Finite Group Databasebeta

← #13140478

Subgroups — #13140478

Order 1728.

506 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 97 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×#3699C211yesmaximal, characteristic
2C2×C9×#214C211yesmaximal, characteristic, complemented
3#11720944C211yesmaximal, characteristic, complemented
4C2×#88427C211yesmaximal, characteristic, complemented
5#11720576C211yesmaximal, characteristic, complemented
6(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7#11720276C211yesmaximal, characteristic, complemented
8#10591461—31nomaximal
9#10588853—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×C3×#611—21no—
12C22×C3×Dic9—21no—
13C3×#3699—22no—
14C3×#3699—22no—
15C2×C3×C4×D9C2211yescharacteristic
16C2×C9×Dic6C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18C4⋊(C3⋊Dic9)C2211yescharacteristic
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20C9×#214—22no—
21C2×C9×C3⋊C8C2211yescharacteristic
22(C2×D9)⋊Dic3—21no—
23#88442—21no—
24#88427—22no—
25C2×C3×#237—31no—
26C2×C3×#214D311yescharacteristic
27#63856—31no—
28C2×#3718—31no—
29C2×#3699D311yescharacteristic
30#63659—31no—
31#63566—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×C9×SD16—31no—
34#63508—31no—
35C2×#3674—31no—
36#63392—31no—
37#63392—61no—
38(C2×D9)⋊C8—31no—
39#63376—31no—
40C23×C3×C9—21no—
41C22×C3×D9D411yescharacteristic
42C3×D4×C9—21no—
43C3×D4×C9D412yes—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3⋊Dic9—21no—
46C3×#611—24no—
47C2×C9×Dic3—21no—
48C2×C3×Dic9—24no—
49C2×C3×Dic9—21no—
50C2×C3×Dic9D411yescharacteristic
51C3×C4×D9—21no—
52C3×C4×D9—21no—
53C9×Dic6—21no—
54C9×Dic6—21no—
55C3×Dic18—21no—
56C3×Dic18D412yes—
57C3⋊(C9⋊C8)—21no—
58C9×C3⋊C8—21no—
59#5239—91no—
60#5173—91no—
61C2×C32×D4C2×D311yescharacteristic
62C2×C3×#67—61no—
63C22×C3×Dic3—61no—
64C3×#237—62no—
65C3×#237—62no—
66C2×C3×C4×D3—31no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6C2×D311yescharacteristic
69C4⋊(C3⋊Dic3)—31no—
70C2×C3⋊(C3⋊C8)—31no—
71C3×#214—22no—
72C2×C3×C3⋊C8C2×D311yescharacteristic
73#3724—61no—
74(C2×D3)⋊Dic3—61no—
75#3718—62no—
76C2×Q8×C9—31no—
77C2×D4×C9—21no—
78C2×D4×C9C2×D311yescharacteristic
79C2×#611—21no—
80C22×Dic9—21no—
81#3699—22no—
82#3699—22no—
83C2×C4×D9C2×D311yescharacteristic
84C2×Dic18C2×D311yescharacteristic
85C9×SD16—62no—
86C2×C8×C9—31no—
87#3674—62no—
88(C2×D9)⋊C4—61no—
89C4⋊Dic9—31no—
90C4⋊Dic9—61no—
91Dic9⋊C4—61no—
92C2×C9⋊C8—31no—
93C2×C9⋊C8—61no—
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×D9—21no—
101C3×C4×C9C2×D411yescharacteristic
102C3×C4×C9C2×D411yescharacteristic
103C3⋊Dic9—41no—
104C9×Dic3—41no—
105C3×Dic9—22no—
106C3×Dic9—21no—
107C2×C3×#41—91no—
108C2×#237—31no—
109C2×C3×SD16—31no—
110#946—91no—
111C2×#216—91no—
112C2×#216—91no—
113C2×#214D911yescharacteristic, complemented
114#924—91no—
115#835—91no—
116#832—91no—
117#832—91no—
118#832—181no—
119#830—91no—
120(C2×D3)⋊C8—91no—
121#816—91no—
122C23×C32—21no—
123C22×C3×D3—31no—
124C32×D4—21no—
125C32×D4#6712yes—
126C2×C32×C4C22×D311yescharacteristic
