Finite Group Databasebeta

← #13140847

Subgroups — #13140847

Order 1728.

620 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 93 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×#88746C211yesmaximal, characteristic
2C2×C9×#216C211yesmaximal, characteristic, complemented
3#11720929C211yesmaximal, characteristic, complemented
4C2×#88426C211yesmaximal, characteristic, complemented
5C3×#63395C211yesmaximal, characteristic, complemented
6#11720284C211yesmaximal, characteristic, complemented
7#11720282C211yesmaximal, characteristic, complemented
8#10591602—31nomaximal
9#10588867—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11#88746—22no—
12#88746—22no—
13C2×C4⋊(C3⋊D9)—21no—
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9—21no—
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C9×#216—22no—
19C3×C4⋊Dic9C2211yescharacteristic
20C2×C9×C3⋊C8C2211yescharacteristic
21C2×C3×C9⋊C8C2211yescharacteristic
22#88445—21no—
23Dic9⋊Dic3—21no—
24#88426—22no—
25C2×#3832—31no—
26C2×C3×#216D311yescharacteristic
27#63871—31no—
28C2×#3717—31no—
29C2×#3701—31no—
30C2×#3701—61no—
31C3×#835—31no—
32#63564—31no—
33#63557—31no—
34C2×C9×Q16—31no—
35#63508—31no—
36C2×#3675—31no—
37#63395D311yescharacteristic, complemented
38#63380—31no—
39(C2×D9)⋊C8—31no—
40C22×C3⋊D9—21no—
41C22×C3⋊D9D411yescharacteristic
42C3×Q8×C9—21no—
43C3×Q8×C9D412yes—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9C2311yescharacteristic
46C2×C3⋊Dic9D411yescharacteristic
47C4⋊(C3⋊D9)—24no—
48C4⋊(C3⋊D9)—21no—
49C4⋊(C3⋊D9)D412yes—
50C4×C3⋊D9—24no—
51C4×C3⋊D9—21no—
52C4×C3⋊D9—21no—
53C2×C9×Dic3—21no—
54C2×C3×Dic9—21no—
55C9×Dic6—21no—
56C9×Dic6—21no—
57C9×C3⋊C8—21no—
58C3×C9⋊C8—21no—
59#5187—91no—
60#4808—91no—
61C2×C32×Q8C2×D311yescharacteristic
62#3832—62no—
63#3832—62no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C4⋊(C3⋊D3)—61no—
66C2×C4×C3⋊D3—31no—
67C2×C4×C3⋊D3—61no—
68C2×C3×Dic6C2×D311yescharacteristic
69C3×#216—22no—
70C3×C4⋊Dic3—31no—
71C2×C3×C3⋊C8—31no—
72C2×C3×C3⋊C8C2×D311yescharacteristic
73Dic3⋊Dic3—61no—
74#3722—61no—
75#3717—62no—
76C2×Q8×C9—21no—
77C2×Q8×C9—31no—
78C2×Q8×C9C2×D311yescharacteristic
79#3701—62no—
80#3701—64no—
81#3701—64no—
82#3701—62no—
83C2×D36—31no—
84C2×D36—61no—
85C2×D36—61no—
86C2×D36—62no—
87C2×C4×D9—31no—
88C2×C4×D9—61no—
89C2×C4×D9—61no—
90C2×C4×D9—62no—
91C9×Q16—62no—
92C2×C8×C9—31no—
93#3675—62no—
94(C2×D9)⋊C4—61no—
95C4⋊Dic9C2×D311yescharacteristic
96Dic9⋊C4—61no—
97C2×C9⋊C8C2×D311yescharacteristic
98C22×C3×C9C2×D411yescharacteristic
99C2×C3⋊D9—21no—
100C2×C3⋊D9—22no—
101C2×C3⋊D9—21no—
102C2×C3⋊D9—21no—
103C2×C3⋊D9—41no—
104C2×C3⋊D9—41no—
105C3×C4×C9—22no—
106C3×C4×C9C2×D411yescharacteristic
107C3×C4×C9C2×D411yescharacteristic
108C3⋊Dic9—21no—
109C9×Dic3—41no—
110C3×Dic9—41no—
111C2×#239—91no—
112C2×#239—91no—
113C2×#239—181no—
114C2×C3×Q16—31no—
115#946—91no—
116C2×#216D911yescharacteristic, complemented
117C2×#215—91no—
118C2×#204—91no—
119#897—91no—
120C3×#103—91no—
121#835—31no—
122#821—91no—
123(C2×D3)⋊C8—91no—
124(C2×D3)⋊C8—91no—
125#809—91no—
126C22×C3⋊D3—31no—
127C22×C3⋊D3—61no—
128C32×Q8—21no—
129C32×Q8#6712yes—
130C2×C32×C4—21no—
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3—31no—
