Finite Group Databasebeta

← #13140476

Subgroups — #13140476

Order 1728.

614 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 101 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×D4×D9C211yesmaximal, characteristic
2C2×C9×#214C211yesmaximal, characteristic, complemented
3#11720944C211yesmaximal, characteristic, complemented
4C2×#88425C211yesmaximal, characteristic, complemented
5#11720576C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic, complemented
7(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591458—31nomaximal
9#10588849—31nomaximal
10C23×C3×D9—21no—
11C2×C3×D4×C9C2211yescharacteristic
12C2×C3×#611—21no—
13C3×D4×D9—22no—
14C3×D4×D9—22no—
15C2×C3×D36C2211yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic, complemented
18C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
19C2×C3⋊(C9⋊C8)C2211yescharacteristic, complemented
20C9×#214—22no—
21C2×C9×C3⋊C8C2211yescharacteristic, complemented
22(C2×D9)⋊Dic3—21no—
23#88442—21no—
24#88425—22no—
25C2×C3×D3×D4—31no—
26C2×C3×#214D311yescharacteristic
27#63856—31no—
28C2×#3715—31no—
29C2×D4×D9D311yescharacteristic
30#63659—31no—
31#63563—31no—
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×C9×SD16—31no—
34#63508—31no—
35C2×#3675—31no—
36#63392—31no—
37#63392—61no—
38#63380—31no—
39(C2×D9)⋊C8—31no—
40C23×C3×C9—21no—
41C22×C3×D9—24no—
42C22×C3×D9—21no—
43C22×C3×D9—42no—
44C22×C3×D9—42no—
45C22×C3×D9D411yescharacteristic
46C3×D4×C9—21no—
47C3×D4×C9D412yes—
48C2×C3×C4×C9C2311yescharacteristic
49C2×C3⋊Dic9—21no—
50C3×#611—24no—
51C2×C9×Dic3—21no—
52C2×C3×Dic9D411yescharacteristic
53C3×D36—21no—
54C3×D36D412yes—
55C3×C4×D9—21no—
56C3×C4×D9—21no—
57C9×Dic6—21no—
58C9×Dic6—21no—
59C3⋊(C9⋊C8)—21no—
60C9×C3⋊C8—21no—
61#5218—91no—
62#5169—91no—
63C23×C3×D3—61no—
64C2×C32×D4C2×D311yescharacteristic
65C2×C3×#67—61no—
66C3×D3×D4—62no—
67C3×D3×D4—62no—
68C2×C3×D12—31no—
69C2×C3×C4×D3—31no—
70C2×C3×Dic6C2×D311yescharacteristic
71C23×D9—21no—
72C4⋊(C3⋊Dic3)—31no—
73C2×C3⋊(C3⋊C8)—31no—
74C3×#214—22no—
75C2×C3×C3⋊C8C2×D311yescharacteristic
76#3724—61no—
77(C2×D3)⋊Dic3—61no—
78#3715—62no—
79C2×Q8×C9—31no—
80C2×D4×C9—21no—
81C2×D4×C9C2×D311yescharacteristic
82C2×#611—21no—
83D4×D9—22no—
84D4×D9—22no—
85C2×D36C2×D311yescharacteristic
86C2×C4×D9C2×D311yescharacteristic
87C9×SD16—62no—
88C2×C8×C9—31no—
89#3675—62no—
90(C2×D9)⋊C4—61no—
91C4⋊Dic9—31no—
92C4⋊Dic9—61no—
93Dic9⋊C4—61no—
94C2×C9⋊C8—31no—
95C2×C9⋊C8—61no—
96C22×C3×C9—22no—
97C22×C3×C9—41no—
98C22×C3×C9—41no—
99C22×C3×C9C2×D411yescharacteristic
100C2×C3×D9—21no—
101C2×C3×D9—21no—
102C2×C3×D9—22no—
103C2×C3×D9—48no—
104C2×C3×D9—41no—
105C2×C3×D9—41no—
106C2×C3×D9SD1612yescomplemented
107C3×C4×C9C2×D411yescharacteristic
108C3×C4×C9C2×D411yescharacteristic
109C3⋊Dic9—41no—
110C9×Dic3—41no—
111C3×Dic9—21no—
112C22×C3×D4—91no—
113C2×D3×D4—31no—
114C2×C3×SD16—31no—
115#946—91no—
116C2×#215—91no—
117C2×#214—91no—
118C2×#214D911yescharacteristic, complemented
119#924—91no—
120#832—91no—
121#832—91no—
122#832—91no—
123#832—181no—
124#830—91no—
125#821—91no—
126(C2×D3)⋊C8—91no—
127C23×C32—21no—
128C22×C3×D3—31no—
129C22×C3×D3—64no—
130C22×C3×D3—61no—
131C22×C3×D3—122no—
132C22×C3×D3—122no—
133C32×D4—21no—
134C32×D4#6712yes—
135C2×C32×C4C22×D311yescharacteristic
136C2×C3⋊Dic3—61no—
137C3×#67—64no—
138C2×C3×Dic3—21no—
