Finite Group Databasebeta

← #13140875

Subgroups — #13140875

Order 1728.

532 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 77 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
1#11722478C211yesmaximal, characteristic
2#11721857C211yesmaximal, characteristic, complemented
3C9×#942C211yesmaximal, characteristic, complemented
4C3×#63505C211yesmaximal, characteristic, complemented
5#11720300C211yesmaximal, characteristic, complemented
6#11720291C211yesmaximal, characteristic, complemented
7#11720288C211yesmaximal, characteristic, complemented
8#10591658—31nomaximal
9#10588883—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic, complemented
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C9×#235C2211yescharacteristic
13C3×#3697C2211yescharacteristic
14#88668—22no—
15D9×D12—21no—
16D3×D36—21no—
17#88642—21no—
18#88635—21no—
19#88536—22no—
20C4×C3⋊Dic9C2211yescharacteristic, complemented
21C9×#216—21no—
22C9×#215—21no—
23C3×#3675—21no—
24C3×#3674—21no—
25C9×#208C2211yescharacteristic, complemented
26C3×#3667C2211yescharacteristic, complemented
27#64308—31no—
28#64156—31no—
29C3×#942—31no—
30C3×#942D311yescharacteristic
31#63715—31no—
32#63568—31no—
33#63568—31no—
34#63559—31no—
35C9×#137—31no—
36#63509—31no—
37#63509—61no—
38#63505D311yescharacteristic, complemented
39#63396—31no—
40#63384—31no—
41#63383—31no—
42C22×C3⋊D9D412yes—
43C2×D3×D9—22no—
44C3×Q8×C9—21no—
45C3×Q8×C9—21no—
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9C2311yescharacteristic
48C2×C3⋊Dic9—21no—
49C4⋊(C3⋊D9)—21no—
50C4⋊(C3⋊D9)—21no—
51C9×#67—21no—
52C3×#611—21no—
53C9×D12D411yescharacteristic
54C4×D3×C9—21no—
55C3×D36D411yescharacteristic
56C3×C4×D9—21no—
57C9×Dic6D411yescharacteristic
58C3×Dic18D411yescharacteristic
59#6195—22no—
60#6192—22no—
61(C3⋊D9)⋊C4—22no—
62C9×C3⋊C8—21no—
63C3×C9⋊C8—21no—
64#5202—91no—
65#5202—91no—
66C2×C32×Q8C2×D311yescharacteristic
67C2×C4⋊(C3⋊D3)—31no—
68C3×#235—31no—
69C3×#235C2×D311yescharacteristic
70#3811—62no—
71D3×D12—61no—
72D3×D12—61no—
73#3799—61no—
74#3799—61no—
75#3752—62no—
76C4×C3⋊Dic3—31no—
77C3×#216—21no—
78C3×#216—61no—
79C3×#215—21no—
80C3×#215—61no—
81C3×#208—31no—
82C3×#208C2×D311yescharacteristic
83C9×#41—31no—
84C2×Q8×C9—21no—
85C2×Q8×C9C2×D311yescharacteristic, complemented
86#3701—61no—
87D4×D9—61no—
88D4×D9—62no—
89#3697—61no—
90#3697C2×D311yescharacteristic
91C2×D36—31no—
92C2×D36—61no—
93C2×D36—61no—
94C9×Q16—61no—
95C9×SD16—61no—
96C9×M16—31no—
97#3675—21no—
98#3674—21no—
99(C2×D9)⋊C4—62no—
100(C2×D9)⋊C4—64no—
101C4×Dic9—31no—
102C4×Dic9—61no—
103#3667C2×D311yescharacteristic, complemented
104C22×C3×C9C2×D411yescharacteristic
105C2×C3⋊D9—22no—
106C2×C3⋊D9—42no—
107C2×D3×C9—21no—
108C2×C3×D9—21no—
109D3×D9—82no—
110C3×C4×C9—41no—
111C3×C4×C9C2×D411yescharacteristic
112C3×C4×C9C2×D411yescharacteristic
113C3⋊Dic9—41no—
114C9×Dic3—21no—
115C3×Dic9—21no—
116#1009—91no—
117#1009—91no—
118C3×#137—31no—
119C3×#137—91no—
120#947—91no—
121#947—91no—
122#947—181no—
123#942—31no—
124#942D911yescharacteristic, complemented
125#837—91no—
126#837—91no—
127#825—91no—
128#825—91no—
129#823—91no—
130#823—91no—
131C22×C3⋊D3—32no—
132C2×D32—62no—
133C32×Q8—21no—
134C32×Q8—21no—
135C2×C32×C4—21no—
