Finite Group Databasebeta

← #13167418

Subgroups — #13167418

Order 1728.

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

normal subgroups only

This group has 618 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
2C3×#64286C211yesmaximal, characteristic, complemented
3C2×#88808C211yesmaximal, characteristic, complemented
4#11722784C212yesmaximal, complemented
5#11722783C212yesmaximal, complemented
6#10593317—31nomaximal
7C3×#3828—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9#89049—21no—
10#89046C412yescomplemented
11C4⋊#1113C2211yescharacteristic
12C4×#1113—21no—
13#89042C412yescomplemented
14#89041—21no—
15#89040C2211yescharacteristic
16C3×C22⋊#157—21no—
17C3×C4⋊#157C2212yes—
18C3×C4×#157—21no—
19C2×C3×C3⋊(C3⋊C8)—21no—
20#88808—21no—
21#88808C2212yes—
22#64306—31no—
23D3×#235—61no—
24#64286D311yescharacteristic, complemented
25C2×#3710—31no—
26C2×D3×C3⋊C8—61no—
27#63730—32no—
28#63729—32no—
29C22×C3×C3⋊D3D411yescharacteristic
30C2×#1113—21no—
31C2×C3×#157—22no—
32C2×C33×C4D411yescharacteristic
33C3×#638—22no—
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C3×C4⋊(C3⋊D3)—21no—
36C3×C4×C3⋊D3—22no—
37C3×C4×C3⋊D3C2×C411yescharacteristic
38C3×C4×C3⋊D3D411yescharacteristic
39C3×C4×C3⋊D3D411yescharacteristic
40C3×C4×C3⋊D3C2311yescharacteristic
41C3×#634—21no—
42#6296C2×C412yes—
43#6295—22no—
44#6295C2×C412yes—
45#6294—21no—
46#6294—22no—
47C3×C3⋊(C3⋊C8)—22no—
48#5002—91nomaximal
49#3828—21no—
50C2×C4×C3⋊D3C2×D311yescharacteristic
51C3×#235—61no—
52C3×#235—61no—
53C3×#235—121no—
54C2×C3×C4×D3—61no—
55C2×C3×C4×D3—61no—
56#3811—61no—
57D3×#67—62no—
58#3805—32no—
59#3804—62no—
60#3802—31no—
61D3×D12—61no—
62C4×D32—62no—
63C4×D32—61no—
64#3799—61no—
65#3798—32no—
66#3798—62no—
67#3797—31no—
68#3796—61no—
69#3795—61no—
70D3×Dic6—61no—
71C22⋊#157—21no—
72C4⋊#157C2×D312yes—
73C4×#157—21no—
74C2×C3⋊(C3⋊C8)—21no—
75C2×C3×C3⋊C8—61no—
76C2×C3×C3⋊C8—61no—
77C2×C3×C3⋊C8—121no—
78#3710—32no—
79#3710—61no—
80D3×C3⋊C8—62no—
81D3×C3⋊C8—62no—
82C22×C33C22⋊C411yescharacteristic
83C2×C3×C3⋊D3—22no—
84C2×C3×C3⋊D3—22no—
85C2×C3×C3⋊D3C22⋊C411yescharacteristic
86C2×C3×C3⋊D3C2×D411yescharacteristic
87#1113—42no—
88C3×#157—42no—
89C33×C4C22⋊C411yescharacteristic
90C33×C4C2×D411yescharacteristic
91C3×C3⋊Dic3—22no—
92C3×C3⋊Dic3C22⋊C411yescharacteristic
93C3×C3⋊Dic3C22×C411yescharacteristic
94D3×#41—181no—
95C2×#235—91no—
96C3×#117—91no—
97C22×C3⋊C8—91no—
98C2×D3×C8—181no—
99#810—92no—
100#809—92no—
101C22×C3⋊D3#6711yescharacteristic
102C22×C3×D3—61no—
103C22×C3×D3—61no—
104C2×D32—62no—
105C2×D32—61no—
106C2×#157—22no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4—41no—
111C2×C32×C4D1211yescharacteristic
112#638—22no—
113C2×C3⋊Dic3C4×D311yescharacteristic
114C4⋊(C3⋊D3)—21no—
