Finite Group Databasebeta

← #13140820

Subgroups — #13140820

Order 1728.

630 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 95 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×Q8×C3⋊D9C211yesmaximal, characteristic, complemented
2C2×C9×#215C211yesmaximal, characteristic, complemented
3#11721045C211yesmaximal, characteristic
4C2×#88424C211yesmaximal, characteristic, complemented
5C3×#63395C211yesmaximal, characteristic, complemented
6#11720282C211yesmaximal, characteristic, complemented
7#11720277C211yesmaximal, characteristic, complemented
8#10591592—31nomaximal
9#10588850—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11Q8×C3⋊D9—22no—
12Q8×C3⋊D9—22no—
13C2×C4×C3⋊D9—21no—
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×#6226—21no—
16C2×#6226C2211yescharacteristic
17C2×C9×D12C2211yescharacteristic
18C2×#6195—21no—
19C9×#215—22no—
20C3×C4⋊Dic9C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23(C2×D3)⋊Dic9—21no—
24#88424—22no—
25C2×Q8×C3⋊D3—31no—
26C2×C3×#215D311yescharacteristic
27#63916—31no—
28C2×#3716—31no—
29C2×Q8×D9—31no—
30C2×Q8×D9—61no—
31C3×#835—31no—
32#63567—31no—
33#63557—31no—
34C2×C9×SD16—31no—
35#63500—31no—
36C2×#3672—31no—
37#63395D311yescharacteristic, complemented
38(C2×D9)⋊C8—31no—
39#63376—31no—
40C22×C3⋊D9D411yescharacteristic
41C22×D3×C9—21no—
42C3×Q8×C9—21no—
43C3×Q8×C9D412yescomplemented
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9C2311yescharacteristic
46C2×C3⋊Dic9—21no—
47C2×C3⋊Dic9D411yescharacteristic
48C4×C3⋊D9—24no—
49C4×C3⋊D9—21no—
50C4×C3⋊D9—21no—
51#6226—24no—
52#6226—21no—
53#6226D412yescomplemented
54C2×C3×Dic9—21no—
55C9×D12—21no—
56C9×D12—21no—
57#6195—42no—
58C9×C3⋊C8—21no—
59C3×C9⋊C8—21no—
60#5170—91no—
61#4806—91no—
62C2×C32×Q8C2×D311yescharacteristic
63Q8×C3⋊D3—62no—
64Q8×C3⋊D3—62no—
65C2×C4×C3⋊D3—31no—
66C2×C4×C3⋊D3—61no—
67C2×#634—31no—
68C2×#634—61no—
69C2×C3×D12C2×D311yescharacteristic
70C2×#626—61no—
71C3×#215—22no—
72C3×C4⋊Dic3—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8C2×D311yescharacteristic
75(C2×D3)⋊Dic3—61no—
76#3716—62no—
77C2×Q8×C9—21no—
78C2×Q8×C9C2×D311yescharacteristic
79C2×D4×C9—31no—
80C2×#611—61no—
81Q8×D9—62no—
82Q8×D9—64no—
83Q8×D9—64no—
84Q8×D9—62no—
85C2×C4×D9—31no—
86C2×C4×D9—61no—
87C2×C4×D9—61no—
88C2×C4×D9—62no—
89C2×Dic18—31no—
90C2×Dic18—61no—
91C2×Dic18—61no—
92C2×Dic18—62no—
93C9×SD16—62no—
94C2×C8×C9—31no—
95#3676—61no—
96#3672—62no—
97C4⋊Dic9C2×D311yescharacteristic
98C2×C9⋊C8C2×D311yescharacteristic
99C22×C3×C9C2×D411yescharacteristic
100C2×C3⋊D9—21no—
101C2×C3⋊D9—21no—
102C2×C3⋊D9SD1612yescomplemented
103C2×D3×C9—41no—
104C2×D3×C9—41no—
105C2×D3×C9—41no—
106C3×C4×C9—22no—
107C3×C4×C9C2×D411yescharacteristic
108C3×C4×C9C2×D411yescharacteristic
109C3⋊Dic9—21no—
110C3⋊Dic9—22no—
111C3×Dic9—41no—
112C2×D3×Q8—91no—
113C2×D3×Q8—91no—
114C2×D3×Q8—181no—
115C2×C3×SD16—31no—
116C2×#215D911yescharacteristic, complemented
117#938—91no—
118C2×#214—91no—
119C2×#204—91no—
120#895—91no—
121C3×#103—91no—
122#835—31no—
123(C2×D3)⋊C8—91no—
124(C2×D3)⋊C8—91no—
125#816—91no—
126#810—91no—
127C22×C3⋊D3—31no—
128C22×C3×D3—21no—
129C32×Q8—21no—
130C32×Q8#6712yes—
131C2×C32×C4—21no—
132C2×C32×C4C22×D311yescharacteristic
133C2×C3⋊Dic3—31no—
