C8—21no—
175#448—271noSylow
176C2×C3×C9—81no—
177C2×C3×C9#12011yescharacteristic
178C2×C3×C9C2×SD1611yescharacteristic
179C2×C3×C9#13711yescharacteristic
180C3⋊D9—22no—
181C2×C3×Q8—31no—
182C2×C3×Q8—91no—
183C2×C3×D4—21no—
184C2×C3×D4C2×D911yescharacteristic
185C2×C3×D4D3211yescharacteristic
186C2×#67—181no—
187C2×#67—181no—
188C2×#67—182no—
189D3×Q8—182no—
190D3×Q8—182no—
191D3×Q8—182no—
192D3×Q8—182no—
193C2×C4×D3—91no—
194C2×C4×D3—91no—
195C2×C4×D3—181no—
196C2×C4×D3—181no—
197C2×C4×D3—181no—
198C2×Dic6—31no—
199C2×Dic6—181no—
200C2×Dic6—181no—
201C2×Dic6C2×D911yescharacteristic
202C3×SD16—62no—
203C3×SD16—182no—
204C2×C3×C8—31no—
205C2×C3×C8—91no—
206#217—181no—
207#217—181no—
208#217—182no—
209#214—22no—
210#214—62no—
211C4⋊Dic3—91no—
212C4⋊Dic3—91no—
213C4⋊Dic3—181no—
214C2×C3⋊C8—31no—
215C2×C3⋊C8C2×D911yescharacteristic
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32D3×D411yescharacteristic
220C2×C3⋊D3—32no—
221C2×C3⋊D3—61no—
222C2×C3⋊D3—61no—
223C32×C4D3×D411yescharacteristic
224C32×C4C2×#6711yescharacteristic
225C3⋊Dic3—61no—
226C3⋊Dic3—121no—
227C3×Dic3—22no—
228C3×Dic3—62no—
229C22×C9—21no—
230C22×C9—42no—
231C22×C9—42no—
232C22×C9—42no—
233C22×C9—41no—
234C22×C9—41no—
235C22×C9—41no—
236C22×C9D3×D411yescharacteristic
237C2×D9—32no—
238C2×D9—62no—
239C2×D9—61no—
240C2×D9—61no—
241C2×D9—121no—
242C2×D9—121no—
243C4×C9—21no—
244C4×C9—21no—
245C4×C9—62no—
246C4×C9D3×D411yescharacteristic
247C4×C9C2×#6711yescharacteristic
248Dic9—22no—
249Dic9—61no—
250Dic9—121no—
251Dic9—122no—
252Dic9—121no—
253C22×Q8—271no—
254C2×SD16—31no—
255C2×SD16—91no—
256#121—271no—
257#103—271no—
258#103—271no—
259C22⋊C8—271no—
260C3×C9#44811yescharacteristic, complemented, Sylow
261C23×C3—22no—
262C23×C3—21no—
263C23×C3—21no—
264C22×D3—91no—
265C22×D3—91no—
266C22×D3—181no—
267C3×Q8—32no—
268C3×Q8—61no—
269C3×Q8—92no—
270C3×Q8—181no—
271C3×D4—21no—
272C3×D4—21no—
273C3×D4—21no—
274C3×D4—21no—
275C3×D4—41no—
276C3×D4—41no—
277C2×C3×C4—21no—
278C2×C3×C4—61no—
279C2×C3×C4—181no—
280C2×C3×C4C22×D911yescharacteristic
281C2×C3×C4C2×D3211yescharacteristic
282#67—362no—
283#67—362no—
284#67—364no—
285C2×Dic3—21no—
286C2×Dic3—61no—
287C2×Dic3—91no—
288C2×Dic3—91no—
289C2×Dic3—181no—
290C2×Dic3—182no—
291C2×Dic3—181no—
292C2×Dic3—181no—
293C4×D3—184no—
294C4×D3—184no—
295C4×D3—181no—
296C4×D3—181no—
297C4×D3—181no—
298C4×D3—181no—
299C4×D3—361no—
300C4×D3—361no—
301Dic6—21no—
302Dic6—32no—
303Dic6—61no—
304Dic6—184no—
305Dic6—184no—
306Dic6#61112yescomplemented
307C3×C8—61no—
308C3×C8—181no—
309C3⋊C8—21no—
310C3⋊C8—61no—
311C2×C32—81no—
312C2×C32D3×SD1611yescharacteristic
313C2×C32#91511yescharacteristic
314C2×C32#93711yescharacteristic
315C3⋊D3—62no—
316C2×C9—21no—
317C2×C9—21no—
318C2×C9—21no—
319C2×C9—82no—
320C2×C9—81no—
321C2×C9D3×SD1611yescharacteristic
322C2×C9#91511yescharacteristic
