Finite Group Databasebeta

← #13139799

Subgroups — #13139799

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×#88736C211yesmaximal, characteristic
2C9×#924C211yesmaximal, characteristic, complemented
3C2×#88428C211yesmaximal, characteristic, complemented
4C2×#88426C211yesmaximal, characteristic, complemented
5C3×#63389C211yesmaximal, characteristic, complemented
6#11720269C211yesmaximal, characteristic, complemented
7#11720267C211yesmaximal, characteristic, complemented
8#10591184—31nomaximal
9#10588730—31nomaximal
10C22×C3×C4×C9C2211yescharacteristic
11C2×#6233—21no—
12#88736—22no—
13#88736—22no—
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9—21no—
16C2×#6226C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C9×#217—21no—
19C9×C4⋊Dic3C2211yescharacteristic
20C9×Dic3⋊C4—21no—
21C2×C3×C9⋊C8C2211yescharacteristic
22C2×C3×C9⋊C8C2211yescharacteristic
23#88428—22no—
24#88426—22no—
25C2×#3828—31no—
26C3×#924D311yescharacteristic
27C2×#3719—31no—
28C2×#3717—31no—
29C2×#3697—31no—
30C2×#3697—61no—
31C3×#830—31no—
32#63567—31no—
33#63564—31no—
34C9×C22⋊Q8—31no—
35C2×#3675—31no—
36C2×#3674—31no—
37#63389D311yescharacteristic, complemented
38#63369—31no—
39#63368—31no—
40C23×C3×C9D411yescharacteristic
41C22×C3⋊D9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9D411yescharacteristic
45C2×C3×C4×C9C2311yescharacteristic
46#6233—24no—
47C2×C3⋊Dic9—21no—
48C4⋊(C3⋊D9)—21no—
49C4⋊(C3⋊D9)D412yes—
50C4×C3⋊D9—24no—
51#6226—21no—
52#6226D412yes—
53C2×C9×Dic3—21no—
54C2×C9×Dic3—21no—
55C9×Dic6—21no—
56C9×Dic6—21no—
57C3×C9⋊C8—21no—
58C3×C9⋊C8—21no—
59#5047—91no—
60#4733—91no—
61C22×C32×C4C2×D311yescharacteristic
62C2×#638—61no—
63#3828—62no—
64#3828—62no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4×C3⋊D3—61no—
67C2×#634—31no—
68C2×C3×Dic6C2×D311yescharacteristic
69C3×#217—21no—
70C3×C4⋊Dic3C2×D311yescharacteristic
71C3×Dic3⋊C4—21no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74#3719—62no—
75#3717—62no—
76C2×Q8×C9—31no—
77C22×C4×C9—21no—
78C22×C4×C9C2×D311yescharacteristic
79C2×#611—62no—
80C2×#611—61no—
81#3697—62no—
82#3697—62no—
83#3697—64no—
84#3697—64no—
85C2×D36—31no—
86C2×D36—61no—
87C2×C4×D9—62no—
88C2×C4×D9—61no—
89C2×Dic18—31no—
90C2×Dic18—61no—
91C9×C4⋊C4—31no—
92C9×C4⋊C4—61no—
93C9×C22⋊C4—61no—
94#3675—62no—
95#3674—62no—
96C2×C9⋊C8C2×D311yescharacteristic
97C2×C9⋊C8C2×D311yescharacteristic
98C22×C3×C9—21no—
99C22×C3×C9—21no—
100C22×C3×C9—21no—
101C22×C3×C9C2×D411yescharacteristic
102C2×C3⋊D9—22no—
103C2×C3⋊D9—41no—
104C2×C3⋊D9—41no—
105C3×C4×C9—21no—
106C3×C4×C9C2×D411yescharacteristic
107C3×C4×C9C2×D411yescharacteristic
108C3⋊Dic9—22no—
109C9×Dic3—41no—
110C9×Dic3—41no—
111C2×#235—91no—
112C2×#235—91no—
113C2×#235—181no—
114C3×C22⋊Q8—31no—
115C2×#216—91no—
116C2×#215—91no—
117#924D911yescharacteristic, complemented
118C2×Dic12—91no—
119C2×#204—91no—
120C3×C22⋊C8—91no—
121#830—31no—
122#821—91no—
123#816—91no—
124#810—91no—
125#809—91no—
126C23×C32#6711yescharacteristic
127C22×C3⋊D3—61no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4#6711yescharacteristic
131C2×C32×C4C22×D311yescharacteristic
132#638—64no—
133C2×C3⋊Dic3—61no—
134C4⋊(C3⋊D3)—32no—
