Finite Group Databasebeta

← #13138672

Subgroups — #13138672

Order 1728.

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

normal subgroups only

This group has 638 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×#88646C211yesmaximal, characteristic
2C2×#88523C211yesmaximal, characteristic, complemented
3#11720922C211yesmaximal, characteristic, complemented
4C9×#821C211yesmaximal, characteristic, complemented
5C3×#63380C211yesmaximal, characteristic, complemented
6#11720294C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic, complemented
8#10590941—31nomaximal
9#10588450—31nomaximal
10C22×D3×D9—21no—
11C2×C4×C3⋊D9C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×C3×D36C2211yescharacteristic
14C2×#6194—21no—
15#88646—22no—
16#88646C414yescomplemented
17C2×C3×C8×C9C2211yescharacteristic, complemented
18C2×C3⋊(C9⋊C8)C2211yescharacteristic, complemented
19#88523—22no—
20C9×C4⋊Dic3C2211yescharacteristic, complemented
21C3×C4⋊Dic9C2211yescharacteristic, complemented
22#88445—21no—
23Dic3×Dic9—21no—
24C2×#3802—31no—
25C2×#3743—31no—
26#63849—31no—
27C2×D4×D9—31no—
28C3×#821—31no—
29C3×#821D311yescharacteristic
30#63562—31no—
31#63562—31no—
32C2×#3663—31no—
33C2×#3663—61no—
34#63454—31no—
35C9×#102—31no—
36#63392—31no—
37#63380D311yescharacteristic, complemented
38#63369—31no—
39C22×C3⋊D9D411yescharacteristic
40C22×D3×C9—21no—
41C22×C3×D9—21no—
42C2×D3×D9—24no—
43C2×D3×D9—42no—
44C2×D3×D9—42no—
45C2×C3×C4×C9C2311yescharacteristic
46C2×C3⋊Dic9D411yescharacteristic
47C4×C3⋊D9C2×C412yes—
48C4×C3⋊D9D412yes—
49C2×C9×Dic3—21no—
50C2×C3×Dic9—21no—
51C9×D12—21no—
52C9×D12C2×C412yes—
53C3×D36—21no—
54C3×D36C2×C412yes—
55#6194—24no—
56C3×C8×C9—21no—
57C3⋊(C9⋊C8)—21no—
58#4770—91no—
59#4770—91no—
60C22×D32—61no—
61C2×C4×C3⋊D3—31no—
62C2×C3×D12—31no—
63C2×C3×D12C2×D311yescharacteristic
64C2×#625—61no—
65#3802—34no—
66#3802—62no—
67C23×D9—61no—
68C2×C32×C8C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)—31no—
70#3743—62no—
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3C2×D311yescharacteristic
73#3722—61no—
74Dic32—61no—
75C2×D4×C9—31no—
76C2×#611—61no—
77D4×D9—34no—
78D4×D9—62no—
79C2×D36C2×D311yescharacteristic
80C2×C4×D9—31no—
81C2×C4×D9—61no—
82C2×C8×C9—21no—
83C2×C8×C9C2×D311yescharacteristic, complemented
84C9×C4⋊C4—31no—
85(C2×D9)⋊C4—61no—
86C4⋊Dic9C2×D311yescharacteristic, complemented
87C4×Dic9—61no—
88C2×C9⋊C8—31no—
89C2×C9⋊C8—61no—
90#3663—62no—
91#3663—64no—
92C22×C3×C9C2×D411yescharacteristic
93C2×C3⋊D9—21no—
94C2×C3⋊D9—21no—
95C2×C3⋊D9C22⋊C412yes—
96C2×D3×C9—22no—
97C2×D3×C9—41no—
98C2×D3×C9—41no—
99C2×C3×D9—22no—
100C2×C3×D9—41no—
101C2×C3×D9—41no—
102D3×D9—48no—
103C3×C4×C9C22×C411yescharacteristic
104C3×C4×C9C2×D411yescharacteristic
105C3⋊Dic9C22⋊C412yes—
106C9×Dic3—41no—
107C3×Dic9—41no—
108C2×D3×D4—91no—
109C2×D3×D4—91no—
110C2×ΓL(1,𝔽25)—91no—
111C2×ΓL(1,𝔽25)—91no—
112C2×ΓL(1,𝔽25)—181no—
113#892—91no—
114#892—91no—
115C3×#102—31no—
116C3×#102—91no—
117#832—91no—
118#832—91no—
119#821—31no—
120#821D911yescharacteristic, complemented
121#809—91no—
122#809—91no—
123C22×C3⋊D3—31no—
124C22×C3×D3—21no—
125C22×C3×D3—61no—
126C2×D32—64no—
127C2×D32—122no—
128C2×D32—122no—
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—31no—
131C4×C3⋊D3—32no—
132C4×C3⋊D3—32no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—61no—