136C2×C32×C4C22×D311yescharacteristic
137C2×C3⋊Dic3—61no—
138C4⋊(C3⋊D3)—61no—
139C4⋊(C3⋊D3)—61no—
140C3×#67—21no—
141C3×#67—61no—
142C3×D12—31no—
143C3×D12#6711yescharacteristic
144C3×C4×D3—21no—
145C3×C4×D3—61no—
146C3×Dic6—31no—
147C3×Dic6#6711yescharacteristic
148#626—62no—
149#626—62no—
150#624—62no—
151C22×D9—32no—
152C22×D9—62no—
153C22×D9—62no—
154C3×C3⋊C8—21no—
155C3×C3⋊C8—61no—
156Q8×C9—21no—
157Q8×C9—21no—
158Q8×C9—31no—
159Q8×C9—41no—
160Q8×C9—41no—
161D4×C9—31no—
162D4×C9—61no—
163C2×C4×C9—22no—
164C2×C4×C9—21no—
165C2×C4×C9—21no—
166C2×C4×C9—61no—
167C2×C4×C9C22×D311yescharacteristic
168#611—21no—
169#611—62no—
170C2×Dic9—62no—
171C2×Dic9—61no—
172D36—62no—
173D36—61no—
174D36—61no—
175D36—121no—
176D36—121no—
177D36#6711yescharacteristic
178C4×D9—21no—
179C4×D9—62no—
180Dic18#6711yescharacteristic
181C8×C9—61no—
182C9⋊C8—21no—
183#455—271noSylow
184C2×C3×C9—21no—
185C2×C3×C9#12011yescharacteristic
186C3⋊D9—42no—
187D3×C9—41no—
188C3×D9—41no—
189C3×#41—31no—
190C3×#41—91no—
191C2×C3×Q8—21no—
192C2×C3×Q8C2×D911yescharacteristic, complemented
193C2×C3×Q8D3211yescharacteristic
194#239—181no—
195#239—181no—
196D3×D4—181no—
197D3×D4—181no—
198D3×D4—182no—
199D3×D4—182no—
200#235—31no—
201#235—181no—
202#235—181no—
203#235C2×D911yescharacteristic
204C2×D12—91no—
205C2×D12—91no—
206C2×D12—181no—
207C2×D12—181no—
208C2×D12—181no—
209C3×Q16—61no—
210C3×Q16—181no—
211C3×SD16—61no—
212C3×SD16—181no—
213C3×M16—31no—
214C3×M16—91no—
215#216—21no—
216#216—61no—
217#215—21no—
218#215—61no—
219(C2×D3)⋊C4—182no—
220(C2×D3)⋊C4—182no—
221(C2×D3)⋊C4—184no—
222C4×Dic3—91no—
223C4×Dic3—91no—
224C4×Dic3—181no—
225#208—31no—
226#208C2×D911yescharacteristic, complemented
227C22×C32D3×D411yescharacteristic
228C2×C3⋊D3—62no—
229C2×C3⋊D3—122no—
230C2×C3×D3—21no—
231C2×C3×D3—61no—
232D32—242no—
233C32×C4—41no—
234C32×C4D3×D411yescharacteristic
235C32×C4C2×#6711yescharacteristic
236C3⋊Dic3—121no—
237C3×Dic3—21no—
238C3×Dic3—61no—
239C22×C9—21no—
240C22×C9—61no—
241C22×C9D3×D411yescharacteristic
242C2×D9—21no—
243C2×D9—62no—
244C2×D9—122no—
245C2×D9—122no—
246C2×D9—124no—
247C2×D9—242no—
248C4×C9—21no—
249C4×C9—21no—
250C4×C9—42no—
251C4×C9—41no—
252C4×C9—61no—
253C4×C9D3×D411yescharacteristic
254C4×C9C2×#6711yescharacteristic
255Dic9—21no—
256Dic9—122no—
257Dic9—121no—
258H(5,𝔽2)—271no—
259#137—31no—
260#137—91no—
261#124—271no—
262#104—271no—
263#104—271no—
264#100—271no—
265C3×C9#45511yescharacteristic, complemented, Sylow
266C22×D3—92no—
267C22×D3—92no—
268C22×D3—182no—
269C22×D3—182no—
270C22×D3—182no—
271C3×Q8—21no—
272C3×Q8—21no—
273C3×Q8—21no—
274C3×Q8—21no—
275C3×Q8—31no—
276C3×Q8—41no—
277C3×Q8—41no—
278C3×Q8—91no—
279C3×D4—31no—
280C3×D4—61no—
281C3×D4—91no—
282C3×D4—181no—
283C2×C3×C4—22no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—61no—
288C2×C3×C4—181no—
289C2×C3×C4C22×D911yescharacteristic
290C2×C3×C4C2×D3211yescharacteristic
291#67—21no—
292#67—61no—
293#67—182no—
294#67—182no—
295C2×Dic3—182no—
296C2×Dic3—181no—
297C2×Dic3—181no—
298D12—31no—
299D12—182no—
300D12—182no—
301D12—181no—
302D12—181no—
303D12—181no—
304D12—181no—
305D12—361no—
306D12—361no—
307D12#61111yescharacteristic
308C4×D3—21no—
309C4×D3—61no—
