Finite Group Databasebeta

← #13167389

Subgroups — #13167389

Order 1728.

608 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 75 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#11724618C211yesmaximal, characteristic
2C2×C3×#3787C211yesmaximal, characteristic, complemented
3#11723656C211yesmaximal, characteristic, complemented
4#11722780C214yesmaximal, complemented
5#10593309—31nomaximal
6C3×#3828—21no—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8#89049C412yes—
9#89046—21no—
10C4⋊#1113C411yescharacteristic
11C4×#1113C2212yes—
12#89042—21no—
13#89041—21no—
14#89040C411yescharacteristic
15C2×C3×#622—21no—
16C3×#3787—21no—
17C3×#3787C2212yes—
18C3×#3748—21no—
19#88809—21no—
20#88808C2212yes—
21#64306—31no—
22D3×#235—61no—
23C2×#3787D311yescharacteristic, complemented
24#63818—31no—
25D3×#208—61no—
26D32⋊C8—34no—
27C22×C3×C3⋊D3C2×C411yescharacteristic
28C2×#1113C2×C412yes—
29C2×C33×C4D411yescharacteristic
30C3×#638—22no—
31C2×C3×C3⋊Dic3D411yescharacteristic
32C3×C4⋊(C3⋊D3)—21no—
33C3×C4×C3⋊D3—22no—
34C3×C4×C3⋊D3C2×C411yescharacteristic
35C3×C4×C3⋊D3D411yescharacteristic
36C3×C4×C3⋊D3D411yescharacteristic
37C3×C4×C3⋊D3C2311yescharacteristic
38C3×#634—21no—
39#6296—21no—
40#6295—22no—
41#6295—21no—
42#6294—22no—
43#6294C2×C412yes—
44C3×#622—22no—
45C3×C3⋊(C3⋊C8)—22no—
46#5110—91nomaximal
47#3828—21no—
48C2×C4×C3⋊D3C2×D311yescharacteristic
49C3×#235—61no—
50C3×#235—61no—
51C3×#235—121no—
52C2×C3×C4×D3—61no—
53C2×C3×C4×D3—61no—
54#3811—32no—
55D3×#67—62no—
56#3805—61no—
57#3804—62no—
58#3802—31no—
59D3×D12—61no—
60C4×D32—32no—
61C4×D32—62no—
62#3799—61no—
63#3798—62no—
64#3798—61no—
65#3797—31no—
66#3796—61no—
67#3795—61no—
68D3×Dic6—61no—
69C2×#622—21no—
70#3787—21no—
71#3787C2×D312yes—
72#3748—21no—
73C3×#208—61no—
74C3×#208—61no—
75C3×#208—121no—
76#3712—61no—
77#3711—62no—
78#3710—32no—
79D3×C3⋊C8—62no—
80C22×C33C22⋊C411yescharacteristic
81C2×C3×C3⋊D3—22no—
82C2×C3×C3⋊D3—22no—
83C2×C3×C3⋊D3C22⋊C411yescharacteristic
84C2×C3×C3⋊D3C22×C411yescharacteristic
85#1113—24no—
86C33×C4C22⋊C411yescharacteristic
87C33×C4C2×D411yescharacteristic
88C3×C3⋊Dic3—22no—
89C3×C3⋊Dic3C22⋊C411yescharacteristic
90C3×C3⋊Dic3C2×D411yescharacteristic
91D3×#41—181no—
92C2×#235—91no—
93C22×C3×C8—91no—
94C2×#208—91no—
95D3×M16—181no—
96(C2×D3)⋊C8—94no—
97C22×C3⋊D3C4×D311yescharacteristic
98C22×C3×D3—61no—
99C22×C3×D3—61no—
100C2×D32—32no—
101C2×D32—62no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4—41no—
105C2×C32×C4—41no—
106C2×C32×C4D1211yescharacteristic
107#638—22no—
108C2×C3⋊Dic3#6711yescharacteristic
109C4⋊(C3⋊D3)—21no—
110C4×C3⋊D3—22no—
111C4×C3⋊D3C4×D311yescharacteristic
112C4×C3⋊D3D1211yescharacteristic
113C4×C3⋊D3#6711yescharacteristic
114C4×C3⋊D3C22×D311yescharacteristic
115#634—21no—
116C3×#67—62no—
117C3×#67—62no—
118C3×#67—122no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C3×D12—61no—
122C3×D12—61no—
123C3×D12—121no—
124C3×C4×D3—61no—
125C3×C4×D3—61no—
126C3×C4×D3—61no—
