Finite Group Databasebeta

← #13167435

Subgroups — #13167435

Order 1728.

602 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 71 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
2C3×#64286C211yesmaximal, characteristic, complemented
3C2×#88809C211yesmaximal, characteristic, complemented
4#11722785C214yesmaximal, complemented
5#10593324—31nomaximal
6C3×#3828—21no—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8#89049—21no—
9#89046C412yescomplemented
10C4⋊#1113C411yescharacteristic, complemented
11C4×#1113—21no—
12#89042C2212yes—
13#89041—21no—
14#89040C411yescharacteristic, complemented
15C3×C22⋊#157—21no—
16C3×C4⋊#157—21no—
17C3×C4×#157C2212yescomplemented
18C2×C3×C3⋊(C3⋊C8)—21no—
19#88809—21no—
20#88809C2212yescomplemented
21#64306—31no—
22D3×#235—61no—
23#64286D311yescharacteristic, complemented
24C2×#3712—31no—
25C2×#3711—61no—
26#63731—34no—
27C22×C3×C3⋊D3D411yescharacteristic
28C2×#1113—21no—
29C2×C3×#157—22no—
30C2×C33×C4D411yescharacteristic
31C3×#638—22no—
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C3×C4⋊(C3⋊D3)—21no—
34C3×C4×C3⋊D3—22no—
35C3×C4×C3⋊D3C2×C411yescharacteristic
36C3×C4×C3⋊D3D411yescharacteristic
37C3×C4×C3⋊D3D411yescharacteristic
38C3×C4×C3⋊D3C2311yescharacteristic
39C3×#634—21no—
40#6296C2×C412yes—
41#6295—22no—
42#6295C2×C412yes—
43#6294—21no—
44#6294—22no—
45C3×C3⋊(C3⋊C8)—22no—
46#5005—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—61no—
55D3×#67—62no—
56#3805—32no—
57#3804—62no—
58#3802—31no—
59D3×D12—61no—
60C4×D32—62no—
61C4×D32—61no—
62#3799—61no—
63#3798—32no—
64#3798—62no—
65#3797—31no—
66#3796—61no—
67#3795—61no—
68D3×Dic6—61no—
69C22⋊#157—21no—
70C4⋊#157—21no—
71C4×#157C2×D312yescomplemented
72C2×C3⋊(C3⋊C8)—21no—
73C2×C3×C3⋊C8—61no—
74C2×C3×C3⋊C8—61no—
75C2×C3×C3⋊C8—121no—
76#3712—32no—
77#3712—61no—
78#3711—62no—
79#3711—62no—
80C22×C33C22⋊C411yescharacteristic
81C2×C3×C3⋊D3—22no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3C22⋊C411yescharacteristic
84C2×C3×C3⋊D3C2×D411yescharacteristic
85#1113—42no—
86C3×#157—42no—
87C33×C4C22⋊C411yescharacteristic
88C33×C4C2×D411yescharacteristic
89C3×C3⋊Dic3—22no—
90C3×C3⋊Dic3C22⋊C411yescharacteristic
91C3×C3⋊Dic3C22×C411yescharacteristic
92D3×#41—181no—
93C2×#235—91no—
94C3×#117—91no—
95C2×#208—91no—
96C2×ΓL(1,𝔽25)—181no—
97#805—94no—
98C22×C3⋊D3#6711yescharacteristic
99C22×C3×D3—61no—
100C22×C3×D3—61no—
101C2×D32—62no—
102C2×D32—61no—
103C2×#157—22no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—41no—
107C2×C32×C4—41no—
108C2×C32×C4D1211yescharacteristic
109#638—22no—
110C2×C3⋊Dic3C4×D311yescharacteristic
111C4⋊(C3⋊D3)—21no—
112C4×C3⋊D3—22no—
113C4×C3⋊D3C4×D311yescharacteristic
114C4×C3⋊D3D1211yescharacteristic
115C4×C3⋊D3#6711yescharacteristic
116C4×C3⋊D3C22×D311yescharacteristic
117#634—21no—
118C3×#67—62no—
119C3×#67—62no—
120C3×#67—122no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C3×D12—61no—
124C3×D12—61no—
125C3×D12—121no—
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—61no—