115C4×C3⋊D3—22no—
116C4×C3⋊D3C4×D311yescharacteristic
117C4×C3⋊D3D1211yescharacteristic
118C4×C3⋊D3#6711yescharacteristic
119C4×C3⋊D3C22×D311yescharacteristic
120#634—21no—
121C3×#67—62no—
122C3×#67—62no—
123C3×#67—122no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
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—
155C3⋊(C3⋊C8)—22no—
156C3×C3⋊C8—62no—
157C3×C3⋊C8—62no—
158C3×C3⋊C8—122no—
159#416—271noSylow
160C2×C33—21no—
161C2×C33C2×C22⋊C411yescharacteristic
162C3×C3⋊D3—21no—
163C3×C3⋊D3—42no—
164C3×C3⋊D3#13512yes—
165C3×#41—181no—
166C22×C3×C4—91no—
167C2×#67—181no—
168#239—181no—
169D3×Q8—181no—
170#237—182no—
171#237—181no—
172D3×D4—181no—
173D3×D4—182no—
174#235—61no—
175#235—61no—
176#235—92no—
177#235—92no—
178#235—121no—
179#235—182no—
180#235—181no—
181C2×D12—91no—
182C2×C4×D3—61no—
183C2×C4×D3—61no—
184C2×C4×D3—181no—
185C2×C4×D3—182no—
186C2×Dic6—91no—
187C2×C3×C8—181no—
188C3×C4⋊C4—92no—
189C3×C22⋊C4—181no—
190C3×C42—181no—
191C2×C3⋊C8—61no—
192C2×C3⋊C8—61no—
193C2×C3⋊C8—92no—
194C2×C3⋊C8—121no—
195C2×C3⋊C8—181no—
196D3×C8—182no—
197D3×C8—182no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32(C2×D3)⋊C411yescharacteristic
203C2×C3⋊D3—22no—
204C2×C3⋊D3—22no—
205C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
206C2×C3⋊D3C2×#6711yescharacteristic
207C2×C3×D3—61no—
208C2×C3×D3—61no—
209C2×C3×D3—61no—
210C2×C3×D3—61no—
211C2×C3×D3—64no—
212C2×C3×D3—62no—
213C2×C3×D3—62no—
214C2×C3×D3—122no—
215C2×C3×D3—122no—
216D32—124no—
217D32—122no—
218#157—42no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4—41no—
226C32×C4—41no—
227C32×C4(C2×D3)⋊C411yescharacteristic
228C32×C4C2×D1211yescharacteristic
229C3⋊Dic3—22no—
230C3⋊Dic3(C2×D3)⋊C411yescharacteristic
231C3⋊Dic3C2×C4×D311yescharacteristic
232C3×Dic3—61no—
233C3×Dic3—61no—
234C3×Dic3—61no—
235C3×Dic3—61no—
236C3×Dic3—62no—
237C3×Dic3—62no—
238C3×Dic3—122no—
239C2×#41—271no—
240C22×C8—271no—
241#117—91no—
242#103—272no—
243#102—272no—
244C33#41611yescharacteristic, 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—181no—
274C2×Dic3—182no—
275D12—61no—
276D12—61no—
277D12—92no—
278D12—121no—
279D12—182no—
280C4×D3—61no—
281C4×D3—61no—
282C4×D3—61no—
283C4×D3—61no—
284C4×D3—61no—
285C4×D3—61no—
286C4×D3—61no—
287C4×D3—61no—
288C4×D3—62no—
289C4×D3—62no—
290C4×D3—122no—
291C4×D3—182no—
292C4×D3—182no—
293C4×D3—182no—
294Dic6—61no—
295Dic6—61no—
296Dic6—92no—
297Dic6—121no—
298Dic6—182no—
299C3×C8—182no—
300C3⋊C8—62no—
301C3⋊C8—62no—
302C3⋊C8—122no—
303C2×C32—22no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—41no—
308C2×C32—41no—
309C2×C32—41no—
310C2×C32—42no—
311C2×C32—81no—
312C2×C32C2×(C2×D3)⋊C411yescharacteristic
313C3⋊D3—21no—
314C3⋊D3—42no—
315C3⋊D3#95012yes—
316C3×D3—62no—
