Finite Group Databasebeta

← #13140506

Subgroups — #13140506

Order 1728.

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

normal subgroups only

This group has 542 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×#88543C211yesmaximal, characteristic, complemented
3C3×#63500C211yesmaximal, characteristic
4C2×C9×#214C211yesmaximal, characteristic, complemented
5#11720297C211yesmaximal, characteristic, complemented
6(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7#11720277C211yesmaximal, characteristic, complemented
8#10591517—31nomaximal
9#10588850—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C3×#611—21no—
13C2×C3×C4×D9C2211yescharacteristic
14C2×C9×Dic6C2211yescharacteristic
15C2×Dic3×D9—21no—
16C2×#6189—21no—
17D9×Dic6—22no—
18D9×Dic6—22no—
19#88543—22no—
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21C3×#3676—21no—
22C9×#214—22no—
23C3×C4⋊Dic9C2211yescharacteristic
24C2×C9×C3⋊C8C2211yescharacteristic
25C2×D3×Dic6—31no—
26C2×#3754—31no—
27C3×#938—31no—
28C2×C3×#214D311yescharacteristic
29C2×Q8×D9—31no—
30#63567—31no—
31#63565—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×C9×SD16—31no—
34#63500D311yescharacteristic
35C2×#3672—31no—
36C2×#3672—61no—
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
45C2×C3⋊Dic9—21no—
46#6226—21no—
47#6226D412yescomplemented
48C3×#611—42no—
49C2×C9×Dic3—21no—
50C2×C3×Dic9—21no—
51C2×C3×Dic9D411yescharacteristic
52C3×C4×D9—21no—
53C3×C4×D9—21no—
54C9×Dic6—21no—
55C9×Dic6D412yescomplemented
56Dic3×D9—24no—
57#6189—24no—
58C3⋊(C9⋊C8)—21no—
59C9×C3⋊C8—21no—
60#5170—91no—
61#5040—91no—
62C2×C32×D4C2×D311yescharacteristic
63C2×#634—31no—
64C2×C3×#67—61no—
65C2×C3×C4×D3—31no—
66C2×C3×Dic6C2×D311yescharacteristic
67C2×#627—61no—
68C2×D3×Dic3—61no—
69D3×Dic6—62no—
70D3×Dic6—62no—
71#3754—62no—
72C2×C3⋊(C3⋊C8)—31no—
73C3×#217—61no—
74C3×#214—22no—
75C3×C4⋊Dic3—31no—
76C2×C3×C3⋊C8C2×D311yescharacteristic
77C2×Q8×C9—31no—
78C2×D4×C9—21no—
79C2×D4×C9C2×D311yescharacteristic
80C2×#611—21no—
81Q8×D9—62no—
82Q8×D9—62no—
83C2×C4×D9—61no—
84C2×C4×D9C2×D311yescharacteristic
85C2×Dic18—31no—
86C2×Dic18—61no—
87C2×Dic18—61no—
88C9×SD16—62no—
89C2×C8×C9—31no—
90#3676—21no—
91#3672—62no—
92#3672—64no—
93C4⋊Dic9C2×D311yescharacteristic
94C2×C9⋊C8—31no—
95C2×C9⋊C8—61no—
96C22×C3×C9—41no—
97C22×C3×C9—41no—
98C22×C3×C9—41no—
99C22×C3×C9C2×D411yescharacteristic
100C2×C3×D9—21no—
101C2×C3×D9—21no—
102C2×C3×D9SD1612yescomplemented
103C3×C4×C9C2×D411yescharacteristic
104C3×C4×C9C2×D411yescharacteristic
105C3⋊Dic9—22no—
106C9×Dic3—22no—
107C3×Dic9—21no—
108C3×Dic9—41no—
109C2×D3×Q8—91no—
110C22×Dic6—91no—
111C2×C3×SD16—31no—
112C3×#121—91no—
113#938—31no—
114C2×#214—91no—
115C2×#214—91no—
116C2×#214—181no—
117C2×#214D911yescharacteristic, complemented
118#835—91no—
119#830—91no—
120(C2×D3)⋊C8—91no—
121#816—91no—
122#810—91no—
123#810—91no—
124C23×C32—21no—
125C22×C3×D3—31no—
126C32×D4—21no—
127C32×D4—21no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3⋊Dic3—61no—
130#634—32no—
131#634—61no—
