Finite Group Databasebeta

← #13167432

Subgroups — #13167432

Order 1728.

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

normal subgroups only

This group has 614 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×#64283C211yesmaximal, characteristic, complemented
3#11723667C211yesmaximal, characteristic, complemented
4#11722795C214yesmaximal, complemented
5#10593381—31nomaximal
6C3×#3828—21no—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8#89049—21no—
9#89046C2212yes—
10C4⋊#1113C411yescharacteristic, complemented
11C4×#1113—21no—
12#89042C412yescomplemented
13#89041—21no—
14#89040C411yescharacteristic, complemented
15C3×#3792C2212yescomplemented
16C3×#3788—21no—
17C3×#3787—21no—
18C3×C4⋊(C3⋊Dic3)—21no—
19#88814—21no—
20#88813C2212yescomplemented
21#64306—31no—
22D3×#235—61no—
23#64283D311yescharacteristic, complemented
24#63849—31no—
25#63841—61no—
26#63741—34no—
27C22×C3×C3⋊D3D411yescharacteristic
28C2×#1113—21no—
29C2×C33×C4D411yescharacteristic
30C3×#638—22no—
31C2×C3×C3⋊Dic3—22no—
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C3×C4⋊(C3⋊D3)—21no—
34C3×C4×C3⋊D3—22no—
35C3×C4×C3⋊D3C2×C411yescharacteristic
36C3×C4×C3⋊D3C2×C411yescharacteristic
37C3×C4×C3⋊D3D411yescharacteristic
38C3×C4×C3⋊D3D411yescharacteristic
39C3×#634—21no—
40#6296C2×C412yes—
41#6295—22no—
42#6295C2×C412yes—
43#6294—21no—
44#6294—22no—
45C3×#622—22no—
46#5139—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—
69#3792C2×D312yescomplemented
70#3788—21no—
71#3787—21no—
72C4⋊(C3⋊Dic3)—21no—
73C3×C4⋊Dic3—61no—
74C3×C4⋊Dic3—61no—
75C3×C4⋊Dic3—121no—
76#3722—61no—
77(C2×D3)⋊Dic3—62no—
78Dic32—32no—
79Dic32—62no—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—22no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3C22⋊C411yescharacteristic
84C2×C3×C3⋊D3C22⋊C411yescharacteristic
85#1113—42no—
86C33×C4C22⋊C411yescharacteristic
87C33×C4C22⋊C411yescharacteristic
88C3×C3⋊Dic3—22no—
89C3×C3⋊Dic3—42no—
90C3×C3⋊Dic3C22×C411yescharacteristic
91C3×C3⋊Dic3C2×D411yescharacteristic
92D3×#41—181no—
93C2×#235—91no—
94C2×C3×M16—91no—
95#926—91no—
96#892—181no—
97#825—94no—
98C22×C3⋊D3D1211yescharacteristic
99C22×C3×D3—61no—
100C22×C3×D3—61no—
101C2×D32—62no—
102C2×D32—61no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4—41no—
106C2×C32×C4—41no—
107C2×C32×C4D1211yescharacteristic
108#638—22no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3C22×D311yescharacteristic
111C4⋊(C3⋊D3)—21no—
112C4×C3⋊D3—22no—
113C4×C3⋊D3C4×D311yescharacteristic
114C4×C3⋊D3C4×D311yescharacteristic
115C4×C3⋊D3#6711yescharacteristic
116C4×C3⋊D3#6711yescharacteristic
117#634—21no—
118C3×#67—62no—
119C3×#67—62no—
120C3×#67—122no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—122no—
126C3×D12—61no—
127C3×D12—61no—
128C3×D12—121no—
129C3×C4×D3—61no—
130C3×C4×D3—61no—
131C3×C4×D3—61no—
132C3×C4×D3—61no—
133C3×C4×D3—61no—
134C3×C4×D3—61no—
135C3×C4×D3—61no—
136C3×C4×D3—61no—
137C3×C4×D3—62no—
138C3×C4×D3—62no—
139C3×C4×D3—122no—
140C3×Dic6—61no—
141C3×Dic6—61no—
142C3×Dic6—121no—
143#627—32no—
144#627—62no—
145#626—62no—
146#626—62no—
147#626—62no—
148#625—32no—
149#625—62no—
150#624—62no—
151#624—61no—
152D3×Dic3—62no—
153D3×Dic3—62no—
154D3×Dic3—62no—
155#622—22no—
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—
184C3×M16—92no—
185C3×M16—181no—
186C2×C3×C8—181no—
187C3×C4⋊C4—181no—
188#217—181no—
189(C2×D3)⋊C4—182no—
190C4⋊Dic3—61no—
191C4⋊Dic3—61no—
192C4⋊Dic3—121no—
193C4×Dic3—92no—
194C4×Dic3—182no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32C2×D1211yescharacteristic
200C2×C3⋊D3—22no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
203C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
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—121no—
213C2×C3×D3—121no—
214D32—124no—
215D32—122no—
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×C4(C2×D3)⋊C411yescharacteristic
226C3⋊Dic3—22no—
227C3⋊Dic3—42no—
228C3⋊Dic3C2×C4×D311yescharacteristic
229C3⋊Dic3C2×#6711yescharacteristic
230C3×Dic3—61no—
231C3×Dic3—61no—
232C3×Dic3—61no—
233C3×Dic3—61no—
234C3×Dic3—62no—
235C3×Dic3—62no—
236C3×Dic3—122no—
237C3×Dic3—122no—
238C3×Dic3—122no—
239C3×Dic3—242no—
240C2×#41—271no—