127C3×C4×D3—61no—
128C3×C4×D3—61no—
129C3×C4×D3—61no—
130C3×C4×D3—61no—
131C3×C4×D3—61no—
132C3×C4×D3—62no—
133C3×C4×D3—62no—
134C3×C4×D3—122no—
135C3×Dic6—61no—
136C3×Dic6—61no—
137C3×Dic6—121no—
138#627—62no—
139#627—61no—
140#626—62no—
141#626—62no—
142#626—62no—
143#625—62no—
144#625—61no—
145#624—32no—
146#624—62no—
147D3×Dic3—62no—
148D3×Dic3—62no—
149D3×Dic3—62no—
150#622—22no—
151C3⋊(C3⋊C8)—22no—
152C3×C3⋊C8—62no—
153C3×C3⋊C8—62no—
154C3×C3⋊C8—122no—
155#408—271noSylow
156C2×C33—21no—
157C2×C33C2×C22⋊C411yescharacteristic
158C3×C3⋊D3—21no—
159C3×C3⋊D3—42no—
160C3×C3⋊D3#13112yes—
161C3×#41—181no—
162C22×C3×C4—91no—
163C2×#67—92no—
164#239—181no—
165D3×Q8—181no—
166#237—181no—
167#237—182no—
168D3×D4—181no—
169D3×D4—182no—
170#235—61no—
171#235—61no—
172#235—121no—
173#235—181no—
174#235—181no—
175#235—182no—
176#235—181no—
177C2×D12—91no—
178C2×C4×D3—61no—
179C2×C4×D3—61no—
180C2×C4×D3—92no—
181C2×C4×D3—182no—
182C2×Dic6—91no—
183C3×M16—181no—
184C2×C3×C8—92no—
185C2×C3×C8—181no—
186C2×C3×C8—181no—
187#208—61no—
188#208—61no—
189#208—121no—
190#208—181no—
191C2×C3⋊C8—92no—
192ΓL(1,𝔽25)—182no—
193D3×C8—182no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32(C2×D3)⋊C411yescharacteristic
199C2×C3⋊D3—22no—
200C2×C3⋊D3—22no—
201C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
202C2×C3⋊D3C2×C4×D311yescharacteristic
203C2×C3×D3—61no—
204C2×C3×D3—61no—
205C2×C3×D3—61no—
206C2×C3×D3—61no—
207C2×C3×D3—62no—
208C2×C3×D3—62no—
209C2×C3×D3—122no—
210C2×C3×D3—122no—
211C2×C3×D3—122no—
212D32—64no—
213D32—124no—
214C32×C4—21no—
215C32×C4—21no—
216C32×C4—21no—
217C32×C4—21no—
218C32×C4—41no—
219C32×C4—41no—
220C32×C4—41no—
221C32×C4—41no—
222C32×C4(C2×D3)⋊C411yescharacteristic
223C32×C4C2×D1211yescharacteristic
224C3⋊Dic3—22no—
225C3⋊Dic3(C2×D3)⋊C411yescharacteristic
226C3⋊Dic3C2×#6711yescharacteristic
227C3×Dic3—61no—
228C3×Dic3—61no—
229C3×Dic3—61no—
230C3×Dic3—61no—
231C3×Dic3—62no—
232C3×Dic3—62no—
233C3×Dic3—122no—
234C2×#41—271no—
235C2×M16—271no—
236C22×C8—91no—
237C22⋊C8—274no—
238C33#40811yescharacteristic, complemented, Sylow
239C23×C3—91no—
240C22×D3—61no—
241C22×D3—61no—
242C22×D3—92no—
243C22×D3—182no—
244C3×Q8—181no—
245C3×D4—181no—
246C3×D4—182no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—91no—
252C2×C3×C4—91no—
253C2×C3×C4—91no—
254C2×C3×C4—91no—
255C2×C3×C4—91no—
256C2×C3×C4—182no—
257C2×C3×C4GO+(4,𝔽2)11yescharacteristic
258#67—62no—
259#67—62no—
260#67—122no—
261#67—182no—
262#67—182no—
263#67—182no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—92no—
267C2×Dic3—182no—
268D12—61no—
269D12—61no—
270D12—121no—
271D12—181no—
272D12—182no—
273C4×D3—61no—
274C4×D3—61no—
275C4×D3—61no—
276C4×D3—61no—
277C4×D3—61no—
278C4×D3—61no—
279C4×D3—61no—
280C4×D3—61no—
281C4×D3—62no—
282C4×D3—62no—
283C4×D3—122no—
284C4×D3—182no—
285C4×D3—182no—
286C4×D3—182no—
287Dic6—61no—
288Dic6—61no—
289Dic6—121no—
290Dic6—181no—
291Dic6—182no—
292C3×C8—182no—
