Finite Group Databasebeta

← #13140791

Subgroups — #13140791

Order 1728.

542 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 107 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×D3×Q8×C9C211yesmaximal, characteristic, complemented
2C2×#88548C211yesmaximal, characteristic, complemented
3#11721038C211yesmaximal, characteristic
4C2×#88426C211yesmaximal, characteristic, complemented
5C3×#63395C211yesmaximal, characteristic, complemented
6#11720297C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C9⋊C8)C211yesmaximal, characteristic, complemented
8#10591584—31nomaximal
9#10588908—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12D3×Q8×C9—22no—
13D3×Q8×C9—22no—
14C2×C4×D3×C9—21no—
15C2×C4×D3×C9C2211yescharacteristic
16C2×C9×Dic6—21no—
17C2×C9×Dic6C2211yescharacteristic
18C2×#6195—21no—
19#88548—22no—
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21C3×C4⋊Dic9C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23#88445—21no—
24#88426—22no—
25C2×C3×D3×Q8D311yescharacteristic
26C2×#3755—31no—
27#63909—31no—
28C2×#3717—31no—
29C22×Q8×C9—31no—
30C3×#835—31no—
31#63565—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×#3675—31no—
34C2×#3675—31no—
35C2×#3675—61no—
36#63492—31no—
37#63395—31no—
38#63395D311yescharacteristic, complemented
39#63389—31no—
40C22×C3⋊D9—21no—
41C22×D3×C9D411yescharacteristic
42C3×Q8×C9—21no—
43C3×Q8×C9D412yescomplemented
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9C2311yescharacteristic
46C4⋊(C3⋊D9)—21no—
47C4⋊(C3⋊D9)—21no—
48C2×C9×Dic3—21no—
49C2×C9×Dic3D411yescharacteristic
50C2×C3×Dic9—21no—
51C4×D3×C9—24no—
52C4×D3×C9—21no—
53C4×D3×C9—21no—
54C9×Dic6—24no—
55C9×Dic6—21no—
56C9×Dic6D412yescomplemented
57#6195—42no—
58C3⋊(C9⋊C8)—21no—
59C3×C9⋊C8—21no—
60#5227—91no—
61#4806—91no—
62C2×C32×Q8C2×D311yescharacteristic
63C2×C4⋊(C3⋊D3)—31no—
64C3×D3×Q8—22no—
65C3×D3×Q8—22no—
66C2×C3×C4×D3—21no—
67C2×C3×C4×D3C2×D311yescharacteristic
68C2×C3×Dic6—21no—
69C2×C3×Dic6C2×D311yescharacteristic
70C2×#626—61no—
71#3755—62no—
72C2×C3⋊(C3⋊C8)—31no—
73C3×C4⋊Dic3—31no—
74C2×C3×C3⋊C8—31no—
75#3722—61no—
76#3717—62no—
77C2×Q8×C9—21no—
78C2×Q8×C9—31no—
79C2×Q8×C9—61no—
80C2×Q8×C9—62no—
81C2×Q8×C9—62no—
82C2×Q8×C9C2×D311yescharacteristic
83C22×C4×C9—31no—
84C22×C4×C9—61no—
85C2×#611—61no—
86C2×D36—31no—
87C2×D36—61no—
88#3675—62no—
89#3675—64no—
90#3675—62no—
91(C2×D9)⋊C4—61no—
92C4⋊Dic9C2×D311yescharacteristic
93C2×C9⋊C8—31no—
94C2×C9⋊C8—61no—
95C2×C9⋊C8C2×D311yescharacteristic
96C22×C3×C9C2×D411yescharacteristic
97C2×C3⋊D9—41no—
98C2×C3⋊D9—41no—
99C2×C3⋊D9—41no—
100C2×D3×C9—21no—
101C2×D3×C9—21no—
102C2×D3×C9SD1612yescomplemented
103C3×C4×C9—22no—
104C3×C4×C9C2×D411yescharacteristic
105C3×C4×C9C2×D411yescharacteristic
106C9×Dic3—21no—
107C9×Dic3—22no—
108C3×Dic9—41no—
109C22×C3×Q8—31no—
110C2×D3×Q8D911yescharacteristic, complemented
111C2×#215—91no—
112C2×#215—91no—
113C2×#215—91no—
114C2×#215—181no—
115#930—91no—
116C2×#204—91no—
117#895—91no—
118C3×#103—91no—
119#835—31no—
120#835—91no—
121#830—91no—
122(C2×D3)⋊C8—91no—
123#810—91no—
124C22×C3⋊D3—61no—
125C22×C3×D3#6711yescharacteristic
126C32×Q8—21no—
127C32×Q8#6712yes—
128C2×C32×C4—21no—
129C2×C32×C4C22×D311yescharacteristic
130C4⋊(C3⋊D3)—61no—