133C4⋊(C3⋊D3)—32no—
134C4⋊(C3⋊D3)—64no—
135C4⋊(C3⋊D3)—61no—
136C4×C3⋊D3—64no—
137C4×C3⋊D3—61no—
138C4×C3⋊D3—61no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—61no—
141C3×Dic6—21no—
142C3×Dic6—21no—
143C22×D9—31no—
144C22×D9—61no—
145C22×D9—62no—
146C22×D9—61no—
147C3×C3⋊C8—21no—
148C3×C3⋊C8—61no—
149Q8×C9—22no—
150Q8×C9—21no—
151Q8×C9—22no—
152Q8×C9—61no—
153Q8×C9—61no—
154Q8×C9D1212yes—
155C2×C4×C9—22no—
156C2×C4×C9—21no—
157C2×C4×C9—21no—
158C2×C4×C9—61no—
159C2×C4×C9C22×D311yescharacteristic
160C2×Dic9—21no—
161C2×Dic9—31no—
162C2×Dic9—61no—
163D36—32no—
164D36—62no—
165D36—68no—
166D36—64no—
167D36—62no—
168D36—61no—
169C4×D9—68no—
170C4×D9—64no—
171C4×D9—62no—
172C4×D9—62no—
173C4×D9—61no—
174C4×D9—61no—
175C8×C9—61no—
176C9⋊C8—21no—
177#452—271noSylow
178C2×C3×C9#12011yescharacteristic
179C2×C3×C9#13511yescharacteristic
180C2×C3×C9#13711yescharacteristic
181C3⋊D9—41no—
182C3⋊D9—42no—
183C2×C3×Q8—21no—
184C2×C3×Q8—31no—
185C2×C3×Q8C2×D911yescharacteristic
186C2×C3×Q8D3211yescharacteristic
187#239—182no—
188#239—182no—
189#239—184no—
190#239—184no—
191#239—182no—
192#239—182no—
193C2×D12—91no—
194C2×D12—91no—
195C2×D12—181no—
196C2×D12—181no—
197C2×D12—181no—
198C2×D12—182no—
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×Dic6C2×D911yescharacteristic
206C3×Q16—62no—
207C2×C3×C8—31no—
208C2×C3×C8—91no—
209C3×C4⋊C4—91no—
210#216—22no—
211#215—182no—
212(C2×D3)⋊C4—181no—
213(C2×D3)⋊C4—181no—
214C4⋊Dic3—31no—
215C4⋊Dic3—181no—
216Dic3⋊C4—181no—
217C2×C3⋊C8—31no—
218C2×C3⋊C8C2×D911yescharacteristic
219#204—182no—
220C22×C32D3×D411yescharacteristic
221C2×C3⋊D3—62no—
222C2×C3⋊D3—61no—
223C2×C3⋊D3—61no—
224C2×C3⋊D3—61no—
225C2×C3⋊D3—121no—
226C2×C3⋊D3—121no—
227C32×C4—22no—
228C32×C4D3×D411yescharacteristic
229C32×C4C2×#6711yescharacteristic
230C3⋊Dic3—61no—
231C3×Dic3—41no—
232C3×Dic3—121no—
233C22×C9—21no—
234C22×C9D3×D411yescharacteristic
235C2×D9—64no—
236C2×D9—62no—
237C2×D9—62no—
238C2×D9—61no—
239C2×D9—61no—
240C2×D9—61no—
241C2×D9—121no—
242C2×D9—121no—
243C2×D9—122no—
244C2×D9—122no—
245C2×D9—121no—
246C2×D9—121no—
247C4×C9—24no—
248C4×C9—22no—
249C4×C9—21no—
250C4×C9—21no—
251C4×C9—121no—
252C4×C9C2×D1211yescharacteristic
253C4×C9D3×D411yescharacteristic
254Dic9—41no—
255Dic9—62no—
256Dic9—61no—
257C2×#41—271no—
258C2×Q16—31no—
259C2×SD16—271no—
260C22⋊Q8—271no—
261#103—91no—
262#102—271no—
263C22⋊C8—271no—
264C3×C9#45211yescharacteristic, complemented, Sylow
265C22×D3—91no—
266C22×D3—91no—
267C22×D3—181no—
268C22×D3—182no—
269C22×D3—181no—
270C22×D3—181no—
271C3×Q8—22no—
272C3×Q8—21no—
273C3×Q8—21no—
274C3×Q8—22no—
275C3×Q8—61no—
276C3×Q8—61no—
277C3×Q8#61112yes—
278C3×Q8#62612yes—
279C2×C3×C4—22no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—61no—
284C2×C3×C4—181no—
285C2×C3×C4C22×D911yescharacteristic
286C2×C3×C4C2×D3211yescharacteristic
287C2×Dic3—21no—
288C2×Dic3—61no—
289C2×Dic3—91no—
290C2×Dic3—91no—
291C2×Dic3—181no—
292D12—92no—
293D12—92no—
294D12—182no—
295D12—184no—
296D12—188no—
297D12—184no—
298D12—182no—
299D12—181no—
300D12—181no—
301C4×D3—184no—
302C4×D3—188no—
303C4×D3—184no—
304C4×D3—182no—
305C4×D3—182no—
306C4×D3—181no—
307C4×D3—181no—
308C4×D3—181no—