135C4⋊(C3⋊D3)—61no—
136C4×C3⋊D3—64no—
137#634—32no—
138#634—61no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—21no—
141C3×Dic6—21no—
142C3×Dic6—21no—
143C23×C9—21no—
144C23×C9D1211yescharacteristic
145C22×D9—62no—
146C22×D9—61no—
147C3×C3⋊C8—61no—
148C3×C3⋊C8—61no—
149Q8×C9—61no—
150Q8×C9—61no—
151C2×C4×C9—22no—
152C2×C4×C9—22no—
153C2×C4×C9—21no—
154C2×C4×C9—21no—
155C2×C4×C9—21no—
156C2×C4×C9—21no—
157C2×C4×C9—61no—
158C2×C4×C9—61no—
159C2×C4×C9D1211yescharacteristic
160C2×C4×C9C22×D311yescharacteristic
161#611—68no—
162#611—64no—
163C2×Dic9—62no—
164C2×Dic9—61no—
165D36—32no—
166D36—62no—
167D36—62no—
168D36—61no—
169C4×D9—68no—
170C4×D9—64no—
171Dic18—32no—
172Dic18—62no—
173Dic18—62no—
174Dic18—61no—
175C9⋊C8—21no—
176C9⋊C8—21no—
177#452—271noSylow
178C2×C3×C9—41no—
179C2×C3×C9#12011yescharacteristic
180C2×C3×C9#13511yescharacteristic
181C2×C3×C9#13711yescharacteristic
182C3⋊D9—42no—
183C2×C3×Q8—31no—
184C22×C3×C4—21no—
185C22×C3×C4C2×D911yescharacteristic
186C22×C3×C4D3211yescharacteristic
187C2×#67—182no—
188C2×#67—181no—
189C2×#67—181no—
190#235—182no—
191#235—182no—
192#235—182no—
193#235—182no—
194#235—184no—
195#235—184no—
196C2×D12—91no—
197C2×D12—91no—
198C2×D12—181no—
199C2×C4×D3—182no—
200C2×C4×D3—181no—
201C2×C4×D3—181no—
202C2×Dic6—91no—
203C2×Dic6—91no—
204C2×Dic6—181no—
205C2×Dic6C2×D911yescharacteristic
206C2×C3×C8—91no—
207C2×C3×C8—91no—
208C3×C4⋊C4—31no—
209C3×C4⋊C4—61no—
210C3×C22⋊C4—61no—
211#217—21no—
212#216—182no—
213#215—182no—
214C4⋊Dic3C2×D911yescharacteristic
215Dic3⋊C4—21no—
216C2×C3⋊C8—31no—
217C2×C3⋊C8—31no—
218Dic12—182no—
219#204—182no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32C2×#6711yescharacteristic
224C2×C3⋊D3—62no—
225C2×C3⋊D3—121no—
226C2×C3⋊D3—121no—
227C32×C4—21no—
228C32×C4D3×D411yescharacteristic
229C32×C4D3×D411yescharacteristic
230C3⋊Dic3—62no—
231C3×Dic3—41no—
232C3×Dic3—41no—
233C22×C9—22no—
234C22×C9—21no—
235C22×C9—21no—
236C22×C9—21no—
237C22×C9—21no—
238C22×C9—41no—
239C22×C9—41no—
240C22×C9C2×D1211yescharacteristic
241C2×D9—64no—
242C2×D9—62no—
243C2×D9—122no—
244C2×D9—122no—
245C2×D9—121no—
246C2×D9—121no—
247C4×C9—22no—
248C4×C9—21no—
249C4×C9—21no—
250C4×C9—21no—
251C4×C9—121no—
252C4×C9—121no—
253C4×C9D3×D411yescharacteristic
254C4×C9D3×D411yescharacteristic
255Dic9—64no—
256Dic9—62no—
257C2×#41—271no—
258C2×Q16—271no—
259C2×SD16—271no—
260C22⋊Q8—31no—
261#103—271no—
262#102—271no—
263C22⋊C8—91no—
264C3×C9#45211yescharacteristic, complemented, Sylow
265C23×C3—21no—
266C23×C3#61111yescharacteristic
267C23×C3#62611yescharacteristic
268C22×D3—182no—
269C22×D3—181no—
270C22×D3—181no—
271C3×Q8—61no—
272C3×Q8—61no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4#61111yescharacteristic
284C2×C3×C4C22×D911yescharacteristic
285C2×C3×C4#62611yescharacteristic
286C2×C3×C4C2×D3211yescharacteristic
287#67—184no—
288#67—188no—
289#67—184no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—182no—
293C2×Dic3—181no—
294C2×Dic3—181no—
295D12—92no—
296D12—92no—
297D12—182no—
298D12—182no—
299D12—181no—
300D12—181no—
301C4×D3—184no—
302C4×D3—188no—
303C4×D3—184no—
304Dic6—21no—
305Dic6—21no—
306Dic6—92no—
307Dic6—92no—