127C2×C3⋊Dic3—61no—
128C3×#67—64no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—31no—
131C2×C3×Dic3—64no—
132C2×C3×Dic3—61no—
133C3×C4×D3—61no—
134C3×C4×D3—61no—
135C3×Dic6—21no—
136C3×Dic6—21no—
137C3×Dic6—32no—
138C3×Dic6—61no—
139C23×C9—21no—
140C23×C9—41no—
141C22×D9#6711yescharacteristic
142C3⋊(C3⋊C8)—61no—
143C3×C3⋊C8—21no—
144Q8×C9—61no—
145Q8×C9—61no—
146D4×C9—21no—
147D4×C9—22no—
148D4×C9—41no—
149D4×C9#6712yes—
150C2×C4×C9—21no—
151C2×C4×C9—61no—
152C2×C4×C9C22×D311yescharacteristic
153#611—24no—
154C2×Dic9—24no—
155C2×Dic9—21no—
156C2×Dic9—61no—
157C2×Dic9—121no—
158C2×Dic9#6711yescharacteristic
159C4×D9—21no—
160C4×D9—21no—
161Dic18—21no—
162Dic18#6712yes—
163C8×C9—61no—
164C9⋊C8—61no—
165C9⋊C8—121no—
166#452—271noSylow
167C2×C3×C9—42no—
168C2×C3×C9#12011yescharacteristic
169C2×C3×C9#13511yescharacteristic
170C2×C3×C9#13711yescharacteristic
171C3×D9—41no—
172C3×#41—182no—
173C3×#41—182no—
174C2×C3×Q8—31no—
175C2×C3×Q8—91no—
176C2×C3×D4—21no—
177C2×C3×D4—181no—
178C2×C3×D4C2×D911yescharacteristic
179C2×C3×D4D3211yescharacteristic
180C22×C3×C4—91no—
181C22×C3×C4—181no—
182C2×#67—61no—
183C22×Dic3—61no—
184#237—62no—
185#237—62no—
186C2×C4×D3—31no—
187C2×Dic6—31no—
188C2×Dic6C2×D911yescharacteristic
189C3×SD16—62no—
190C2×C3×C8—31no—
191#217—181no—
192#216—182no—
193#216—182no—
194#214—22no—
195(C2×D3)⋊C4—181no—
196C4⋊Dic3—91no—
197C4⋊Dic3—91no—
198C4⋊Dic3—181no—
199Dic3⋊C4—181no—
200Dic3⋊C4—181no—
201C2×C3⋊C8—91no—
202C2×C3⋊C8—91no—
203C2×C3⋊C8—181no—
204C2×C3⋊C8C2×D911yescharacteristic
205C22×C32—22no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32D3×D411yescharacteristic
209C2×C3×D3—61no—
210C2×C3×D3—61no—
211C2×C3×D3—61no—
212C32×C4D3×D411yescharacteristic
213C32×C4C2×#6711yescharacteristic
214C3⋊Dic3—121no—
215C3×Dic3—41no—
216C3×Dic3—62no—
217C3×Dic3—61no—
218C22×C9—22no—
219C22×C9—21no—
220C22×C9—42no—
221C22×C9—41no—
222C22×C9—41no—
223C22×C9—81no—
224C22×C9—81no—
225C22×C9C2×#6711yescharacteristic
226C2×D9—21no—
227C2×D9—21no—
228C2×D9—21no—
229C4×C9—21no—
230C4×C9—21no—
231C4×C9—121no—
232C4×C9C2×#6711yescharacteristic
233C4×C9C2×#6711yescharacteristic
234Dic9—22no—
235Dic9—21no—
236Dic9—121no—
237Dic9—241no—
238C2×#41—91no—
239C2×Q16—271no—
240C2×SD16—31no—
241C22⋊Q8—271no—
242#103—271no—
243#102—271no—
244C22⋊C8—271no—
245C3×C9#45211yescharacteristic, complemented, Sylow
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—41no—
249C23×C3—91no—
250C22×D3—31no—
251C3×Q8—61no—
252C3×Q8—61no—
253C3×Q8—92no—
254C3×Q8—181no—
255C3×D4—21no—
256C3×D4—21no—
257C3×D4—22no—
258C3×D4—41no—
259C3×D4—184no—
260C3×D4#61112yes—
261C3×D4#62512yes—
262C2×C3×C4—21no—
263C2×C3×C4—61no—
264C2×C3×C4—91no—
265C2×C3×C4—184no—
266C2×C3×C4—181no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4C22×D911yescharacteristic
270C2×C3×C4C2×D3211yescharacteristic
271#67—64no—
272C2×Dic3—21no—
273C2×Dic3—31no—
274C2×Dic3—64no—
275C2×Dic3—61no—
276C2×Dic3—181no—
277C2×Dic3—181no—
278C2×Dic3—361no—
279C4×D3—61no—
280C4×D3—61no—
