Finite Group Databasebeta

← #13138697

Subgroups — #13138697

Order 1728.

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

normal subgroups only

This group has 516 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
1C2×D9×Dic6C211yesmaximal, characteristic, complemented
2C2×C9×#204C211yesmaximal, characteristic, complemented
3#11721036C211yesmaximal, characteristic
4C2×#88424C211yesmaximal, characteristic, complemented
5#11720539C211yesmaximal, characteristic, complemented
6C3×(C2×D9)⋊C8C211yesmaximal, characteristic, complemented
7#11720298C211yesmaximal, characteristic, complemented
8#10590822—31nomaximal
9#10588850—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×C3×C4×D9C2211yescharacteristic
13C2×C9×Dic6C2211yescharacteristic
14C2×#6194—21no—
15C2×Dic3×D9—21no—
16C2×#6189—21no—
17D9×Dic6—22no—
18D9×Dic6—22no—
19C2×C3×C8×C9C2211yescharacteristic
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21C2×C3×C9⋊C8C2211yescharacteristic
22C9×#204—22no—
23(C2×D3)⋊Dic9—21no—
24#88424—22no—
25C2×D3×Dic6—31no—
26C2×C3×#204D311yescharacteristic
27#63908—31no—
28C2×#3716—31no—
29C2×Q8×D9—31no—
30#63643—31no—
31C3×(C2×D3)⋊C8—31no—
32#63566—31no—
33C2×C9×SD16—31no—
34#63500—31no—
35C2×#3672—31no—
36#63395—31no—
37(C2×D9)⋊C8D311yescharacteristic, complemented
38#63376—31no—
39#63376—61no—
40C22×D3×C9—21no—
41C22×C3×D9D411yescharacteristic
42C2×C3×C4×C9C2311yescharacteristic
43C2×C3⋊Dic9—21no—
44C2×C3⋊Dic9—21no—
45#6226—21no—
46#6226D412yescomplemented
47C2×C9×Dic3—21no—
48C2×C3×Dic9D411yescharacteristic
49C9×D12—21no—
50C9×D12—21no—
51C3×C4×D9—21no—
52C3×C4×D9—21no—
53C9×Dic6—21no—
54C9×Dic6D412yescomplemented
55#6194—42no—
56Dic3×D9—24no—
57#6189—24no—
58C3×C8×C9—21no—
59C3×C9⋊C8—21no—
60#5170—91no—
61#4738—91no—
62C2×#634—31no—
63C2×C3×D12C2×D311yescharacteristic
64C2×C3×C4×D3—31no—
65C2×C3×Dic6C2×D311yescharacteristic
66C2×#627—61no—
67C2×#625—61no—
68C2×D3×Dic3—61no—
69D3×Dic6—62no—
70D3×Dic6—62no—
71C2×C32×C8C2×D311yescharacteristic
72C4⋊(C3⋊Dic3)—31no—
73C2×C3×C3⋊C8—31no—
74C3×#204—22no—
75(C2×D3)⋊Dic3—61no—
76#3716—62no—
77C2×Q8×C9—31no—
78C2×D4×C9—31no—
79C2×#611—61no—
80Q8×D9—62no—
81Q8×D9—62no—
82C2×C4×D9—61no—
83C2×C4×D9C2×D311yescharacteristic
84C2×Dic18—31no—
85C2×Dic18—61no—
86C2×Dic18—61no—
87C9×SD16—62no—
88C2×C8×C9—21no—
89C2×C8×C9C2×D311yescharacteristic
90#3676—61no—
91#3672—62no—
92C4⋊Dic9—31no—
93C4⋊Dic9—61no—
94C2×C9⋊C8C2×D311yescharacteristic
95C22×C3×C9C2×D411yescharacteristic
96C2×D3×C9—41no—
97C2×D3×C9—41no—
98C2×D3×C9—41no—
99C2×C3×D9—21no—
100C2×C3×D9—21no—
101C2×C3×D9SD1612yescomplemented
102C3×C4×C9C2×D411yescharacteristic
103C3×C4×C9C2×D411yescharacteristic
104C3⋊Dic9—22no—
105C3⋊Dic9—41no—
106C9×Dic3—22no—
107C3×Dic9—21no—
108C2×D3×Q8—91no—
109C22×Dic6—91no—
110C2×C3×SD16—31no—
111#938—91no—
112C2×#214—91no—
113#930—91no—
114C2×#204—91no—
115C2×#204D911yescharacteristic, complemented
116C3×C22⋊C8—91no—
117#835—91no—
118(C2×D3)⋊C8—31no—
119#816—91no—
120#816—91no—
121#816—91no—
122#816—181no—
123C22×C3×D3—21no—
124C22×C3×D3—31no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3—61no—
127C2×C3⋊Dic3—61no—
128#634—32no—