308Dic6—182no—
309Dic6—182no—
310Dic6—181no—
311Dic6—181no—
312C3×C8—181no—
313C3×C8—181no—
314C3⋊C8—61no—
315C3⋊C8—61no—
316C2×C32—41no—
317C2×C32#93711yescharacteristic
318C2×C32#94211yescharacteristic
319C2×C32#95011yescharacteristic
320C3⋊D3—122no—
321C2×C9—21no—
322C2×C9—21no—
323C2×C9—21no—
324C2×C9—42no—
325C2×C9—41no—
326C2×C9#88211yescharacteristic
327C2×C9#90411yescharacteristic
328C2×C9#90911yescharacteristic
329D9—124no—
330D9—122no—
331#41—542no—
332#41—542no—
333C2×Q8—31no—
334C2×Q8—271no—
335C2×D4—271no—
336C2×D4—541no—
337C22×C4—541no—
338C22×C4D3×D911yescharacteristic
339Q16—542no—
340SD16—542no—
341C2×C8—91no—
342C2×C8—91no—
343C4⋊C4—31no—
344C4⋊C4—61no—
345C22⋊C4—61no—
346C22×C3—22no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3C2×#61111yescharacteristic
357C22×C3C2×#62611yescharacteristic
358C2×D3—184no—
359C2×D3—182no—
360C2×D3—182no—
361C2×D3—362no—
362C2×D3—362no—
363C2×D3—361no—
364C2×D3—361no—
365C2×D3—361no—
366C2×D3—361no—
367C3×C4—22no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—21no—
371C3×C4—21no—
372C3×C4—121no—
373C3×C4—121no—
374C3×C4D4×D911yescharacteristic
375C3×C4D4×D911yescharacteristic
376C3×C4#381111yescharacteristic
377C3×C4#381111yescharacteristic
378Dic3—41no—
379Dic3—41no—
380Dic3—184no—
381Dic3—182no—
382Dic3—182no—
383C32#504711yescharacteristic
384C9—21no—
385C9#473311yescharacteristic, complemented
386C23—541no—
387C23#619511yescharacteristic
388Q8—61no—
389Q8—61no—
390Q8—272no—
391Q8—541no—
392D4—272no—
393D4—544no—
394D4—541no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—544no—
400C2×C4—541no—
401C2×C4#619511yescharacteristic
402C2×C4C2×D3×D911yescharacteristic
403C8—181no—
404C8—181no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—42no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3#6349911yescharacteristic
412C2×C3#6350511yescharacteristic
413C2×C3#6351311yescharacteristic
414C2×C3#6382911yescharacteristic
415C2×C3#6383311yescharacteristic
416C2×C3#6398111yescharacteristic
417D3—364no—
418D3—362no—
419D3—362no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—542no—
424C22—1081no—
425C22—1081no—
426C22C2×#619511yescharacteristic
427C4—21no—
428C4—121no—
429C4—121no—
430C4—542no—
431C4#8866811yescharacteristic
432C4#8866811yescharacteristic
433C3—21no—
434C3#1058873011yescharacteristic, complemented
435C3#1059118411yescharacteristic
436C2—41no—
437C2—1082no—
438C2#1172088911yescharacteristic
439C2#1172089511yescharacteristic
440C2#1172129111yescharacteristic
subgroupG/H
centerC22C2×#6195
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×#67
prime radicalC2×#88736C2
hypercenterC22×C4D3×D9
hypocenterC3×C9#452
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C9×#924C2semidirect
3. C2×#88428C2semidirect
4. C2×#88426C2semidirect
5. C3×#63389C2semidirect
6. #11720269C2semidirect
7. #11720267C2semidirect
37. #63389D3semidirect
117. #924D9semidirect
264. C3×C9#452semidirect
385. C9#4733semidirect
434. C3#10588730semidirect

Maximal subgroups

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

subgroupindex
C2×#887362
C9×#9242
C2×#884282
C2×#884262
C3×#633892
#117202692
#117202672
#105911843
#105887303