281Dic6—21no—
282Dic6—21no—
283Dic6—32no—
284Dic6—61no—
285C3×C8—61no—
286C3⋊C8—21no—
287C3⋊C8—181no—
288C3⋊C8—181no—
289C3⋊C8—361no—
290C2×C32—42no—
291C2×C32#91511yescharacteristic
292C2×C32#91611yescharacteristic
293C2×C32#93711yescharacteristic
294C3×D3—121no—
295C2×C9—21no—
296C2×C9—21no—
297C2×C9—21no—
298C2×C9—42no—
299C2×C9—82no—
300C2×C9#95011yescharacteristic
301C2×C9#95111yescharacteristic
302C2×C9#95311yescharacteristic
303D9—41no—
304#41—182no—
305#41—182no—
306C2×Q8—31no—
307C2×Q8—91no—
308C2×D4—181no—
309C2×D4D3×D911yescharacteristic
310C22×C4—91no—
311C22×C4—181no—
312Q16—542no—
313SD16—62no—
314C2×C8—31no—
315C2×C8—271no—
316C4⋊C4—271no—
317C4⋊C4—541no—
318C22⋊C4—541no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—21no—
322C22×C3—42no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—181no—
330C22×C3—181no—
331C22×C3—181no—
332C22×C3D4×D911yescharacteristic
333C22×C3D3×#6711yescharacteristic
334C2×D3—61no—
335C2×D3—61no—
336C2×D3—61no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—121no—
340C3×C4—181no—
341C3×C4—182no—
342C3×C4D4×D911yescharacteristic
343C3×C4C2×#61111yescharacteristic
344C3×C4C2×#62511yescharacteristic
345C3×C4D3×#6711yescharacteristic
346Dic3—41no—
347Dic3—62no—
348Dic3—61no—
349Dic3—361no—
350Dic3—361no—
351Dic3—721no—
352C32#517311yescharacteristic
353C9—21no—
354C9#523911yescharacteristic, complemented
355C23—21no—
356C23—91no—
357Q8—61no—
358Q8—61no—
359Q8—92no—
360Q8—181no—
361D4—21no—
362D4—184no—
363D4#619412yes—
364C2×C4—61no—
365C2×C4—91no—
366C2×C4—184no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4—541no—
371C2×C4C2×D3×D911yescharacteristic
372C8—61no—
373C8—541no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—42no—
378C2×C3—42no—
379C2×C3—82no—
380C2×C3—361no—
381C2×C3#6347711yescharacteristic
382C2×C3#6347811yescharacteristic
383C2×C3#6349911yescharacteristic
384C2×C3#6392911yescharacteristic
385C2×C3#6393511yescharacteristic
386C2×C3#6397611yescharacteristic
387D3—121no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—181no—
392C22—181no—
393C22—181no—
394C22D9×#6711yescharacteristic
395C4—121no—
396C4—181no—
397C4—182no—
398C4—1081no—
399C4C2×#619411yescharacteristic
400C4D9×#6711yescharacteristic
401C3—21no—
402C3#1058885311yescharacteristic, complemented
403C3#1059146111yescharacteristic
404C2—42no—
405C2—361no—
406C2#1172118111yescharacteristic
407C2#1172118511yescharacteristic
408C2#1172128511yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×#3699C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#452
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
2. C2×C9×#214C2semidirect
3. #11720944C2semidirect
4. C2×#88427C2semidirect
5. #11720576C2semidirect
6. (C2×D9)⋊(C3⋊C8)C2semidirect
7. #11720276C2semidirect
113. C2×#214D9semidirect
245. C3×C9#452semidirect
354. C9#5239semidirect
402. C3#10588853semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36992
C2×C9×#2142
#117209442
C2×#884272
#117205762
(C2×D9)⋊(C3⋊C8)2
#117202762
#105914613
#105888533