Finite Group Databasebeta

← #13137841

Subgroups — #13137841

Order 1728.

528 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 89 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×#88638C211yesmaximal, characteristic, complemented
2C3×#63457C211yesmaximal, characteristic, complemented
3C2×#88421C211yesmaximal, characteristic, complemented
4C2×#88415C211yesmaximal, characteristic, complemented
5#11720527C211yesmaximal, characteristic, complemented
6C9×#809C211yesmaximal, characteristic, complemented
7(C2×D9)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10590613—31nomaximal
9#10588461—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C9×D12C2211yescharacteristic
12C2×C3×D36—21no—
13C2×C3×D36C2211yescharacteristic
14C2×C3×C4×D9C2211yescharacteristic
15C2×#6194—21no—
16C2×D3×Dic9—21no—
17#88638—22no—
18#88638—22no—
19C3×C9×C4⋊C4C2211yescharacteristic
20C2×C3⋊(C9⋊C8)C2211yescharacteristic
21C3×(C2×D9)⋊C4—21no—
22C2×C9×C3⋊C8C2211yescharacteristic
23#88421—22no—
24#88415—22no—
25C2×#3795—31no—
26C3×#895—31no—
27C2×#3716—31no—
28C2×#3713—31no—
29C2×#3699—31no—
30#63637—31no—
31C3×#809D311yescharacteristic
32(C2×D3)⋊(C3⋊C8)—31no—
33C2×#3673—31no—
34C2×#3664—31no—
35#63457D311yescharacteristic, complemented
36C9×#102—31no—
37(C2×D9)⋊C8—31no—
38#63368—31no—
39#63368—61no—
40C22×D3×C9—21no—
41C22×C3×D9—21no—
42C22×C3×D9D411yescharacteristic
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3⋊Dic9—21no—
46#6226—21no—
47#6226D412yescomplemented
48C2×C3×Dic9D411yescharacteristic
49C9×D12—21no—
50C9×D12D412yescomplemented
51C3×D36—21no—
52C3×D36—21no—
53C3×D36—42no—
54C3×C4×D9—21no—
55C3×C4×D9—21no—
56#6194—24no—
57D3×Dic9—24no—
58C3⋊(C9⋊C8)—21no—
59C9×C3⋊C8—21no—
60#5037—91no—
61#4781—91no—
62C2×#634—31no—
63C2×C3×D12—31no—
64C2×C3×D12—61no—
65C2×C3×D12C2×D311yescharacteristic
66C2×C3×C4×D3—31no—
67C2×#625—61no—
68C2×D3×Dic3—61no—
69#3795—62no—
70#3795—62no—
71C32×C4⋊C4C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)—31no—
73C3×(C2×D3)⋊C4—61no—
74C2×C3×C3⋊C8C2×D311yescharacteristic
75#3716—62no—
76#3713—62no—
77C2×D4×C9—31no—
78C2×#611—61no—
79C22×Dic9—61no—
80#3699—62no—
81#3699—62no—
82C2×D36—21no—
83C2×D36C2×D311yescharacteristic
84C2×C4×D9C2×D311yescharacteristic
85C2×Dic18—31no—
86C2×Dic18—61no—
87C2×C8×C9—31no—
88C9×C4⋊C4—21no—
89C9×C4⋊C4C2×D311yescharacteristic
90#3673—62no—
91(C2×D9)⋊C4—21no—
92C2×C9⋊C8—31no—
93C2×C9⋊C8—61no—
94#3664—62no—
95C22×C3×C9C2×D411yescharacteristic
96C2×D3×C9—22no—
97C2×D3×C9—41no—
98C2×D3×C9—41no—
99C2×C3×D9—21no—
100C2×C3×D9—21no—
101C2×C3×D9—21no—
102C2×C3×D9—41no—
103C2×C3×D9—41no—
104C2×C3×D9—41no—
105C3×C4×C9—41no—
106C3×C4×C9C2×D411yescharacteristic
107C3×C4×C9C2×D411yescharacteristic
108C3⋊Dic9—22no—
109C3×Dic9—21no—
110C2×#237—91no—
111C2×#235—91no—
112C3×#121—91no—
113C2×#214—91no—
114C2×#213—91no—
115C2×#213—91no—
116C2×#204—91no—
117#895—31no—
118C3×#102—31no—
119#830—91no—
120(C2×D3)⋊C8—91no—
121#810—91no—
122#810—91no—
123#810—181no—
124#809D911yescharacteristic, complemented
125C22×C3×D3—21no—
126C22×C3×D3—31no—
127C22×C3×D3—61no—
128C2×C32×C4—21no—
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3—61no—
131#634—32no—