293C3×C8—182no—
294C3⋊C8—62no—
295C3⋊C8—62no—
296C3⋊C8—122no—
297C2×C32—21no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32—41no—
301C2×C32—41no—
302C2×C32—41no—
303C2×C32—41no—
304C2×C32—81no—
305C2×C32—81no—
306C2×C32C2×(C2×D3)⋊C411yescharacteristic
307C3⋊D3—21no—
308C3⋊D3—42no—
309C3⋊D3#90112yes—
310C3×D3—62no—
311C3×D3—62no—
312C3×D3—121no—
313C3×D3—121no—
314C3×D3—122no—
315C3×D3—122no—
316C3×D3—242no—
317#41—181no—
318#41—541no—
319#41—541no—
320#41—541no—
321C2×Q8—271no—
322C2×D4—271no—
323C2×D4—272no—
324C22×C4—91no—
325C22×C4—272no—
326M16—181no—
327M16—541no—
328C2×C8—92no—
329C2×C8—181no—
330C2×C8—181no—
331C2×C8—272no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—91no—
337C22×C3—91no—
338C22×C3—182no—
339C22×C3—182no—
340C22×C3D32⋊C411yescharacteristic
341C2×D3—61no—
342C2×D3—61no—
343C2×D3—61no—
344C2×D3—61no—
345C2×D3—62no—
346C2×D3—62no—
347C2×D3—122no—
348C2×D3—122no—
349C2×D3—122no—
350C2×D3—184no—
351C2×D3—364no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—41no—
357C3×C4—41no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—91no—
361C3×C4—91no—
362C3×C4—182no—
363C3×C4D32⋊C411yescharacteristic
364C3×C4C2×GO+(4,𝔽2)11yescharacteristic
365Dic3—61no—
366Dic3—61no—
367Dic3—61no—
368Dic3—61no—
369Dic3—62no—
370Dic3—62no—
371Dic3—122no—
372C32—21no—
373C32—21no—
374C32—41no—
375C32—41no—
376C32#511011yescharacteristic, complemented
377C23—91no—
378C23—272no—
379Q8—181no—
380Q8—541no—
381D4—182no—
382D4—181no—
383D4—541no—
384D4—542no—
385C2×C4—91no—
386C2×C4—91no—
387C2×C4—91no—
388C2×C4—91no—
389C2×C4—91no—
390C2×C4—182no—
391C2×C4—272no—
392C2×C4—542no—
393C2×C4#157⋊D311yescharacteristic
394C8—182no—
395C8—182no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—41no—
400C2×C3—41no—
401C2×C3—41no—
402C2×C3—41no—
403C2×C3—81no—
404C2×C3—81no—
405C2×C3—92no—
406C2×C3—181no—
407C2×C3—362no—
408C2×C3C2×D32⋊C411yescharacteristic
409D3—62no—
410D3—62no—
411D3—121no—
412D3—121no—
413D3—122no—
414D3—122no—
415D3—242no—
416C22—91no—
417C22—91no—
418C22—182no—
419C22—182no—
420C22—544no—
421C22#8893911yescharacteristic
422C4—91no—
423C4—91no—
424C4—182no—
425C4#8893911yescharacteristic
426C4C2×#157⋊D311yescharacteristic
427C3—21no—
428C3—21no—
429C3—41no—
430C3—41no—
431C3#1059330911yescharacteristic, complemented
432C2—21no—
433C2—92no—
434C2—181no—
435C2—362no—
436C2C2×#8893911yescharacteristic
subgroupG/H
center, FrattiniC4C2×#157⋊D3
commutatorC2×C3×C3⋊D3C22×C4
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radical#11724618C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#408
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3787C2semidirect
3. #11723656C2semidirect
4. #11722780C2semidirect
23. C2×#3787D3semidirect
238. C33#408semidirect
376. C32#5110semidirect
431. C3#10593309semidirect

Maximal subgroups

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

subgroupindex
#117246182
C2×C3×#37872
#117236562
#117227802
#105933093
#51109