134C2×C3⋊Dic3—61no—
135C4×C3⋊D3—64no—
136C4×C3⋊D3—61no—
137C4×C3⋊D3—61no—
138#634—32no—
139#634—64no—
140#634—61no—
141C2×C3×Dic3—61no—
142C3×D12—21no—
143C3×D12—21no—
144#626—122no—
145C23×C9—61no—
146C22×D9—31no—
147C22×D9—61no—
148C3×C3⋊C8—21no—
149C3×C3⋊C8—61no—
150Q8×C9—22no—
151Q8×C9—21no—
152Q8×C9—22no—
153Q8×C9D1212yescomplemented
154D4×C9—61no—
155D4×C9—61no—
156C2×C4×C9—22no—
157C2×C4×C9—21no—
158C2×C4×C9—21no—
159C2×C4×C9C22×D311yescharacteristic
160#611—122no—
161C2×Dic9—21no—
162C2×Dic9—31no—
163C2×Dic9—61no—
164C2×Dic9—62no—
165C2×Dic9—61no—
166C4×D9—68no—
167C4×D9—64no—
168C4×D9—62no—
169C4×D9—62no—
170C4×D9—61no—
171C4×D9—61no—
172Dic18—32no—
173Dic18—62no—
174Dic18—68no—
175Dic18—64no—
176Dic18—62no—
177Dic18—61no—
178C8×C9—61no—
179C9⋊C8—21no—
180#448—271noSylow
181C2×C3×C9#12011yescharacteristic
182C2×C3×C9C2×SD1611yescharacteristic
183C2×C3×C9#13711yescharacteristic
184C3⋊D9—22no—
185D3×C9—81no—
186C2×C3×Q8—21no—
187C2×C3×Q8C2×D911yescharacteristic
188C2×C3×Q8D3211yescharacteristic
189C2×C3×D4—31no—
190C2×#67—181no—
191D3×Q8—182no—
192D3×Q8—182no—
193D3×Q8—184no—
194D3×Q8—184no—
195D3×Q8—182no—
196D3×Q8—182no—
197C2×D12—181no—
198C2×D12C2×D911yescharacteristic
199C2×C4×D3—91no—
200C2×C4×D3—91no—
201C2×C4×D3—181no—
202C2×C4×D3—181no—
203C2×C4×D3—181no—
204C2×C4×D3—182no—
205C2×Dic6—91no—
206C2×Dic6—91no—
207C2×Dic6—181no—
208C2×Dic6—181no—
209C2×Dic6—181no—
210C2×Dic6—182no—
211C3×SD16—62no—
212C2×C3×C8—31no—
213C2×C3×C8—91no—
214C3×C4⋊C4—91no—
215#217—181no—
216#215—22no—
217#214—182no—
218(C2×D3)⋊C4—181no—
219C4⋊Dic3—31no—
220C2×C3⋊C8—31no—
221C2×C3⋊C8C2×D911yescharacteristic
222#204—182no—
223C22×C32D3×D411yescharacteristic
224C2×C3⋊D3—32no—
225C2×C3⋊D3—61no—
226C2×C3⋊D3—61no—
227C2×C3×D3—41no—
228C2×C3×D3—41no—
229C2×C3×D3—41no—
230C32×C4—22no—
231C32×C4D3×D411yescharacteristic
232C32×C4C2×#6711yescharacteristic
233C3⋊Dic3—62no—
234C3⋊Dic3—61no—
235C3×Dic3—121no—
236C22×C9—21no—
237C22×C9—121no—
238C22×C9—121no—
239C22×C9—121no—
240C22×C9D3×D411yescharacteristic
241C2×D9—32no—
242C2×D9—62no—
243C2×D9—62no—
244C2×D9—62no—
245C2×D9—61no—
246C2×D9—61no—
247C4×C9—24no—
248C4×C9—22no—
249C4×C9—21no—
250C4×C9—21no—
251C4×C9C2×D1211yescharacteristic
252C4×C9D3×D411yescharacteristic
253Dic9—41no—
254Dic9—64no—
255Dic9—62no—
256Dic9—62no—
257Dic9—61no—
258C22×Q8—271no—
259C2×SD16—31no—
260C2×SD16—271no—
261#121—271no—
262#103—91no—
263#103—271no—
264C22⋊C8—271no—
265C3×C9#44811yescharacteristic, complemented, Sylow
266C23×C3—61no—
267C22×D3—21no—
268C22×D3—91no—
269C22×D3—91no—
270C22×D3—181no—
271C3×Q8—22no—
272C3×Q8—21no—
273C3×Q8—21no—
274C3×Q8—22no—
275C3×Q8#61112yescomplemented
276C3×Q8#62612yes—
277C3×D4—61no—
278C3×D4—61no—
279C2×C3×C4—22no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—181no—
284C2×C3×C4C22×D911yescharacteristic
285C2×C3×C4C2×D3211yescharacteristic
286#67—362no—
287C2×Dic3—61no—
288C2×Dic3—91no—
289C2×Dic3—91no—
290C2×Dic3—181no—
291C2×Dic3—182no—
292C2×Dic3—181no—
293C2×Dic3—181no—
294D12—21no—
295D12—21no—
296D12—362no—
297C4×D3—184no—
298C4×D3—188no—
299C4×D3—184no—
300C4×D3—182no—
301C4×D3—182no—
302C4×D3—181no—
303C4×D3—181no—
304C4×D3—181no—
305C4×D3—181no—
306Dic6—92no—
307Dic6—92no—
308Dic6—182no—
