Finite Group Databasebeta

← #13167298

Subgroups — #13167298

Order 1728.

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

normal subgroups only

This group has 636 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
1C2×#89046C211yesmaximal, characteristic
2C2×#88993C211yesmaximal, characteristic, complemented
3C3×#63957C211yesmaximal, characteristic, complemented
4#11722776C214yesmaximal, complemented
5#10593506—31nomaximal
6C2×C3×#638—21no—
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×#6296C411yescharacteristic, complemented
9C2×#6295C411yescharacteristic, complemented
10#89046—21no—
11#89046—21no—
12#89046C412yescomplemented
13#89046C2212yes—
14C2×#6294—21no—
15#88993—21no—
16#88993C2212yescomplemented
17C2×C3⋊#622—21no—
18C32×#217—21no—
19C3×#3724—21no—
20C3×Dic32C2212yescomplemented
21C2×#3805—31no—
22C2×#3804—61no—
23C2×#3792—31no—
24C3×#877—61no—
25#63957D311yescharacteristic, complemented
26#63741—34no—
27C23×C33D411yescharacteristic
28C22×C3×C3⋊D3—21no—
29C3×#638—22no—
30C3×#638—22no—
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D411yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3C2311yescharacteristic
38C2×C32×Dic3—22no—
39#6296—21no—
40#6296C2×C412yes—
41#6295—21no—
42#6295C2×C412yes—
43#6294—22no—
44#6294—22no—
45C3⋊#622—22no—
46#5005—91nomaximal
47C2×#638—21no—
48C22×C3⋊Dic3C2×D311yescharacteristic
49C2×C3×#67—61no—
50C2×C3×#67—61no—
51C2×C3×#67—121no—
52C22×C3×Dic3—61no—
53C22×C3×Dic3—61no—
54C2×#627—31no—
55C2×#627—61no—
56C2×#626—61no—
57C2×#625—31no—
58C2×#624—61no—
59#3805—32no—
60#3805—32no—
61#3805—61no—
62#3805—61no—
63#3804—62no—
64#3804—62no—
65#3804—62no—
66#3804—62no—
67C2×D3×Dic3—61no—
68C2×D3×Dic3—61no—
69#3792—32no—
70#3792—61no—
71C2×#622—61no—
72C32×C22⋊C4—61no—
73C3×#217—21no—
74C3×#217—21no—
75C3×#217—41no—
76C3×Dic3⋊C4—62no—
77C3×C4×Dic3—62no—
78#3724—21no—
79Dic32C2×D312yescomplemented
80C22×C33—41no—
81C22×C33C22⋊C411yescharacteristic
82C22×C33C22⋊C411yescharacteristic
83C22×C33C2×D411yescharacteristic
84C2×C3×C3⋊D3—22no—
85C2×C3×C3⋊D3—42no—
86C3×C3⋊Dic3—22no—
87C3×C3⋊Dic3C22⋊C411yescharacteristic
88C3×C3⋊Dic3C22⋊C411yescharacteristic
89C3×C3⋊Dic3C22×C411yescharacteristic
90C3×C3⋊Dic3C2×D411yescharacteristic
91C32×Dic3—42no—
92C2×#237—181no—
93C2×#235—91no—
94C3×#117—91no—
95C2×#208—91no—
96#877—61no—
97#805—94no—
98C23×C32—21no—
99C23×C32—21no—
100C23×C32—41no—
101C23×C32—41no—
102C23×C32#6711yescharacteristic
103C22×C3⋊D3—21no—
104C22×C3×D3—61no—
105C22×C3×D3—61no—
106C22×C3×D3—121no—
107C2×C32×C4—62no—
108#638—22no—
109#638—22no—
110C2×C3⋊Dic3—21no—
111C2×C3⋊Dic3C4×D311yescharacteristic
112C2×C3⋊Dic3C4×D311yescharacteristic
113C2×C3⋊Dic3D1211yescharacteristic