132#634—61no—
133C2×C3×Dic3—31no—
134C3×D12—21no—
135C3×D12—61no—
136C3×D12—61no—
137C3×D12—122no—
138C3×D12D1212yes—
139C3×C4×D3—61no—
140C3×C4×D3—61no—
141#625—64no—
142D3×Dic3—64no—
143C23×C9—61no—
144C22×D9—21no—
145C22×D9#6711yescharacteristic
146C3⋊(C3⋊C8)—61no—
147C3×C3⋊C8—21no—
148D4×C9—32no—
149D4×C9—61no—
150C2×C4×C9—21no—
151C2×C4×C9—21no—
152C2×C4×C9—41no—
153C2×C4×C9C22×D311yescharacteristic
154#611—64no—
155C2×Dic9—64no—
156C2×Dic9—61no—
157C2×Dic9—121no—
158C2×Dic9#6711yescharacteristic
159D36—21no—
160D36—21no—
161D36—42no—
162C4×D9—21no—
163C4×D9—21no—
164Dic18—32no—
165Dic18—62no—
166Dic18—61no—
167Dic18—121no—
168C8×C9—61no—
169C9⋊C8—61no—
170C9⋊C8—121no—
171#449—271noSylow
172C2×C3×C9#12011yescharacteristic
173C2×C3×C9#13511yescharacteristic
174C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
175D3×C9—42no—
176C3×D9—41no—
177C3×D9—81no—
178C2×C3×D4—31no—
179C2×C3×D4—91no—
180C2×C3×D4—181no—
181C22×C3×C4—91no—
182C2×#67—181no—
183C2×#67—181no—
184C22×Dic3—181no—
185#237—182no—
186#237—182no—
187#235—182no—
188#235—182no—
189C2×D12—31no—
190C2×D12—61no—
191C2×D12C2×D911yescharacteristic
192C2×C4×D3—31no—
193C2×C4×D3—181no—
194C2×Dic6—91no—
195C2×Dic6—91no—
196C2×Dic6—181no—
197C2×C3×C8—31no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4C2×D911yescharacteristic
200C3×C4⋊C4D3211yescharacteristic
201C3×C22⋊C4—181no—
202#214—182no—
203#213—182no—
204#213—182no—
205(C2×D3)⋊C4—61no—
206C2×C3⋊C8—91no—
207C2×C3⋊C8—91no—
208C2×C3⋊C8—181no—
209C2×C3⋊C8C2×D911yescharacteristic
210#204—182no—
211C22×C32D3×D411yescharacteristic
212C2×C3×D3—22no—
213C2×C3×D3—41no—
214C2×C3×D3—41no—
215C2×C3×D3—61no—
216C2×C3×D3—61no—
217C2×C3×D3—61no—
218C2×C3×D3—121no—
219C2×C3×D3—121no—
220C2×C3×D3—121no—
221C32×C4—41no—
222C32×C4C2×D1211yescharacteristic
223C32×C4D3×D411yescharacteristic
224C3⋊Dic3—62no—
225C3×Dic3—61no—
226C22×C9—21no—
227C22×C9—62no—
228C22×C9—121no—
229C22×C9—121no—
230C22×C9C2×#6711yescharacteristic
231C2×D9—21no—
232C2×D9—21no—
233C2×D9—21no—
234C2×D9—41no—
235C2×D9—41no—
236C2×D9—41no—
237C4×C9—21no—
238C4×C9—21no—
239C4×C9—41no—
240C4×C9—81no—
241C4×C9D3×D411yescharacteristic
242C4×C9D3×D411yescharacteristic
243Dic9—21no—
244Dic9—62no—
245Dic9—122no—
246C2×#41—271no—
247C2×SD16—271no—
248C2×D8—271no—
249#121—91no—
250#103—271no—
251#102—31no—
252C22⋊C8—271no—
253C3×C9#44911yescharacteristic, complemented, Sylow
254C23×C3—61no—
255C23×C3—91no—
256C23×C3—181no—
257C22×D3—21no—
258C22×D3—31no—
259C22×D3—61no—
260C3×D4—32no—
261C3×D4—61no—
262C3×D4—181no—
263C3×D4—181no—
264C3×D4—362no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—41no—
269C2×C3×C4—91no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4C22×D911yescharacteristic
273C2×C3×C4C2×D3211yescharacteristic
274#67—184no—
275#67—184no—
276C2×Dic3—31no—
277C2×Dic3—184no—
278C2×Dic3—181no—
279C2×Dic3—181no—
280C2×Dic3—361no—
281D12—21no—
282D12—61no—
283D12—61no—
284D12—122no—
285D12D3612yescomplemented
286C4×D3—61no—
287C4×D3—61no—
288C4×D3—184no—
289Dic6—92no—
290Dic6—92no—
291Dic6—182no—
292Dic6—181no—
293Dic6—181no—
294Dic6—361no—
295C3×C8—61no—
296C3⋊C8—21no—
297C3⋊C8—181no—
298C3⋊C8—181no—