132C3×#67—122no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—61no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138C3×Dic6—21no—
139C3×Dic6#6712yes—
140#627—64no—
141D3×Dic3—64no—
142C23×C9—21no—
143C23×C9—41no—
144C22×D9#6711yescharacteristic
145C3⋊(C3⋊C8)—61no—
146C3×C3⋊C8—21no—
147Q8×C9—32no—
148Q8×C9—61no—
149D4×C9—22no—
150D4×C9—21no—
151D4×C9—21no—
152D4×C9—41no—
153C2×C4×C9—21no—
154C2×C4×C9—61no—
155C2×C4×C9C22×D311yescharacteristic
156#611—42no—
157C2×Dic9—21no—
158C2×Dic9—61no—
159C2×Dic9—121no—
160C2×Dic9#6711yescharacteristic
161C4×D9—21no—
162C4×D9—21no—
163C4×D9—64no—
164Dic18—32no—
165Dic18—62no—
166Dic18—64no—
167Dic18—61no—
168Dic18—121no—
169C8×C9—61no—
170C9⋊C8—62no—
171C9⋊C8—61no—
172#448—271noSylow
173C2×C3×C9—81no—
174C2×C3×C9#12011yescharacteristic
175C2×C3×C9C2×SD1611yescharacteristic
176C2×C3×C9#13711yescharacteristic
177C3×D9—22no—
178C2×C3×Q8—31no—
179C2×C3×D4—21no—
180C2×C3×D4—181no—
181C2×C3×D4C2×D911yescharacteristic
182C2×C3×D4D3211yescharacteristic
183C22×C3×C4—91no—
184C2×#67—61no—
185C22×Dic3—181no—
186D3×Q8—182no—
187D3×Q8—182no—
188C2×C4×D3—31no—
189C2×C4×D3—181no—
190C2×Dic6—91no—
191C2×Dic6—91no—
192C2×Dic6—181no—
193C2×Dic6—182no—
194C2×Dic6—181no—
195C2×Dic6—181no—
196C2×Dic6—182no—
197C2×Dic6C2×D911yescharacteristic
198C3×SD16—62no—
199C2×C3×C8—31no—
200C3×C4⋊C4—91no—
201C3×C22⋊C4—181no—
202#217—61no—
203#214—22no—
204#214—182no—
205#214—182no—
206#214—184no—
207C4⋊Dic3—31no—
208C2×C3⋊C8—91no—
209C2×C3⋊C8—91no—
210C2×C3⋊C8—181no—
211C2×C3⋊C8C2×D911yescharacteristic
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32D3×D411yescharacteristic
216C2×C3×D3—32no—
217C2×C3×D3—61no—
218C2×C3×D3—61no—
219C32×C4D3×D411yescharacteristic
220C32×C4C2×#6711yescharacteristic
221C3⋊Dic3—62no—
222C3×Dic3—22no—
223C3×Dic3—61no—
224C3×Dic3—121no—
225C22×C9—21no—
226C22×C9—42no—
227C22×C9—41no—
228C22×C9—41no—
229C22×C9—41no—
230C22×C9—81no—
231C22×C9—81no—
232C22×C9C2×#6711yescharacteristic
233C2×D9—21no—
234C2×D9—21no—
235C2×D9#21412yescomplemented
236C4×C9—21no—
237C4×C9—21no—
238C4×C9—62no—
239C4×C9D3×D411yescharacteristic
240C4×C9D3×D411yescharacteristic
241Dic9—21no—
242Dic9—41no—
243Dic9—62no—
244Dic9—122no—
245C22×Q8—271no—
246C2×SD16—31no—
247C2×SD16—271no—
248#121—91no—
249#103—271no—
250#103—271no—
251C22⋊C8—271no—
252C3×C9#44811yescharacteristic, complemented, Sylow
253C23×C3—21no—
254C23×C3—21no—
255C23×C3—41no—
256C23×C3—91no—
257C22×D3—31no—
258C3×Q8—32no—
259C3×Q8—61no—
260C3×D4—22no—
261C3×D4—21no—
262C3×D4—21no—
263C3×D4—21no—
264C3×D4—21no—
265C3×D4—41no—
266C3×D4—362no—
267C2×C3×C4—21no—
268C2×C3×C4—61no—
269C2×C3×C4—91no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4—181no—
273C2×C3×C4C22×D911yescharacteristic
274C2×C3×C4C2×D3211yescharacteristic
275#67—122no—
276C2×Dic3—21no—
277C2×Dic3—31no—
278C2×Dic3—61no—
279C2×Dic3—184no—
280C2×Dic3—181no—
281C2×Dic3—181no—
282C2×Dic3—361no—
283C4×D3—61no—
284C4×D3—61no—
285C4×D3—184no—
286Dic6—21no—
287Dic6—92no—
288Dic6—92no—
289Dic6—182no—
290Dic6—184no—
291Dic6—184no—
292Dic6—181no—
293Dic6—181no—