131C4⋊(C3⋊D3)—61no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3#6711yescharacteristic
135C3×C4×D3—24no—
136C3×C4×D3—21no—
137C3×C4×D3—21no—
138C3×Dic6—24no—
139C3×Dic6—21no—
140C3×Dic6#6712yes—
141#626—122no—
142C23×C9—31no—
143C22×D9—61no—
144C22×D9—121no—
145C3⋊(C3⋊C8)—61no—
146C3×C3⋊C8—61no—
147Q8×C9—21no—
148Q8×C9—22no—
149Q8×C9—32no—
150Q8×C9—41no—
151Q8×C9—64no—
152Q8×C9—61no—
153Q8×C9D1212yescomplemented
154C2×C4×C9—21no—
155C2×C4×C9—21no—
156C2×C4×C9—31no—
157C2×C4×C9—41no—
158C2×C4×C9—64no—
159C2×C4×C9—61no—
160C2×C4×C9—61no—
161C2×C4×C9—61no—
162C2×C4×C9C22×D311yescharacteristic
163#611—122no—
164C2×Dic9—21no—
165D36—62no—
166D36—61no—
167D36—61no—
168D36—121no—
169C9⋊C8—21no—
170C9⋊C8—62no—
171C9⋊C8—61no—
172#448—271noSylow
173C2×C3×C9#12011yescharacteristic
174C2×C3×C9C2×SD1611yescharacteristic
175C2×C3×C9#13711yescharacteristic
176C3⋊D9—81no—
177D3×C9—22no—
178C2×C3×Q8—21no—
179C2×C3×Q8—31no—
180C2×C3×Q8—61no—
181C2×C3×Q8—62no—
182C2×C3×Q8—62no—
183C2×C3×Q8C2×D911yescharacteristic
184C2×C3×Q8D3211yescharacteristic
185C22×C3×C4—31no—
186C22×C3×C4—61no—
187C2×#67—181no—
188D3×Q8—22no—
189D3×Q8—22no—
190C2×D12—91no—
191C2×D12—91no—
192C2×D12—181no—
193C2×D12—181no—
194C2×C4×D3—21no—
195C2×C4×D3C2×D911yescharacteristic
196C2×Dic6—21no—
197C2×Dic6C2×D911yescharacteristic
198C2×C3×C8—91no—
199C3×C4⋊C4—91no—
200#215—182no—
201#215—184no—
202#215—182no—
203#215—182no—
204(C2×D3)⋊C4—181no—
205(C2×D3)⋊C4—181no—
206C4⋊Dic3—31no—
207C2×C3⋊C8—31no—
208C2×C3⋊C8—91no—
209C2×C3⋊C8—91no—
210C2×C3⋊C8—181no—
211#204—182no—
212C22×C32C2×#6711yescharacteristic
213C2×C3⋊D3—121no—
214C2×C3⋊D3—121no—
215C2×C3⋊D3—121no—
216C2×C3×D3—21no—
217C2×C3×D3—21no—
218C2×C3×D3#21512yes—
219C32×C4—22no—
220C32×C4C2×#6711yescharacteristic
221C32×C4C2×#6711yescharacteristic
222C3×Dic3—21no—
223C3×Dic3—22no—
224C3×Dic3—121no—
225C22×C9—21no—
226C22×C9—32no—
227C22×C9—61no—
228C22×C9—61no—
229C22×C9D3×D411yescharacteristic
230C2×D9—122no—
231C2×D9—121no—
232C2×D9—121no—
233C2×D9—121no—
234C2×D9—241no—
235C2×D9—241no—
236C4×C9—22no—
237C4×C9—21no—
238C4×C9—21no—
239C4×C9—42no—
240C4×C9—61no—
241C4×C9—62no—
242C4×C9C2×D1211yescharacteristic
243C4×C9D3×D411yescharacteristic
244Dic9—41no—
245C22×Q8—31no—
246C2×SD16—271no—
247C2×SD16—271no—
248#121—271no—
249#103—91no—
250#103—271no—
251C22⋊C8—271no—
252C3×C9#44811yescharacteristic, complemented, Sylow
253C23×C3—31no—
254C22×D3—181no—
255C22×D3—181no—
256C22×D3—361no—
257C22×D3#61111yescharacteristic
258C3×Q8—21no—
259C3×Q8—21no—
260C3×Q8—22no—
261C3×Q8—32no—
262C3×Q8—41no—
263C3×Q8—64no—
264C3×Q8—61no—
265C3×Q8#61112yescomplemented
266C3×Q8#62612yes—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—31no—
271C2×C3×C4—41no—
272C2×C3×C4—64no—
273C2×C3×C4—61no—
274C2×C3×C4—61no—
275C2×C3×C4—61no—
276C2×C3×C4—181no—
277C2×C3×C4C22×D911yescharacteristic
278C2×C3×C4C2×D3211yescharacteristic
279#67—362no—
280C2×Dic3—21no—
281C2×Dic3—61no—
282C2×Dic3#61111yescharacteristic
283D12—182no—
284D12—181no—
285D12—181no—
286D12—181no—
287D12—181no—
288D12—361no—
289D12—362no—
290C4×D3—24no—
291C4×D3—21no—
292C4×D3—21no—
293Dic6—24no—
294Dic6—21no—
295Dic6#61112yescomplemented
296C3×C8—181no—
297C3⋊C8—61no—
298C3⋊C8—182no—
299C3⋊C8—181no—