323C2×C9#93711yescharacteristic
324D9—62no—
325D9—122no—
326C2×Q8—31no—
327C2×Q8—91no—
328C2×Q8—541no—
329C2×Q8—542no—
330C2×Q8—542no—
331C2×D4—541no—
332C2×D4D3×D911yescharacteristic
333C22×C4—271no—
334C22×C4—541no—
335SD16—62no—
336SD16—182no—
337C2×C8—31no—
338C2×C8—91no—
339C4⋊C4—271no—
340C22⋊C4—541no—
341C22×C3—21no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3D4×D911yescharacteristic
352C22×C3#381111yescharacteristic
353C2×D3—92no—
354C2×D3—92no—
355C2×D3—182no—
356C2×D3—181no—
357C2×D3—181no—
358C2×D3—181no—
359C2×D3—181no—
360C2×D3—361no—
361C2×D3—361no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—62no—
365C3×C4—182no—
366C3×C4D4×D911yescharacteristic
367C3×C4C2×#61111yescharacteristic
368C3×C4D3×#6711yescharacteristic
369C3×C4D3×#6711yescharacteristic
370Dic3—22no—
371Dic3—62no—
372Dic3—181no—
373Dic3—181no—
374Dic3—361no—
375Dic3—362no—
376Dic3—361no—
377Dic3—361no—
378C32#517011yescharacteristic
379C9—21no—
380C9#517011yescharacteristic, complemented
381C23—21no—
382C23—271no—
383Q8—32no—
384Q8—61no—
385Q8—92no—
386Q8—181no—
387Q8—544no—
388D4—21no—
389D4—21no—
390D4—1082no—
391C2×C4—61no—
392C2×C4—181no—
393C2×C4—271no—
394C2×C4—544no—
395C2×C4—541no—
396C2×C4—541no—
397C2×C4—541no—
398C2×C4C2×D3×D911yescharacteristic
399C8—61no—
400C8—181no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—82no—
405C2×C3—81no—
406C2×C3—81no—
407C2×C3SD16×D911yescharacteristic
408C2×C3#6347711yescharacteristic
409C2×C3#6349911yescharacteristic
410C2×C3#6393011yescharacteristic
411C2×C3#6393111yescharacteristic
412C2×C3#6398311yescharacteristic
413D3—182no—
414D3—182no—
415D3—362no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—272no—
420C22—541no—
421C22—541no—
422C22#8866811yescharacteristic
423C4—62no—
424C4—182no—
425C4—541no—
426C4—1081no—
427C4D3×#61111yescharacteristic
428C4D9×#6711yescharacteristic
429C3—21no—
430C3#1058885011yescharacteristic, complemented
431C3#1059151811yescharacteristic
432C2—81no—
433C2—542no—
434C2#1172118211yescharacteristic
435C2#1172118311yescharacteristic
436C2#1172129211yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radical#11721838C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#448
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. C2×(C3⋊D9)⋊Q8C2semidirect
3. C2×C9×#214C2semidirect
4. C2×C3×#3672C2semidirect
5. #11720298C2semidirect
6. #11720282C2semidirect
7. #11720276C2semidirect
36. C2×#3672D3semidirect
53. C9×Dic6D4semidirect
55. C3×Dic18D4semidirect
103. C2×C3⋊D9SD16semidirect
118. C2×#214D9semidirect
172. Dic18#67semidirect
260. C3×C9#448semidirect
306. Dic6#611semidirect
380. C9#5170semidirect
430. C3#10588850semidirect

Maximal subgroups

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

subgroupindex
C2×(C3⋊D9)⋊Q82
#117218382
C2×C9×#2142
C2×C3×#36722
#117202982
#117202822
#117202762
#105915183
#105888503