139C2×C3×Dic3—31no—
140C3×D12—32no—
141C3×D12—61no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144C3×Dic6—21no—
145C3×Dic6—21no—
146C23×C9—21no—
147C23×C9—41no—
148C22×D9—24no—
149C22×D9—21no—
150C22×D9—42no—
151C22×D9—42no—
152C22×D9#6711yescharacteristic
153C3⋊(C3⋊C8)—61no—
154C3×C3⋊C8—21no—
155Q8×C9—61no—
156Q8×C9—61no—
157D4×C9—21no—
158D4×C9—22no—
159D4×C9—41no—
160D4×C9#6712yes—
161C2×C4×C9—21no—
162C2×C4×C9—61no—
163C2×C4×C9C22×D311yescharacteristic
164#611—24no—
165C2×Dic9—61no—
166C2×Dic9—121no—
167C2×Dic9#6711yescharacteristic
168D36—21no—
169D36#6712yes—
170C4×D9—21no—
171C4×D9—21no—
172C8×C9—61no—
173C9⋊C8—61no—
174C9⋊C8—121no—
175#450—271noSylow
176C2×C3×C9—42no—
177C2×C3×C9#12011yescharacteristic
178C2×C3×C9C2×SD1611yescharacteristic
179C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
180C3×D9—22no—
181C3×D9—42no—
182C24×C3—181no—
183C23×D3—61no—
184C2×C3×Q8—31no—
185C2×C3×D4—21no—
186C2×C3×D4—91no—
187C2×C3×D4—181no—
188C2×C3×D4—182no—
189C2×C3×D4—182no—
190C2×C3×D4C2×D911yescharacteristic
191C2×C3×D4D3211yescharacteristic
192C22×C3×C4—91no—
193C2×#67—61no—
194D3×D4—62no—
195D3×D4—62no—
196C2×D12—31no—
197C2×C4×D3—31no—
198C2×Dic6C2×D911yescharacteristic, complemented
199C3×SD16—62no—
200C2×C3×C8—31no—
201#217—181no—
202#215—182no—
203#214—22no—
204#214—182no—
205(C2×D3)⋊C4—181no—
206C4⋊Dic3—91no—
207C4⋊Dic3—91no—
208C4⋊Dic3—181no—
209Dic3⋊C4—181no—
210Dic3⋊C4—181no—
211C2×C3⋊C8—91no—
212C2×C3⋊C8—91no—
213C2×C3⋊C8—181no—
214C2×C3⋊C8C2×D911yescharacteristic, complemented
215C22×C32—22no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32D3×D411yescharacteristic
219C2×C3×D3—32no—
220C2×C3×D3—61no—
221C2×C3×D3—61no—
222C2×C3×D3—62no—
223C2×C3×D3—128no—
224C2×C3×D3—121no—
225C2×C3×D3—121no—
226C32×C4D3×D411yescharacteristic
227C32×C4C2×#6711yescharacteristic
228C3⋊Dic3—121no—
229C3×Dic3—41no—
230C3×Dic3—61no—
231C22×C9—22no—
232C22×C9—21no—
233C22×C9—42no—
234C22×C9—41no—
235C22×C9—41no—
236C22×C9—81no—
237C22×C9—81no—
238C22×C9C2×#6711yescharacteristic
239C2×D9—21no—
240C2×D9—21no—
241C2×D9—22no—
242C2×D9—48no—
243C2×D9—41no—
244C2×D9—41no—
245C2×D9#21412yescomplemented
246C4×C9—21no—
247C4×C9—21no—
248C4×C9—121no—
249C4×C9C2×#6711yescharacteristic
250C4×C9C2×#6711yescharacteristic
251Dic9—21no—
252Dic9—121no—
253Dic9—241no—
254C22×D4—91no—
255C2×SD16—31no—
256C2×SD16—271no—
257C22⋊Q8—271no—
258#102—271no—
259#102—271no—
260C22⋊C8—271no—
261C3×C9#45011yescharacteristic, complemented, Sylow
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—41no—
265C23×C3—91no—
266C23×C3—184no—
267C23×C3—181no—
268C23×C3—362no—
269C23×C3—362no—
270C22×D3—31no—
271C22×D3—64no—
272C22×D3—61no—
273C22×D3—122no—
274C22×D3—122no—
275C3×Q8—61no—
276C3×Q8—61no—
277C3×D4—21no—
278C3×D4—21no—
279C3×D4—22no—
280C3×D4—41no—
281C3×D4—92no—
282C3×D4—184no—
283C3×D4—181no—
284C3×D4#61112yes—
285C3×D4#62512yes—
286C2×C3×C4—21no—
287C2×C3×C4—61no—
288C2×C3×C4—91no—
289C2×C3×C4—181no—
290C2×C3×C4—181no—
291C2×C3×C4C22×D911yescharacteristic
292C2×C3×C4C2×D3211yescharacteristic
293#67—64no—
294C2×Dic3—21no—
295C2×Dic3—31no—
296C2×Dic3—181no—
297C2×Dic3—181no—
298C2×Dic3—361no—
299D12—32no—
300D12—61no—
301C4×D3—61no—
302C4×D3—61no—
303Dic6—21no—
304Dic6—21no—
305C3×C8—61no—
306C3⋊C8—21no—
307C3⋊C8—181no—
308C3⋊C8—181no—