133C3×C4×D3—61no—
134C3×C4×D3—62no—
135C3×C4×D3—62no—
136C3×C4×D3—122no—
137C3×Dic6—61no—
138C3×Dic6—61no—
139C3×Dic6—121no—
140#627—32no—
141#627—62no—
142#626—62no—
143#626—62no—
144#626—62no—
145#625—32no—
146#625—62no—
147#624—62no—
148#624—61no—
149D3×Dic3—62no—
150D3×Dic3—62no—
151D3×Dic3—62no—
152C3⋊(C3⋊C8)—22no—
153C3×C3⋊C8—62no—
154C3×C3⋊C8—62no—
155C3×C3⋊C8—122no—
156#421—271noSylow
157C2×C33—21no—
158C2×C33C2×C22⋊C411yescharacteristic
159C3×C3⋊D3—21no—
160C3×C3⋊D3—21no—
161C3×C3⋊D3—42no—
162C3×#41—181no—
163C22×C3×C4—91no—
164C2×#67—181no—
165#239—181no—
166D3×Q8—181no—
167#237—182no—
168#237—181no—
169D3×D4—181no—
170D3×D4—182no—
171#235—61no—
172#235—61no—
173#235—92no—
174#235—92no—
175#235—121no—
176#235—182no—
177#235—181no—
178C2×D12—91no—
179C2×C4×D3—61no—
180C2×C4×D3—61no—
181C2×C4×D3—181no—
182C2×C4×D3—182no—
183C2×Dic6—91no—
184C2×C3×C8—181no—
185C3×C4⋊C4—181no—
186C3×C22⋊C4—181no—
187C3×C42—92no—
188#208—92no—
189#208—181no—
190C2×C3⋊C8—61no—
191C2×C3⋊C8—61no—
192C2×C3⋊C8—121no—
193ΓL(1,𝔽25)—182no—
194ΓL(1,𝔽25)—182no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32(C2×D3)⋊C411yescharacteristic
200C2×C3⋊D3—22no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
203C2×C3⋊D3C2×#6711yescharacteristic
204C2×C3×D3—61no—
205C2×C3×D3—61no—
206C2×C3×D3—61no—
207C2×C3×D3—61no—
208C2×C3×D3—62no—
209C2×C3×D3—62no—
210C2×C3×D3—122no—
211C2×C3×D3—122no—
212C2×C3×D3—122no—
213D32—124no—
214D32—122no—
215#157—42no—
216C32×C4—21no—
217C32×C4—21no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—41no—
221C32×C4—41no—
222C32×C4—41no—
223C32×C4—41no—
224C32×C4(C2×D3)⋊C411yescharacteristic
225C32×C4C2×D1211yescharacteristic
226C3⋊Dic3—22no—
227C3⋊Dic3(C2×D3)⋊C411yescharacteristic
228C3⋊Dic3C2×C4×D311yescharacteristic
229C3×Dic3—61no—
230C3×Dic3—61no—
231C3×Dic3—61no—
232C3×Dic3—61no—
233C3×Dic3—62no—
234C3×Dic3—62no—
235C3×Dic3—122no—
236C2×#41—271no—
237C2×M16—271no—
238#117—91no—
239#104—274no—
240C33#42111yescharacteristic, complemented, Sylow
241C23×C3—91no—
242C22×D3—61no—
243C22×D3—61no—
244C22×D3—181no—
245C22×D3—182no—
246C3×Q8—181no—
247C3×D4—181no—
248C3×D4—182no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—91no—
254C2×C3×C4—91no—
255C2×C3×C4—91no—
256C2×C3×C4—91no—
257C2×C3×C4—91no—
258C2×C3×C4—182no—
259C2×C3×C4—182no—
260C2×C3×C4GO+(4,𝔽2)11yescharacteristic
261#67—62no—
262#67—62no—
263#67—122no—
264#67—182no—
265#67—182no—
266#67—182no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—181no—
270C2×Dic3—182no—
271D12—61no—
272D12—61no—
273D12—92no—
274D12—121no—
275D12—182no—
276C4×D3—61no—
277C4×D3—61no—
278C4×D3—61no—
279C4×D3—61no—
280C4×D3—61no—
281C4×D3—61no—
282C4×D3—61no—
283C4×D3—61no—
284C4×D3—62no—
285C4×D3—62no—
286C4×D3—122no—
287C4×D3—182no—
288C4×D3—182no—
289C4×D3—182no—
290Dic6—61no—
291Dic6—61no—
292Dic6—92no—
293Dic6—121no—
294Dic6—182no—
295C3×C8—182no—
296C3⋊C8—62no—
297C3⋊C8—62no—
298C3⋊C8—122no—