114C2×C3⋊Dic3D1211yescharacteristic
115C2×C3⋊Dic3#6711yescharacteristic
116C2×C3⋊Dic3C22×D311yescharacteristic
117C3×#67—62no—
118C3×#67—62no—
119C3×#67—62no—
120C3×#67—62no—
121C3×#67—122no—
122C3×#67—122no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—42no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—121no—
141#627—32no—
142#627—62no—
143#627—62no—
144#627—61no—
145#626—62no—
146#626—62no—
147#625—32no—
148#625—61no—
149#624—62no—
150#624—62no—
151D3×Dic3—62no—
152D3×Dic3—62no—
153D3×Dic3—62no—
154D3×Dic3—62no—
155#622—62no—
156#421—271noSylow
157C2×C33—21no—
158C2×C33—21no—
159C2×C33—21no—
160C2×C33C2×C22⋊C411yescharacteristic
161C3×C3⋊D3—42no—
162C2×C3×D4—181no—
163C22×C3×C4—91no—
164C2×#67—61no—
165C2×#67—61no—
166C2×#67—121no—
167C2×#67—181no—
168C22×Dic3—61no—
169C22×Dic3—61no—
170C22×Dic3—181no—
171#237—182no—
172#237—182no—
173#237—182no—
174#237—182no—
175#235—92no—
176#235—92no—
177#235—181no—
178#235—181no—
179C2×D12—91no—
180C2×C4×D3—181no—
181C2×C4×D3—181no—
182C2×Dic6—91no—
183C2×Dic6—181no—
184C3×C4⋊C4—181no—
185C3×C22⋊C4—61no—
186C3×C22⋊C4—61no—
187C3×C22⋊C4—121no—
188C3×C42—92no—
189#217—21no—
190Dic3⋊C4—62no—
191C4×Dic3—62no—
192#208—92no—
193#208—181no—
194C2×C3⋊C8—181no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32—41no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32—41no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32—42no—
209C22×C32—42no—
210C22×C32—82no—
211C22×C32—82no—
212C22×C32(C2×D3)⋊C411yescharacteristic
213C22×C32(C2×D3)⋊C411yescharacteristic
214C22×C32C2×#6711yescharacteristic
215C2×C3⋊D3—22no—
216C2×C3⋊D3—42no—
217C2×C3×D3—62no—
218C2×C3×D3—62no—
219C2×C3×D3—122no—
220C2×C3×D3—122no—
221C2×C3×D3—122no—
222C2×C3×D3—242no—
223C32×C4—122no—
224C3⋊Dic3—22no—
225C3⋊Dic3(C2×D3)⋊C411yescharacteristic
226C3⋊Dic3(C2×D3)⋊C411yescharacteristic
227C3⋊Dic3C2×C4×D311yescharacteristic
228C3⋊Dic3C2×D1211yescharacteristic
229C3×Dic3—42no—
230C3×Dic3—42no—
231C3×Dic3—61no—
232C3×Dic3—61no—
233C3×Dic3—61no—
234C3×Dic3—61no—
235C3×Dic3—61no—
236C3×Dic3—61no—
237C3×Dic3—61no—
238C3×Dic3—61no—
239C3×Dic3—62no—
240C3×Dic3—62no—
241C3×Dic3—82no—
242C3×Dic3—122no—
243C2×#41—271no—
244C2×M16—271no—
245#117—91no—
246#104—274no—
247C33#42111yescharacteristic, complemented, Sylow
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—181no—
253C23×C3GO+(4,𝔽2)11yescharacteristic
254C22×D3—61no—
255C22×D3—61no—
256C22×D3—121no—
257C3×D4—182no—
258C3×D4—182no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4—91no—
262C2×C3×C4—91no—
263C2×C3×C4—91no—
264C2×C3×C4—91no—
265C2×C3×C4—91no—
266C2×C3×C4—91no—
267C2×C3×C4—122no—
268C2×C3×C4—181no—
269#67—62no—
270#67—62no—
271#67—62no—
272#67—62no—
273#67—122no—
274#67—122no—