294Dic6—361no—
295Dic6#61112yescomplemented
296C3×C8—61no—
297C3⋊C8—21no—
298C3⋊C8—182no—
299C3⋊C8—181no—
300C3⋊C8—181no—
301C2×C32—81no—
302C2×C32D3×SD1611yescharacteristic
303C2×C32#91511yescharacteristic
304C2×C32#93711yescharacteristic
305C3×D3—62no—
306C2×C9—21no—
307C2×C9—21no—
308C2×C9—21no—
309C2×C9—82no—
310C2×C9—81no—
311C2×C9C2×#21411yescharacteristic
312C2×C9#93711yescharacteristic
313C2×C9#95111yescharacteristic
314D9—22no—
315C2×Q8—31no—
316C2×Q8—271no—
317C2×Q8—541no—
318C2×Q8—542no—
319C2×Q8—542no—
320C2×D4—181no—
321C2×D4D3×D911yescharacteristic
322C22×C4—91no—
323C22×C4—541no—
324SD16—62no—
325SD16—542no—
326C2×C8—31no—
327C2×C8—271no—
328C4⋊C4—91no—
329C22⋊C4—181no—
330C22×C3—21no—
331C22×C3—42no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—81no—
339C22×C3—81no—
340C22×C3—92no—
341C22×C3—181no—
342C22×C3—181no—
343C22×C3D4×D911yescharacteristic
344C22×C3D3×#6711yescharacteristic
345C2×D3—32no—
346C2×D3—61no—
347C2×D3—61no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—62no—
351C3×C4—181no—
352C3×C4—361no—
353C3×C4D4×D911yescharacteristic
354C3×C4C2×#61111yescharacteristic
355C3×C4D3×#6711yescharacteristic
356C3×C4#381111yescharacteristic
357Dic3—22no—
358Dic3—61no—
359Dic3—121no—
360Dic3—182no—
361Dic3—182no—
362Dic3—362no—
363C32#517011yescharacteristic
364C9—21no—
365C9#504011yescharacteristic, complemented
366C23—21no—
367C23—91no—
368Q8—32no—
369Q8—61no—
370Q8—272no—
371Q8—544no—
372Q8—541no—
373D4—21no—
374D4—21no—
375D4—362no—
376C2×C4—61no—
377C2×C4—91no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—544no—
382C2×C4—541no—
383C2×C4C2×D3×D911yescharacteristic
384C8—61no—
385C8—541no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—82no—
390C2×C3—81no—
391C2×C3—81no—
392C2×C3—182no—
393C2×C3SD16×D911yescharacteristic
394C2×C3#6347711yescharacteristic
395C2×C3#6349911yescharacteristic
396C2×C3D3×#21411yescharacteristic
397C2×C3#6392911yescharacteristic
398C2×C3#6398311yescharacteristic
399D3—62no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—92no—
404C22—181no—
405C22—181no—
406C22D9×#6711yescharacteristic
407C4—62no—
408C4—181no—
409C4—361no—
410C4—542no—
411C4D3×#61111yescharacteristic
412C4#8866811yescharacteristic
413C3—21no—
414C3#1058885011yescharacteristic, complemented
415C3#1059151711yescharacteristic
416C2—81no—
417C2—182no—
418C2D9×#21411yescharacteristic
419C2#1172118111yescharacteristic
420C2#1172129211yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC3×#63500C2
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×D9×Dic6C2semidirect
2. C2×#88543C2semidirect
4. C2×C9×#214C2semidirect
5. #11720297C2semidirect
6. (C2×D9)⋊(C3⋊C8)C2semidirect
7. #11720277C2semidirect
47. #6226D4semidirect
55. C9×Dic6D4semidirect
102. C2×C3×D9SD16semidirect
117. C2×#214D9semidirect
235. C2×D9#214semidirect
252. C3×C9#448semidirect
295. Dic6#611semidirect
365. C9#5040semidirect
414. 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×#885432
C3×#635002
C2×C9×#2142
#117202972
(C2×D9)⋊(C3⋊C8)2
#117202772
#105915173
#105888503