129#634—61no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—31no—
132C3×D12—21no—
133C3×D12—21no—
134C3×C4×D3—61no—
135C3×C4×D3—61no—
136C3×Dic6—21no—
137C3×Dic6#6712yes—
138#627—64no—
139#625—122no—
140D3×Dic3—64no—
141C23×C9—61no—
142C22×D9D1211yescharacteristic
143C32×C8—21no—
144C3×C3⋊C8—61no—
145Q8×C9—32no—
146Q8×C9—61no—
147D4×C9—61no—
148D4×C9—61no—
149C2×C4×C9—21no—
150C2×C4×C9—61no—
151C2×C4×C9C22×D311yescharacteristic
152#611—122no—
153C2×Dic9—61no—
154C2×Dic9—61no—
155C2×Dic9—121no—
156C2×Dic9—121no—
157C2×Dic9D1211yescharacteristic
158C4×D9—21no—
159C4×D9—21no—
160C4×D9—64no—
161Dic18—32no—
162Dic18—62no—
163Dic18—64no—
164Dic18—61no—
165Dic18—121no—
166C8×C9—21no—
167C8×C9—41no—
168C9⋊C8—21no—
169#448—271noSylow
170C2×C3×C9#12011yescharacteristic
171C2×C3×C9C2×SD1611yescharacteristic
172C2×C3×C9#13711yescharacteristic
173D3×C9—81no—
174C3×D9—22no—
175C2×C3×Q8—31no—
176C2×C3×D4—31no—
177C22×C3×C4—91no—
178C2×#67—181no—
179C2×#67—181no—
180C22×Dic3—181no—
181D3×Q8—182no—
182D3×Q8—182no—
183C2×D12C2×D911yescharacteristic
184C2×C4×D3—31no—
185C2×C4×D3—181no—
186C2×Dic6—91no—
187C2×Dic6—91no—
188C2×Dic6—181no—
189C2×Dic6—182no—
190C2×Dic6—181no—
191C2×Dic6—181no—
192C2×Dic6—182no—
193C2×Dic6C2×D911yescharacteristic
194C3×SD16—62no—
195C2×C3×C8—21no—
196C2×C3×C8—91no—
197C2×C3×C8C2×D911yescharacteristic
198C2×C3×C8D3211yescharacteristic
199#217—181no—
200#214—182no—
201(C2×D3)⋊C4—181no—
202C4⋊Dic3—91no—
203C4⋊Dic3—91no—
204C4⋊Dic3—181no—
205C2×C3⋊C8—31no—
206#204—22no—
207#204—182no—
208C22×C32D3×D411yescharacteristic
209C2×C3×D3—32no—
210C2×C3×D3—41no—
211C2×C3×D3—41no—
212C2×C3×D3—41no—
213C2×C3×D3—61no—
214C2×C3×D3—61no—
215C32×C4D3×D411yescharacteristic
216C32×C4C2×#6711yescharacteristic
217C3⋊Dic3—62no—
218C3⋊Dic3—121no—
219C3×Dic3—22no—
220C3×Dic3—61no—
221C22×C9—21no—
222C22×C9—121no—
223C22×C9—121no—
224C22×C9—121no—
225C22×C9C2×D1211yescharacteristic
226C2×D9—21no—
227C2×D9—21no—
228C2×D9#20412yescomplemented
229C4×C9—21no—
230C4×C9—21no—
231C4×C9—62no—
232C4×C9D3×D411yescharacteristic
233C4×C9D3×D411yescharacteristic
234Dic9—21no—
235Dic9—62no—
236Dic9—122no—
237Dic9—121no—
238Dic9—241no—
239C22×Q8—271no—
240C2×SD16—31no—
241C2×SD16—271no—
242#121—271no—
243#103—271no—
244#103—271no—
245C22⋊C8—91no—
246C3×C9#44811yescharacteristic, complemented, Sylow
247C23×C3—61no—
248C23×C3—91no—
249C22×D3—21no—
250C22×D3—31no—
251C3×Q8—32no—
252C3×Q8—61no—
253C3×D4—61no—
254C3×D4—61no—
255C2×C3×C4—21no—
256C2×C3×C4—61no—
257C2×C3×C4—91no—
258C2×C3×C4—181no—
259C2×C3×C4—181no—
260C2×C3×C4C22×D911yescharacteristic
261C2×C3×C4C2×D3211yescharacteristic
262#67—362no—
263#67—362no—
264C2×Dic3—21no—
265C2×Dic3—31no—
266C2×Dic3—184no—
267C2×Dic3—181no—
268C2×Dic3—181no—
269C2×Dic3—181no—
270C2×Dic3—181no—
271C2×Dic3—361no—
272C2×Dic3—361no—
273D12—21no—
274D12—21no—
275C4×D3—61no—
276C4×D3—61no—
277C4×D3—184no—
278Dic6—21no—
279Dic6—92no—
280Dic6—92no—
281Dic6—182no—
282Dic6—184no—
283Dic6—184no—
284Dic6—181no—
285Dic6—181no—
286Dic6—361no—
287Dic6#61112yescomplemented
288C3×C8—21no—
289C3×C8—21no—
290C3×C8—41no—
291C3×C8—181no—
292C3⋊C8—61no—