135C3×D12—21no—
136C3×D12—32no—
137C3×D12—61no—
138C3×D12C4×D312yes—
139#625—64no—
140C23×C9—61no—
141C22×D9—21no—
142C22×D9—31no—
143C22×D9—61no—
144C22×D9—64no—
145C22×D9—122no—
146C22×D9—122no—
147C32×C8—21no—
148C3⋊(C3⋊C8)—61no—
149D4×C9—32no—
150D4×C9—61no—
151C2×C4×C9—21no—
152C2×C4×C9—61no—
153C2×C4×C9C22×D311yescharacteristic
154#611—64no—
155C2×Dic9—21no—
156C2×Dic9—31no—
157C2×Dic9—61no—
158D36—21no—
159D36C4×D312yes—
160C4×D9—32no—
161C4×D9—32no—
162C4×D9—62no—
163C4×D9—62no—
164C8×C9—22no—
165C8×C9—21no—
166C9⋊C8—62no—
167C9⋊C8—61no—
168#418—271noSylow
169C2×C3×C9C2×C22⋊C411yescharacteristic
170C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
171C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
172C3⋊D9—22no—
173D3×C9—42no—
174C3×D9—42no—
175C23×D3—181no—
176C23×D3—181no—
177C2×C3×D4—31no—
178C2×C3×D4—91no—
179C2×#67—181no—
180C2×#67—181no—
181D3×D4—94no—
182D3×D4—94no—
183D3×D4—182no—
184D3×D4—182no—
185C2×D12—31no—
186C2×D12C2×D911yescharacteristic
187C2×C4×D3—91no—
188C2×C4×D3—91no—
189C2×C4×D3—181no—
190C2×C3×C8—21no—
191C2×C3×C8C2×D911yescharacteristic, complemented
192C2×C3×C8D3211yescharacteristic
193C3×C4⋊C4—31no—
194C3×C4⋊C4—91no—
195(C2×D3)⋊C4—181no—
196(C2×D3)⋊C4—181no—
197C4⋊Dic3—31no—
198C4⋊Dic3C2×D911yescharacteristic, complemented
199C4×Dic3—181no—
200C4×Dic3—181no—
201C2×C3⋊C8—91no—
202C2×C3⋊C8—91no—
203C2×C3⋊C8—181no—
204ΓL(1,𝔽25)—182no—
205ΓL(1,𝔽25)—182no—
206ΓL(1,𝔽25)—184no—
207C22×C32D3×D411yescharacteristic
208C2×C3⋊D3—32no—
209C2×C3⋊D3—61no—
210C2×C3⋊D3—61no—
211C2×C3×D3—22no—
212C2×C3×D3—41no—
213C2×C3×D3—41no—
214C2×C3×D3—62no—
215C2×C3×D3—121no—
216C2×C3×D3—121no—
217D32—128no—
218C32×C4C2×C4×D311yescharacteristic
219C32×C4D3×D411yescharacteristic
220C3⋊Dic3—32no—
221C3×Dic3—41no—
222C3×Dic3—121no—
223C22×C9—21no—
224C22×C9—62no—
225C22×C9—121no—
226C22×C9—121no—
227C22×C9D3×D411yescharacteristic
228C2×D9—22no—
229C2×D9—32no—
230C2×D9—41no—
231C2×D9—41no—
232C2×D9—62no—
233C2×D9—61no—
234C2×D9—61no—
235C2×D9—121no—
236C2×D9—121no—
237C2×D9—128no—
238C4×C9—21no—
239C4×C9—21no—
240C4×C9—121no—
241C4×C9C2×C4×D311yescharacteristic
242C4×C9D3×D411yescharacteristic
243Dic9—32no—
244Dic9—41no—
245Dic9—62no—
246C22×D4—271no—
247C2×M16—271no—
248#117—271no—
249#102—31no—
250#102—91no—
251#102—271no—
252#102—271no—
253C3×C9#41811yescharacteristic, complemented, Sylow
254C23×C3—61no—
255C23×C3—181no—
256C22×D3—21no—
257C22×D3—61no—
258C22×D3—91no—
259C22×D3—91no—
260C22×D3—181no—
261C22×D3—184no—
262C22×D3—184no—
263C22×D3—362no—
264C22×D3—362no—
265C22×D3—362no—
266C22×D3—362no—
267C3×D4—32no—
268C3×D4—61no—
269C3×D4—92no—
270C3×D4—181no—
271C2×C3×C4—21no—
272C2×C3×C4—61no—
273C2×C3×C4—181no—
274C2×C3×C4C22×D911yescharacteristic
275C2×C3×C4C2×D3211yescharacteristic
276#67—184no—
277#67—184no—
278C2×Dic3—21no—
279C2×Dic3—61no—
280C2×Dic3—91no—
281C2×Dic3—91no—
282C2×Dic3—181no—
283D12—21no—
284D12—32no—
285D12—61no—
286D12C4×D912yes—
287C4×D3—92no—
288C4×D3—92no—
289C4×D3—92no—
290C4×D3—92no—
291C4×D3—182no—
292C4×D3—182no—
293C3×C8—22no—
294C3×C8—21no—
295C3×C8—21no—
296C3⋊C8—182no—
297C3⋊C8—181no—
298C3⋊C8—181no—
299C2×C32D3×C22⋊C411yescharacteristic
300C2×C32#91111yescharacteristic
301C2×C32#91411yescharacteristic
302C3⋊D3—62no—
303C3×D3—42no—
304C3×D3—122no—
305C2×C9—21no—
306C2×C9—21no—