300C3⋊C8—181no—
301C2×C32C2×#21511yescharacteristic
302C2×C32#94211yescharacteristic
303C2×C32#95311yescharacteristic
304C3⋊D3—241no—
305C3×D3—22no—
306C2×C9—21no—
307C2×C9—21no—
308C2×C9—21no—
309C2×C9—62no—
310C2×C9#88211yescharacteristic
311C2×C9D3×SD1611yescharacteristic
312C2×C9#91811yescharacteristic
313D9—242no—
314D9—241no—
315C2×Q8—31no—
316C2×Q8—61no—
317C2×Q8—62no—
318C2×Q8—62no—
319C2×Q8D3×D911yescharacteristic
320C2×D4—271no—
321C2×D4—541no—
322C22×C4—31no—
323C22×C4—61no—
324SD16—542no—
325SD16—542no—
326C2×C8—91no—
327C2×C8—271no—
328C4⋊C4—91no—
329C22⋊C4—541no—
330C22×C3—21no—
331C22×C3—32no—
332C22×C3—61no—
333C22×C3—61no—
334C22×C3C2×#61111yescharacteristic
335C22×C3D3×#6711yescharacteristic
336C2×D3—21no—
337C2×D3—21no—
338C2×D3—362no—
339C2×D3—361no—
340C2×D3—361no—
341C2×D3—361no—
342C2×D3—361no—
343C2×D3—361no—
344C2×D3—361no—
345C2×D3—721no—
346C2×D3—721no—
347C2×D3#367512yescomplemented
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—42no—
353C3×C4—61no—
354C3×C4—62no—
355C3×C4—361no—
356C3×C4C2×#61111yescharacteristic
357C3×C4C2×#61111yescharacteristic
358C3×C4C2×#62611yescharacteristic
359C3×C4D3×#6711yescharacteristic
360Dic3—21no—
361Dic3—22no—
362Dic3—121no—
363C32#522711yescharacteristic
364C9—21no—
365C9#480611yescharacteristic, complemented
366C23—31no—
367C23—541no—
368Q8—21no—
369Q8—32no—
370Q8—64no—
371Q8—61no—
372Q8#619512yescomplemented
373D4—541no—
374D4—541no—
375D4—1082no—
376C2×C4—21no—
377C2×C4—31no—
378C2×C4—64no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—181no—
383C2×C4C2×D3×D911yescharacteristic
384C8—181no—
385C8—541no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—62no—
390C2×C3C2×#367511yescharacteristic
391C2×C3#6350511yescharacteristic
392C2×C3#6351511yescharacteristic
393C2×C3D3×#21511yescharacteristic
394C2×C3#6394811yescharacteristic
395C2×C3#6397711yescharacteristic
396D3—22no—
397D3—722no—
398D3—721no—
399D3—721no—
400C22—32no—
401C22—61no—
402C22—61no—
403C22—1081no—
404C22—1081no—
405C22—1081no—
406C22D3×#61111yescharacteristic
407C4—22no—
408C4—61no—
409C4—62no—
410C4—361no—
411C4C2×#619511yescharacteristic
412C4D3×#61111yescharacteristic
413C3—21no—
414C3#1058890811yescharacteristic, complemented
415C3#1059158411yescharacteristic
416C2—62no—
417C2—2161no—
418C2D3×#367511yescharacteristic
419C2#1172120811yescharacteristic
420C2#1172128711yescharacteristic
subgroupG/H
centerC22D3×#611
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32C2×#67
prime radical#11721038C2
hypercenterC2×Q8D3×D9
hypocenterC3×C9#448
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×D3×Q8×C9C2semidirect
2. C2×#88548C2semidirect
4. C2×#88426C2semidirect
5. C3×#63395C2semidirect
6. #11720297C2semidirect
7. (C2×D3)⋊(C9⋊C8)C2semidirect
38. #63395D3semidirect
43. C3×Q8×C9D4semidirect
56. C9×Dic6D4semidirect
102. C2×D3×C9SD16semidirect
110. C2×D3×Q8D9semidirect
153. Q8×C9D12semidirect
252. C3×C9#448semidirect
265. C3×Q8#611semidirect
295. Dic6#611semidirect
347. C2×D3#3675semidirect
365. C9#4806semidirect
372. Q8#6195semidirect
414. C3#10588908semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Q8×C92
C2×#885482
#117210382
C2×#884262
C3×#633952
#117202972
(C2×D3)⋊(C9⋊C8)2
#105915843
#105889083