293C2×C32D3×SD1611yescharacteristic
294C2×C32#91511yescharacteristic
295C2×C32#93711yescharacteristic
296C3×D3—62no—
297C3×D3—81no—
298C2×C9—21no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9—241no—
302C2×C9#88211yescharacteristic
303C2×C9C2×#20411yescharacteristic
304C2×C9#90911yescharacteristic
305D9—22no—
306C2×Q8—31no—
307C2×Q8—271no—
308C2×Q8—541no—
309C2×Q8—542no—
310C2×Q8—542no—
311C2×D4—31no—
312C2×D4—541no—
313C22×C4—91no—
314C22×C4—541no—
315SD16—62no—
316SD16—542no—
317C2×C8—91no—
318C2×C8D3×D911yescharacteristic
319C4⋊C4—271no—
320C22⋊C4—541no—
321C22×C3—21no—
322C22×C3—92no—
323C22×C3—121no—
324C22×C3—121no—
325C22×C3—121no—
326C22×C3—181no—
327C22×C3—181no—
328C22×C3D4×D911yescharacteristic
329C22×C3D3×D1211yescharacteristic
330C2×D3—32no—
331C2×D3—41no—
332C2×D3—41no—
333C2×D3—41no—
334C2×D3—61no—
335C2×D3—61no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—62no—
339C3×C4—181no—
340C3×C4D4×D911yescharacteristic
341C3×C4C2×#61111yescharacteristic
342C3×C4#380211yescharacteristic
343C3×C4D3×#6711yescharacteristic
344Dic3—22no—
345Dic3—61no—
346Dic3—182no—
347Dic3—182no—
348Dic3—362no—
349Dic3—361no—
350Dic3—361no—
351Dic3—721no—
352C32#517011yescharacteristic
353C9—21no—
354C9#473811yescharacteristic, complemented
355C23—61no—
356C23—91no—
357Q8—32no—
358Q8—61no—
359Q8—272no—
360Q8—544no—
361Q8—541no—
362D4—61no—
363D4—61no—
364D4—1082no—
365C2×C4—61no—
366C2×C4—91no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—544no—
370C2×C4—541no—
371C2×C4—541no—
372C2×C4C2×D3×D911yescharacteristic
373C8—21no—
374C8—181no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—182no—
379C2×C3—241no—
380C2×C3SD16×D911yescharacteristic
381C2×C3#6347711yescharacteristic
382C2×C3#6349911yescharacteristic
383C2×C3D3×#20411yescharacteristic
384C2×C3#6380011yescharacteristic
385C2×C3#6392211yescharacteristic
386D3—62no—
387D3—81no—
388C22—92no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—181no—
393C22—181no—
394C22D9×D1211yescharacteristic
395C4—62no—
396C4—181no—
397C4—542no—
398C4—1081no—
399C4#8864611yescharacteristic
400C4D3×#61111yescharacteristic
401C3—21no—
402C3#1058885011yescharacteristic, complemented
403C3#1059082211yescharacteristic
404C2—182no—
405C2—241no—
406C2D9×#20411yescharacteristic
407C2#1172082711yescharacteristic
408C2#1172105011yescharacteristic
subgroupG/H
centerC22D9×D12
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radical#11721036C2
hypercenterC2×C8D3×D9
hypocenterC3×C9#448
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×D9×Dic6C2semidirect
2. C2×C9×#204C2semidirect
4. C2×#88424C2semidirect
5. #11720539C2semidirect
6. C3×(C2×D9)⋊C8C2semidirect
7. #11720298C2semidirect
37. (C2×D9)⋊C8D3semidirect
46. #6226D4semidirect
54. C9×Dic6D4semidirect
101. C2×C3×D9SD16semidirect
115. C2×#204D9semidirect
228. C2×D9#204semidirect
246. C3×C9#448semidirect
287. Dic6#611semidirect
354. C9#4738semidirect
402. C3#10588850semidirect

Maximal subgroups

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

subgroupindex
C2×D9×Dic62
C2×C9×#2042
#117210362
C2×#884242
#117205392
C3×(C2×D9)⋊C82
#117202982
#105908223
#105888503