241C2×M16—91no—
242#117—271no—
243#104—274no—
244C33#42111yescharacteristic, complemented, Sylow
245C23×C3—91no—
246C22×D3—61no—
247C22×D3—61no—
248C22×D3—181no—
249C22×D3—182no—
250C3×Q8—181no—
251C3×D4—181no—
252C3×D4—182no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—91no—
258C2×C3×C4—91no—
259C2×C3×C4—91no—
260C2×C3×C4—91no—
261C2×C3×C4—91no—
262C2×C3×C4—182no—
263C2×C3×C4—182no—
264C2×C3×C4GO+(4,𝔽2)11yescharacteristic
265#67—62no—
266#67—62no—
267#67—122no—
268#67—182no—
269#67—182no—
270#67—182no—
271C2×Dic3—61no—
272C2×Dic3—61no—
273C2×Dic3—62no—
274C2×Dic3—62no—
275C2×Dic3—122no—
276C2×Dic3—181no—
277C2×Dic3—182no—
278D12—61no—
279D12—61no—
280D12—92no—
281D12—121no—
282D12—182no—
283C4×D3—61no—
284C4×D3—61no—
285C4×D3—61no—
286C4×D3—61no—
287C4×D3—61no—
288C4×D3—61no—
289C4×D3—61no—
290C4×D3—61no—
291C4×D3—62no—
292C4×D3—62no—
293C4×D3—122no—
294C4×D3—182no—
295C4×D3—182no—
296C4×D3—182no—
297Dic6—61no—
298Dic6—61no—
299Dic6—92no—
300Dic6—121no—
301Dic6—182no—
302C3×C8—182no—
303C2×C32—21no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—21no—
308C2×C32—41no—
309C2×C32—41no—
310C2×C32—41no—
311C2×C32—41no—
312C2×C32—41no—
313C2×C32—81no—
314C2×C32C2×(C2×D3)⋊C411yescharacteristic
315C3⋊D3—21no—
316C3⋊D3—21no—
317C3⋊D3—42no—
318C3×D3—62no—
319C3×D3—121no—
320C3×D3—121no—
321C3×D3—121no—
322C3×D3—122no—
323C3×D3—122no—
324C3×D3—242no—
325#41—181no—
326#41—272no—
327#41—272no—
328#41—541no—
329C2×Q8—271no—
330C2×D4—271no—
331C2×D4—541no—
332C22×C4—91no—
333C22×C4—541no—
334M16—92no—
335M16—181no—
336C2×C8—181no—
337C4⋊C4—181no—
338C22⋊C4—541no—
339C42—272no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—91no—
345C22×C3—91no—
346C22×C3—182no—
347C22×C3—361no—
348C22×C3C2×GO+(4,𝔽2)11yescharacteristic
349C2×D3—61no—
350C2×D3—61no—
351C2×D3—61no—
352C2×D3—61no—
353C2×D3—62no—
354C2×D3—62no—
355C2×D3—122no—
356C2×D3—122no—
357C2×D3—121no—
358C2×D3—121no—
359C2×D3—364no—
360C2×D3—362no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—41no—
366C3×C4—41no—
367C3×C4—41no—
368C3×C4—41no—
369C3×C4—91no—
370C3×C4—91no—
371C3×C4—182no—
372C3×C4—362no—
373C3×C4D32⋊C411yescharacteristic
374C3×C4D32⋊C411yescharacteristic
375Dic3—61no—
376Dic3—61no—
377Dic3—61no—
378Dic3—61no—
379Dic3—62no—
380Dic3—62no—
381Dic3—122no—
382Dic3—122no—
383Dic3—122no—
384Dic3—242no—
385C32—21no—
386C32—21no—
387C32—41no—
388C32—41no—
389C32#513911yescharacteristic, complemented
390C23—91no—
391C23—541no—
392Q8—181no—
393Q8—272no—
394D4—182no—
395D4—181no—
396D4—272no—
397D4—542no—
398C2×C4—91no—
399C2×C4—91no—
400C2×C4—91no—
401C2×C4—91no—
402C2×C4—91no—
403C2×C4—182no—
404C2×C4—182no—
405C2×C4—542no—
406C2×C4—541no—
407C2×C4#157⋊D311yescharacteristic
408C8—182no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—41no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3—41no—
419C2×C3—81no—
420C2×C3—181no—
421C2×C3—181no—
422C2×C3—362no—
423C2×C3C2×D32⋊C411yescharacteristic
424D3—62no—
425D3—121no—
426D3—121no—
427D3—121no—
428D3—122no—
429D3—122no—
430D3—242no—
431C22—91no—
432C22—91no—
433C22—182no—
434C22—361no—
435C22—1082no—
436C22C2×#157⋊D311yescharacteristic
437C4—91no—
438C4—91no—
439C4—182no—
440C4—362no—
441C4#8893911yescharacteristic
442C4#8893911yescharacteristic
443C3—21no—
444C3—21no—
445C3—41no—
446C3—41no—
447C3#1059338111yescharacteristic, complemented
448C2—21no—
449C2—181no—
450C2—181no—
451C2—362no—
452C2C2×#8893911yescharacteristic
subgroupG/H
centerC2C2×#88939
commutatorC3×C3⋊Dic3C22×C4
FrattiniC22C2×#157⋊D3
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radical#11724618C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#421
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64283C2semidirect
3. #11723667C2semidirect
4. #11722795C2semidirect
10. C4⋊#1113C4semidirect
12. #89042C4semidirect
14. #89040C4semidirect
15. C3×#3792C22semidirect
20. #88813C22semidirect
23. #64283D3semidirect
69. #3792C2×D3semidirect
244. C33#421semidirect
389. C32#5139semidirect
447. C3#10593381semidirect

Maximal subgroups

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

subgroupindex
#117246182
C3×#642832
#117236672
#117227952
#105933813
#51399