309C4×D3—181no—
310Dic6—21no—
311Dic6—21no—
312C3×C8—61no—
313C3×C8—181no—
314C3⋊C8—21no—
315C3⋊C8—61no—
316C2×C32#91811yescharacteristic
317C2×C32#91911yescharacteristic
318C2×C32#93711yescharacteristic
319C3⋊D3—121no—
320C3⋊D3—122no—
321C2×C9—21no—
322C2×C9—21no—
323C2×C9—21no—
324C2×C9#88211yescharacteristic
325C2×C9#91611yescharacteristic
326C2×C9#91811yescharacteristic
327D9—122no—
328D9—124no—
329D9—121no—
330D9—122no—
331#41—542no—
332#41—542no—
333C2×Q8—31no—
334C2×Q8D3×D911yescharacteristic
335C2×D4—271no—
336C2×D4—541no—
337C22×C4—271no—
338C22×C4—541no—
339Q16—62no—
340SD16—542no—
341C2×C8—31no—
342C2×C8—91no—
343C4⋊C4—91no—
344C4⋊C4—541no—
345C22⋊C4—541no—
346C22×C3—21no—
347C22×C3D4×D911yescharacteristic
348C22×C3#381111yescharacteristic
349C2×D3—184no—
350C2×D3—182no—
351C2×D3—182no—
352C2×D3—181no—
353C2×D3—181no—
354C2×D3—181no—
355C2×D3—182no—
356C2×D3—181no—
357C2×D3—181no—
358C2×D3—181no—
359C2×D3—361no—
360C2×D3—361no—
361C2×D3—362no—
362C2×D3—362no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—361no—
366C2×D3—361no—
367C3×C4—24no—
368C3×C4—22no—
369C3×C4—22no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—121no—
373C3×C4—361no—
374C3×C4D4×D911yescharacteristic
375C3×C4C2×#61111yescharacteristic
376C3×C4C2×#62611yescharacteristic
377C3×C4#381111yescharacteristic
378Dic3—41no—
379Dic3—121no—
380Dic3—182no—
381Dic3—181no—
382Dic3—181no—
383C32#518711yescharacteristic
384C9—21no—
385C9#480811yescharacteristic, complemented
386C23—271no—
387C23—541no—
388Q8—21no—
389Q8—61no—
390Q8—61no—
391Q8#619512yes—
392D4—272no—
393D4—544no—
394D4—541no—
395C2×C4—21no—
396C2×C4—61no—
397C2×C4—181no—
398C2×C4—271no—
399C2×C4—544no—
400C2×C4—541no—
401C2×C4—541no—
402C2×C4C2×D3×D911yescharacteristic
403C8—61no—
404C8—181no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3#6348011yescharacteristic
409C2×C3#6348111yescharacteristic
410C2×C3#6349911yescharacteristic
411C2×C3#6394511yescharacteristic
412C2×C3#6395011yescharacteristic
413C2×C3#6398111yescharacteristic
414D3—362no—
415D3—364no—
416D3—361no—
417D3—362no—
418D3—361no—
419D3—362no—
420C22—541no—
421C22—541no—
422C22—542no—
423C22—541no—
424C22—1081no—
425C22—1081no—
426C22#8866811yescharacteristic
427C4—22no—
428C4—121no—
429C4—361no—
430C4—541no—
431C4C2×#619511yescharacteristic
432C4#8866811yescharacteristic
433C3—21no—
434C3#1058886711yescharacteristic, complemented
435C3#1059160211yescharacteristic
436C2—1081no—
437C2—1082no—
438C2#1172122111yescharacteristic
439C2#1172122411yescharacteristic
440C2#1172129111yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC2×#88746C2
hypercenterC2×Q8D3×D9
hypocenterC3×C9#452
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
2. C2×C9×#216C2semidirect
3. #11720929C2semidirect
4. C2×#88426C2semidirect
5. C3×#63395C2semidirect
6. #11720284C2semidirect
7. #11720282C2semidirect
37. #63395D3semidirect
116. C2×#216D9semidirect
264. C3×C9#452semidirect
385. C9#4808semidirect
434. C3#10588867semidirect

Maximal subgroups

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

subgroupindex
C2×#887462
C2×C9×#2162
#117209292
C2×#884262
C3×#633952
#117202842
#117202822
#105916023
#105888673