317C3×D3—62no—
318C3×D3—62no—
319C3×D3—121no—
320C3×D3—122no—
321C3×D3—122no—
322C3×D3—242no—
323#41—181no—
324#41—272no—
325#41—272no—
326#41—541no—
327C2×Q8—271no—
328C2×D4—271no—
329C2×D4—541no—
330C22×C4—91no—
331C22×C4—541no—
332C2×C8—181no—
333C2×C8—272no—
334C2×C8—541no—
335C4⋊C4—92no—
336C22⋊C4—181no—
337C42—181no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—91no—
343C22×C3—91no—
344C22×C3—182no—
345C22×C3—182no—
346C22×C3D32⋊C411yescharacteristic
347C2×D3—61no—
348C2×D3—61no—
349C2×D3—61no—
350C2×D3—61no—
351C2×D3—64no—
352C2×D3—62no—
353C2×D3—62no—
354C2×D3—122no—
355C2×D3—122no—
356C2×D3—364no—
357C2×D3—362no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—41no—
363C3×C4—41no—
364C3×C4—41no—
365C3×C4—41no—
366C3×C4—91no—
367C3×C4—91no—
368C3×C4—182no—
369C3×C4—362no—
370C3×C4D32⋊C411yescharacteristic
371C3×C4C2×GO+(4,𝔽2)11yescharacteristic
372Dic3—61no—
373Dic3—61no—
374Dic3—61no—
375Dic3—61no—
376Dic3—62no—
377Dic3—62no—
378Dic3—122no—
379C32—21no—
380C32—21no—
381C32—41no—
382C32—41no—
383C32#500211yescharacteristic, complemented
384C23—91no—
385C23—541no—
386Q8—181no—
387Q8—272no—
388D4—182no—
389D4—181no—
390D4—272no—
391D4—542no—
392C2×C4—91no—
393C2×C4—91no—
394C2×C4—91no—
395C2×C4—91no—
396C2×C4—91no—
397C2×C4—182no—
398C2×C4—182no—
399C2×C4—542no—
400C2×C4—541no—
401C2×C4#157⋊D311yescharacteristic
402C8—182no—
403C2×C3—22no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—41no—
408C2×C3—42no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3—81no—
412C2×C3—92no—
413C2×C3—181no—
414C2×C3—362no—
415C2×C3C2×D32⋊C411yescharacteristic
416D3—62no—
417D3—62no—
418D3—62no—
419D3—121no—
420D3—122no—
421D3—122no—
422D3—242no—
423C22—91no—
424C22—91no—
425C22—182no—
426C22—182no—
427C22—1082no—
428C22#8893911yescharacteristic
429C4—91no—
430C4—91no—
431C4—182no—
432C4—362no—
433C4#8893911yescharacteristic
434C4C2×#157⋊D311yescharacteristic
435C3—21no—
436C3—21no—
437C3—41no—
438C3—41no—
439C3#1059331711yescharacteristic, complemented
440C2—21no—
441C2—92no—
442C2—181no—
443C2—362no—
444C2C2×#8893911yescharacteristic
subgroupG/H
centerC4#88939
commutatorC3×C3⋊Dic3C22×C4
FrattiniC4C2×#157⋊D3
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radical#11724618C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#416
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×#88808C2semidirect
4. #11722784C2semidirect
5. #11722783C2semidirect
10. #89046C4semidirect
13. #89042C4semidirect
24. #64286D3semidirect
244. C33#416semidirect
383. C32#5002semidirect
439. C3#10593317semidirect

Maximal subgroups

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

subgroupindex
#117246182
C3×#642862
C2×#888082
#117227842
#117227832
#105933173
#50029