310C4×D3—182no—
311C4×D3—182no—
312Dic6—31no—
313Dic6#61111yescharacteristic
314C3×C8—61no—
315C3×C8—181no—
316C3⋊C8—21no—
317C3⋊C8—61no—
318C2×C32—21no—
319C2×C32#93711yescharacteristic
320C3⋊D3—122no—
321C3×D3—41no—
322C3×D3—121no—
323C2×C9—22no—
324C2×C9—21no—
325C2×C9—21no—
326C2×C9—121no—
327C2×C9#93711yescharacteristic
328D9—41no—
329D9—122no—
330D9—242no—
331#41—31no—
332#41—91no—
333#41—541no—
334#41—541no—
335C2×Q8D3×D911yescharacteristic, complemented
336C2×D4—271no—
337C2×D4—541no—
338C2×D4—541no—
339C2×D4—542no—
340Q16—61no—
341Q16—181no—
342SD16—61no—
343SD16—181no—
344M16—31no—
345M16—91no—
346C22⋊C4—542no—
347C42—271no—
348C22×C3—21no—
349C22×C3—61no—
350C22×C3—181no—
351C22×C3D4×D911yescharacteristic
352C22×C3#381111yescharacteristic
353C2×D3—21no—
354C2×D3—61no—
355C2×D3—182no—
356C2×D3—182no—
357C2×D3—362no—
358C2×D3—362no—
359C2×D3—362no—
360C2×D3—364no—
361C2×D3—722no—
362C2×D3—722no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—42no—
366C3×C4—41no—
367C3×C4—41no—
368C3×C4—61no—
369C3×C4—181no—
370C3×C4D4×D911yescharacteristic
371C3×C4C2×#61111yescharacteristic
372C3×C4D3×#6711yescharacteristic
373C3×C4D3×#6711yescharacteristic
374Dic3—21no—
375Dic3—61no—
376Dic3—361no—
377Dic3—362no—
378Dic3—361no—
379C32#520211yescharacteristic
380C9—21no—
381C9#520211yescharacteristic, complemented
382C23—272no—
383C23—542no—
384Q8—21no—
385Q8—21no—
386Q8—31no—
387Q8—91no—
388D4—31no—
389D4—61no—
390D4—91no—
391D4—181no—
392D4—542no—
393D4—542no—
394D4—541no—
395D4—541no—
396C2×C4—21no—
397C2×C4—61no—
398C2×C4—181no—
399C2×C4—542no—
400C2×C4—541no—
401C2×C4C2×D3×D911yescharacteristic
402C8—61no—
403C8—181no—
404C2×C3—22no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—121no—
409C2×C3—361no—
410C2×C3#6349911yescharacteristic
411C2×C3#6398311yescharacteristic
412D3—41no—
413D3—121no—
414D3—362no—
415D3—362no—
416D3—722no—
417C22—61no—
418C22—181no—
419C22—542no—
420C22—1082no—
421C22—2162no—
422C22#8866811yescharacteristic
423C4—41no—
424C4—61no—
425C4—181no—
426C4—1081no—
427C4D3×#61111yescharacteristic
428C4D9×#6711yescharacteristic
429C3—21no—
430C3#1058888311yescharacteristic, complemented
431C3#1059165811yescharacteristic
432C2—21no—
433C2—121no—
434C2—361no—
435C2—1082no—
436C2#1172129211yescharacteristic
subgroupG/H
centerC2#11721292
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC2×C32#937
prime radical#11722478C2
hypercenterC2×Q8D3×D9
hypocenterC3×C9#455
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. #11721857C2semidirect
3. C9×#942C2semidirect
4. C3×#63505C2semidirect
5. #11720300C2semidirect
6. #11720291C2semidirect
7. #11720288C2semidirect
10. C2×C3×Q8×C9C22semidirect
20. C4×C3⋊Dic9C22semidirect
25. C9×#208C22semidirect
26. C3×#3667C22semidirect
38. #63505D3semidirect
85. C2×Q8×C9C2×D3semidirect
103. #3667C2×D3semidirect
124. #942D9semidirect
192. C2×C3×Q8C2×D9semidirect
226. #208C2×D9semidirect
265. C3×C9#455semidirect
335. C2×Q8D3×D9semidirect
381. C9#5202semidirect
430. C3#10588883semidirect

Maximal subgroups

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

subgroupindex
#117224782
#117218572
C9×#9422
C3×#635052
#117203002
#117202912
#117202882
#105916583
#105888833