309Dic6—184no—
310Dic6—188no—
311Dic6—184no—
312Dic6—182no—
313Dic6—181no—
314Dic6—181no—
315C3×C8—61no—
316C3×C8—181no—
317C3⋊C8—21no—
318C3⋊C8—61no—
319C2×C32D3×SD1611yescharacteristic
320C2×C32#91511yescharacteristic
321C2×C32#93711yescharacteristic
322C3⋊D3—62no—
323C3×D3—81no—
324C2×C9—21no—
325C2×C9—21no—
326C2×C9—21no—
327C2×C9—241no—
328C2×C9#88211yescharacteristic
329C2×C9D3×SD1611yescharacteristic
330C2×C9#91811yescharacteristic
331D9—64no—
332D9—62no—
333C2×Q8—271no—
334C2×Q8—541no—
335C2×Q8—542no—
336C2×Q8—542no—
337C2×Q8D3×D911yescharacteristic
338C2×D4—31no—
339C2×D4—541no—
340C22×C4—271no—
341C22×C4—541no—
342SD16—62no—
343SD16—542no—
344C2×C8—31no—
345C2×C8—91no—
346C4⋊C4—91no—
347C22⋊C4—541no—
348C22×C3—21no—
349C22×C3—121no—
350C22×C3—121no—
351C22×C3—121no—
352C22×C3D4×D911yescharacteristic
353C22×C3#381111yescharacteristic
354C2×D3—41no—
355C2×D3—41no—
356C2×D3—41no—
357C2×D3—92no—
358C2×D3—92no—
359C2×D3—182no—
360C2×D3—182no—
361C2×D3—182no—
362C2×D3—181no—
363C2×D3—181no—
364C2×D3—181no—
365C2×D3—181no—
366C3×C4—24no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—361no—
372C3×C4D4×D911yescharacteristic
373C3×C4C2×#61111yescharacteristic
374C3×C4C2×#62611yescharacteristic
375C3×C4#381111yescharacteristic
376Dic3—121no—
377Dic3—182no—
378Dic3—184no—
379Dic3—182no—
380Dic3—181no—
381Dic3—182no—
382Dic3—181no—
383C32#517011yescharacteristic
384C9—21no—
385C9#480611yescharacteristic, complemented
386C23—61no—
387C23—271no—
388Q8—21no—
389Q8—272no—
390Q8—544no—
391Q8—541no—
392Q8#619512yescomplemented
393D4—61no—
394D4—61no—
395D4—1082no—
396C2×C4—21no—
397C2×C4—181no—
398C2×C4—271no—
399C2×C4—544no—
400C2×C4—541no—
401C2×C4—541no—
402C2×C4—541no—
403C2×C4C2×D3×D911yescharacteristic
404C8—61no—
405C8—181no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—241no—
410C2×C3SD16×D911yescharacteristic
411C2×C3#6347711yescharacteristic
412C2×C3#6349911yescharacteristic
413C2×C3#6394011yescharacteristic
414C2×C3#6395111yescharacteristic
415C2×C3#6398111yescharacteristic
416D3—81no—
417D3—184no—
418D3—182no—
419D3—182no—
420C22—121no—
421C22—121no—
422C22—121no—
423C22—272no—
424C22—541no—
425C22—541no—
426C22#8866811yescharacteristic
427C4—22no—
428C4—361no—
429C4—541no—
430C4—542no—
431C4C2×#619511yescharacteristic
432C4#8866811yescharacteristic
433C3—21no—
434C3#1058885011yescharacteristic, complemented
435C3#1059159211yescharacteristic
436C2—241no—
437C2—542no—
438C2#1172121211yescharacteristic
439C2#1172121711yescharacteristic
440C2#1172129111yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radical#11721045C2
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×Q8×C3⋊D9C2semidirect
2. C2×C9×#215C2semidirect
4. C2×#88424C2semidirect
5. C3×#63395C2semidirect
6. #11720282C2semidirect
7. #11720277C2semidirect
37. #63395D3semidirect
43. C3×Q8×C9D4semidirect
53. #6226D4semidirect
102. C2×C3⋊D9SD16semidirect
116. C2×#215D9semidirect
153. Q8×C9D12semidirect
265. C3×C9#448semidirect
275. C3×Q8#611semidirect
385. C9#4806semidirect
392. Q8#6195semidirect
434. C3#10588850semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C3⋊D92
C2×C9×#2152
#117210452
C2×#884242
C3×#633952
#117202822
#117202772
#105915923
#105888503