307C2×C9—21no—
308C2×C9—122no—
309C2×C9D3×C22⋊C411yescharacteristic
310C2×C9#91111yescharacteristic
311C2×C9#91411yescharacteristic
312D9—42no—
313D9—62no—
314D9—122no—
315C24—541no—
316C2×D4—31no—
317C2×D4—91no—
318C2×D4—274no—
319C2×D4—542no—
320C2×D4—541no—
321C22×C4—271no—
322M16—542no—
323C2×C8—271no—
324C2×C8D3×D911yescharacteristic, complemented
325C4⋊C4—31no—
326C4⋊C4—91no—
327C22⋊C4—541no—
328C42—541no—
329C22×C3—21no—
330C22×C3—62no—
331C22×C3—121no—
332C22×C3—121no—
333C22×C3—182no—
334C22×C3—361no—
335C22×C3—361no—
336C22×C3D4×D911yescharacteristic
337C22×C3#380211yescharacteristic
338C2×D3—22no—
339C2×D3—41no—
340C2×D3—41no—
341C2×D3—62no—
342C2×D3—92no—
343C2×D3—92no—
344C2×D3—121no—
345C2×D3—121no—
346C2×D3—182no—
347C2×D3—181no—
348C2×D3—181no—
349C2×D3—181no—
350C2×D3—181no—
351C2×D3—361no—
352C2×D3—361no—
353C2×D3—368no—
354C2×D3—368no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—121no—
358C3×C4—361no—
359C3×C4C2×C4×D911yescharacteristic
360C3×C4D4×D911yescharacteristic
361C3×C4C4×D3211yescharacteristic
362C3×C4#381111yescharacteristic
363Dic3—41no—
364Dic3—92no—
365Dic3—92no—
366Dic3—121no—
367Dic3—182no—
368C32#477011yescharacteristic
369C9—21no—
370C9#477011yescharacteristic, complemented
371C23—61no—
372C23—181no—
373C23—271no—
374C23—544no—
375C23—1082no—
376C23—1082no—
377D4—32no—
378D4—61no—
379D4—92no—
380D4—181no—
381D4—544no—
382C2×C4—61no—
383C2×C4—181no—
384C2×C4—271no—
385C2×C4—272no—
386C2×C4—272no—
387C2×C4C2×D3×D911yescharacteristic
388C8—21no—
389C8—541no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—122no—
394C2×C3—362no—
395C2×C3C22⋊C4×D911yescharacteristic
396C2×C3#6347311yescharacteristic
397C2×C3#6347611yescharacteristic
398C2×C3#6379811yescharacteristic
399C2×C3#6379811yescharacteristic
400C2×C3#6392111yescharacteristic
401D3—42no—
402D3—122no—
403D3—182no—
404D3—182no—
405D3—362no—
406C22—62no—
407C22—121no—
408C22—121no—
409C22—182no—
410C22—272no—
411C22—361no—
412C22—361no—
413C22—541no—
414C22—541no—
415C22—1088no—
416C22#8864611yescharacteristic
417C4—121no—
418C4—272no—
419C4—361no—
420C4C4×D3×D911yescharacteristic
421C4#8866811yescharacteristic
422C3—21no—
423C3#1058845011yescharacteristic, complemented
424C3#1059094111yescharacteristic
425C2—122no—
426C2—362no—
427C2—542no—
428C2#1172081911yescharacteristic
429C2#1172082011yescharacteristic
430C2#1172104811yescharacteristic
subgroupG/H
centerC22#88646
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×#88646C2
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9#418
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#88523C2semidirect
3. #11720922C2semidirect
4. C9×#821C2semidirect
5. C3×#63380C2semidirect
6. #11720294C2semidirect
7. #11720266C2semidirect
16. #88646C4semidirect
17. C2×C3×C8×C9C22semidirect
18. C2×C3⋊(C9⋊C8)C22semidirect
20. C9×C4⋊Dic3C22semidirect
21. C3×C4⋊Dic9C22semidirect
37. #63380D3semidirect
83. C2×C8×C9C2×D3semidirect
86. C4⋊Dic9C2×D3semidirect
120. #821D9semidirect
191. C2×C3×C8C2×D9semidirect
198. C4⋊Dic3C2×D9semidirect
253. C3×C9#418semidirect
324. C2×C8D3×D9semidirect
370. C9#4770semidirect
423. C3#10588450semidirect

Maximal subgroups

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

subgroupindex
C2×#886462
C2×#885232
#117209222
C9×#8212
C3×#633802
#117202942
#117202662
#105909413
#105884503