309C3⋊C8—361no—
310C2×C32—42no—
311C2×C32D3×SD1611yescharacteristic
312C2×C32#91411yescharacteristic
313C2×C32#93711yescharacteristic
314C3×D3—62no—
315C3×D3—122no—
316C2×C9—21no—
317C2×C9—21no—
318C2×C9—21no—
319C2×C9—42no—
320C2×C9—82no—
321C2×C9#93211yescharacteristic
322C2×C9C2×#21411yescharacteristic
323C2×C9#95311yescharacteristic
324D9—22no—
325D9—42no—
326C24—181no—
327C2×Q8—31no—
328C2×D4—91no—
329C2×D4—181no—
330C2×D4—182no—
331C2×D4—182no—
332C2×D4D3×D911yescharacteristic
333C22×C4—91no—
334SD16—62no—
335SD16—542no—
336C2×C8—31no—
337C2×C8—271no—
338C4⋊C4—271no—
339C4⋊C4—541no—
340C22⋊C4—541no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—21no—
344C22×C3—42no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3—92no—
352C22×C3—181no—
353C22×C3—181no—
354C22×C3—182no—
355C22×C3—368no—
356C22×C3—361no—
357C22×C3—361no—
358C22×C3D4×D911yescharacteristic
359C22×C3D3×#6711yescharacteristic
360C2×D3—32no—
361C2×D3—61no—
362C2×D3—61no—
363C2×D3—62no—
364C2×D3—128no—
365C2×D3—121no—
366C2×D3—121no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—121no—
370C3×C4—181no—
371C3×C4D4×D911yescharacteristic
372C3×C4C2×#61111yescharacteristic
373C3×C4C2×#62511yescharacteristic
374C3×C4D3×#6711yescharacteristic
375Dic3—41no—
376Dic3—61no—
377Dic3—361no—
378Dic3—361no—
379Dic3—721no—
380C32#516911yescharacteristic
381C9—21no—
382C9#521811yescharacteristic, complemented
383C23—21no—
384C23—91no—
385C23—184no—
386C23—181no—
387C23—362no—
388C23—362no—
389Q8—61no—
390Q8—61no—
391D4—21no—
392D4—92no—
393D4—184no—
394D4—181no—
395D4#619412yes—
396C2×C4—61no—
397C2×C4—91no—
398C2×C4—181no—
399C2×C4—181no—
400C2×C4—541no—
401C2×C4C2×D3×D911yescharacteristic
402C8—61no—
403C8—541no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—42no—
408C2×C3—42no—
409C2×C3—82no—
410C2×C3—182no—
411C2×C3—362no—
412C2×C3SD16×D911yescharacteristic
413C2×C3#6347611yescharacteristic
414C2×C3#6349911yescharacteristic
415C2×C3D3×#21411yescharacteristic
416C2×C3#6393411yescharacteristic
417C2×C3#6397611yescharacteristic
418D3—62no—
419D3—122no—
420C22—22no—
421C22—41no—
422C22—41no—
423C22—92no—
424C22—181no—
425C22—181no—
426C22—182no—
427C22—368no—
428C22—361no—
429C22—361no—
430C22D9×#6711yescharacteristic
431C4—121no—
432C4—181no—
433C4—1081no—
434C4C2×#619411yescharacteristic
435C4D9×#6711yescharacteristic
436C3—21no—
437C3#1058884911yescharacteristic, complemented
438C3#1059145811yescharacteristic
439C2—42no—
440C2—182no—
441C2—362no—
442C2D9×#21411yescharacteristic
443C2#1172118411yescharacteristic
444C2#1172128511yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×D4×D9C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#450
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×#88425C2semidirect
5. #11720576C2semidirect
6. #11720283C2semidirect
7. (C2×D9)⋊(C3⋊C8)C2semidirect
17. C2×C9×Dic6C22semidirect
18. C4⋊(C3⋊Dic9)C22semidirect
19. C2×C3⋊(C9⋊C8)C22semidirect
21. C2×C9×C3⋊C8C22semidirect
106. C2×C3×D9SD16semidirect
118. C2×#214D9semidirect
198. C2×Dic6C2×D9semidirect
214. C2×C3⋊C8C2×D9semidirect
245. C2×D9#214semidirect
261. C3×C9#450semidirect
382. C9#5218semidirect
437. C3#10588849semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×D92
C2×C9×#2142
#117209442
C2×#884252
#117205762
#117202832
(C2×D9)⋊(C3⋊C8)2
#105914583
#105888493