299C3⋊C8—361no—
300C2×C32#88211yescharacteristic
301C2×C32#91111yescharacteristic
302C2×C32#91611yescharacteristic
303C3×D3—42no—
304C3×D3—121no—
305C3×D3—241no—
306C2×C9—21no—
307C2×C9—21no—
308C2×C9—21no—
309C2×C9—122no—
310C2×C9#93211yescharacteristic
311C2×C9#93711yescharacteristic
312C2×C9#95011yescharacteristic
313D9—41no—
314D9—81no—
315#41—542no—
316#41—542no—
317C2×Q8—271no—
318C2×D4—31no—
319C2×D4—91no—
320C2×D4—181no—
321C2×D4—541no—
322C22×C4—91no—
323C22×C4—541no—
324SD16—542no—
325D8—542no—
326C2×C8—31no—
327C2×C8—271no—
328C4⋊C4D3×D911yescharacteristic
329C22⋊C4—181no—
330C22×C3—21no—
331C22×C3—62no—
332C22×C3—121no—
333C22×C3—121no—
334C22×C3—181no—
335C22×C3—181no—
336C22×C3—181no—
337C22×C3—361no—
338C22×C3—361no—
339C22×C3—361no—
340C22×C3D4×D911yescharacteristic
341C22×C3D3×#6711yescharacteristic
342C2×D3—22no—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—61no—
346C2×D3—61no—
347C2×D3—61no—
348C2×D3—121no—
349C2×D3—121no—
350C2×D3—121no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—41no—
354C3×C4—41no—
355C3×C4—81no—
356C3×C4—181no—
357C3×C4C2×D3611yescharacteristic
358C3×C4D4×D911yescharacteristic
359C3×C4D3×D1211yescharacteristic
360C3×C4#380211yescharacteristic
361Dic3—61no—
362Dic3—182no—
363Dic3—182no—
364Dic3—362no—
365C32#478111yescharacteristic
366C9—21no—
367C9#503711yescharacteristic, complemented
368C23—61no—
369C23—91no—
370C23—181no—
371Q8—272no—
372Q8—541no—
373D4—32no—
374D4—61no—
375D4—181no—
376D4—181no—
377D4—362no—
378D4—544no—
379C2×C4—21no—
380C2×C4—91no—
381C2×C4—181no—
382C2×C4—181no—
383C2×C4—544no—
384C2×C4—541no—
385C2×C4C2×D3×D911yescharacteristic
386C8—61no—
387C8—541no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—122no—
392C2×C3—361no—
393C2×C3—721no—
394C2×C3#6344411yescharacteristic
395C2×C3#6347311yescharacteristic
396C2×C3#6347811yescharacteristic
397C2×C3#6392211yescharacteristic
398C2×C3#6393211yescharacteristic
399C2×C3#6394711yescharacteristic
400D3—42no—
401D3—121no—
402D3—241no—
403C22—62no—
404C22—121no—
405C22—121no—
406C22—181no—
407C22—181no—
408C22—181no—
409C22—361no—
410C22—361no—
411C22—361no—
412C22D9×#6711yescharacteristic
413C4—41no—
414C4—181no—
415C4—542no—
416C4D3×D3611yescharacteristic
417C4#8864611yescharacteristic
418C3—21no—
419C3#1058846111yescharacteristic, complemented
420C3#1059061311yescharacteristic
421C2—122no—
422C2—361no—
423C2—721no—
424C2#1172104911yescharacteristic
425C2#1172117611yescharacteristic
426C2#1172121511yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#449
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#88638C2semidirect
2. C3×#63457C2semidirect
3. C2×#88421C2semidirect
4. C2×#88415C2semidirect
5. #11720527C2semidirect
6. C9×#809C2semidirect
7. (C2×D9)⋊(C3⋊C8)C2semidirect
35. #63457D3semidirect
47. #6226D4semidirect
50. C9×D12D4semidirect
124. #809D9semidirect
253. C3×C9#449semidirect
285. D12D36semidirect
367. C9#5037semidirect
419. C3#10588461semidirect

Maximal subgroups

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

subgroupindex
C2×#886382
C3×#634572
C2×#884212
C2×#884152
#117205272
C9×#8092
(C2×D9)⋊(C3⋊C8)2
#105906133
#105884613