299C2×C32—22no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32—41no—
304C2×C32—41no—
305C2×C32—41no—
306C2×C32—42no—
307C2×C32—81no—
308C2×C32C2×(C2×D3)⋊C411yescharacteristic
309C3⋊D3—21no—
310C3⋊D3—21no—
311C3⋊D3—42no—
312C3×D3—62no—
313C3×D3—121no—
314C3×D3—121no—
315C3×D3—121no—
316C3×D3—122no—
317C3×D3—122no—
318C3×D3—242no—
319#41—181no—
320#41—272no—
321#41—272no—
322#41—541no—
323C2×Q8—271no—
324C2×D4—271no—
325C2×D4—541no—
326C22×C4—91no—
327C22×C4—541no—
328M16—272no—
329M16—541no—
330C2×C8—181no—
331C4⋊C4—181no—
332C22⋊C4—181no—
333C42—92no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—91no—
339C22×C3—91no—
340C22×C3—182no—
341C22×C3—361no—
342C22×C3D32⋊C411yescharacteristic
343C2×D3—61no—
344C2×D3—61no—
345C2×D3—61no—
346C2×D3—61no—
347C2×D3—62no—
348C2×D3—62no—
349C2×D3—122no—
350C2×D3—122no—
351C2×D3—122no—
352C2×D3—364no—
353C2×D3—362no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—41no—
361C3×C4—41no—
362C3×C4—91no—
363C3×C4—91no—
364C3×C4—182no—
365C3×C4—362no—
366C3×C4D32⋊C411yescharacteristic
367C3×C4C2×GO+(4,𝔽2)11yescharacteristic
368Dic3—61no—
369Dic3—61no—
370Dic3—61no—
371Dic3—61no—
372Dic3—62no—
373Dic3—62no—
374Dic3—122no—
375C32—21no—
376C32—21no—
377C32—41no—
378C32—41no—
379C32#500511yescharacteristic, complemented
380C23—91no—
381C23—541no—
382Q8—181no—
383Q8—272no—
384D4—182no—
385D4—181no—
386D4—272no—
387D4—542no—
388C2×C4—91no—
389C2×C4—91no—
390C2×C4—91no—
391C2×C4—91no—
392C2×C4—91no—
393C2×C4—182no—
394C2×C4—182no—
395C2×C4—542no—
396C2×C4—541no—
397C2×C4#157⋊D311yescharacteristic
398C8—182no—
399C2×C3—22no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—21no—
403C2×C3—41no—
404C2×C3—42no—
405C2×C3—41no—
406C2×C3—41no—
407C2×C3—81no—
408C2×C3—181no—
409C2×C3—181no—
410C2×C3—362no—
411C2×C3C2×D32⋊C411yescharacteristic
412D3—62no—
413D3—121no—
414D3—121no—
415D3—121no—
416D3—122no—
417D3—122no—
418D3—242no—
419C22—91no—
420C22—91no—
421C22—182no—
422C22—361no—
423C22—1082no—
424C22#8893911yescharacteristic
425C4—91no—
426C4—91no—
427C4—182no—
428C4—362no—
429C4#8893911yescharacteristic
430C4C2×#157⋊D311yescharacteristic
431C3—21no—
432C3—21no—
433C3—41no—
434C3—41no—
435C3#1059332411yescharacteristic, complemented
436C2—21no—
437C2—181no—
438C2—181no—
439C2—362no—
440C2C2×#8893911yescharacteristic
subgroupG/H
center, FrattiniC4C2×#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radical#11724618C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#421
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64286C2semidirect
3. C2×#88809C2semidirect
4. #11722785C2semidirect
9. #89046C4semidirect
10. C4⋊#1113C4semidirect
14. #89040C4semidirect
17. C3×C4×#157C22semidirect
20. #88809C22semidirect
23. #64286D3semidirect
71. C4×#157C2×D3semidirect
240. C33#421semidirect
379. C32#5005semidirect
435. C3#10593324semidirect

Maximal subgroups

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

subgroupindex
#117246182
C3×#642862
C2×#888092
#117227852
#105933243
#50059