275#67—182no—
276#67—182no—
277C2×Dic3—22no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—61no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—61no—
286C2×Dic3—61no—
287C2×Dic3—61no—
288C2×Dic3—61no—
289C2×Dic3—61no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—121no—
293C2×Dic3—182no—
294C2×Dic3—182no—
295D12—92no—
296D12—181no—
297C4×D3—182no—
298C4×D3—182no—
299C4×D3—182no—
300C4×D3—182no—
301Dic6—92no—
302Dic6—182no—
303Dic6—181no—
304Dic6—182no—
305C3⋊C8—182no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—41no—
314C2×C32—41no—
315C2×C32—41no—
316C2×C32—41no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—42no—
320C2×C32—42no—
321C2×C32—81no—
322C2×C32—81no—
323C2×C32—81no—
324C2×C32—81no—
325C2×C32C2×(C2×D3)⋊C411yescharacteristic
326C3⋊D3—42no—
327C3×D3—122no—
328C3×D3—122no—
329C3×D3—242no—
330#41—272no—
331#41—272no—
332#41—541no—
333#41—541no—
334C2×Q8—271no—
335C2×D4—181no—
336C2×D4—271no—
337C22×C4—91no—
338C22×C4—541no—
339M16—272no—
340M16—541no—
341C2×C8—541no—
342C4⋊C4—181no—
343C22⋊C4—61no—
344C42—92no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—41no—
352C22×C3—42no—
353C22×C3—42no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3—182no—
363C22×C3—362no—
364C22×C3D32⋊C411yescharacteristic
365C22×C3D32⋊C411yescharacteristic
366C22×C3C2×GO+(4,𝔽2)11yescharacteristic
367C2×D3—62no—
368C2×D3—62no—
369C2×D3—122no—
370C2×D3—122no—
371C2×D3—122no—
372C2×D3—242no—
373C3×C4—91no—
374C3×C4—91no—
375C3×C4—91no—
376C3×C4—91no—
377C3×C4—122no—
378C3×C4—122no—
379C3×C4—182no—
380C3×C4—242no—
381Dic3—42no—
382Dic3—61no—
383Dic3—61no—
384Dic3—61no—
385Dic3—61no—
386Dic3—61no—
387Dic3—61no—
388Dic3—61no—
389Dic3—61no—
390Dic3—62no—
391Dic3—62no—
392Dic3—122no—
393C32—21no—
394C32—21no—
395C32—41no—
396C32—41no—
397C32#500511yescharacteristic, complemented
398C23—181no—
399C23D32⋊D311yescharacteristic
400Q8—272no—
401Q8—541no—
402D4—182no—
403D4—182no—
404D4—272no—
405D4—541no—
406C2×C4—62no—
407C2×C4—91no—
408C2×C4—91no—
409C2×C4—91no—
410C2×C4—91no—
411C2×C4—91no—
412C2×C4—91no—
413C2×C4—181no—
414C2×C4—542no—
415C2×C4—542no—
416C8—542no—
417C2×C3—22no—
418C2×C3—22no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—41no—
425C2×C3—41no—
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—42no—
429C2×C3—42no—
430C2×C3—41no—
431C2×C3—41no—
432C2×C3—81no—
433C2×C3—81no—
434C2×C3—81no—
435C2×C3—81no—
436C2×C3—362no—
437C2×C3C2×D32⋊C411yescharacteristic
438D3—122no—
439D3—122no—
440D3—242no—
441C22—41no—
442C22—182no—
443C22—362no—
444C22#8893211yescharacteristic
445C22#8893211yescharacteristic
446C22C2×D32⋊D311yescharacteristic
447C4—91no—
448C4—91no—
449C4—91no—
450C4—91no—
451C4—122no—
452C4—182no—
453C3—21no—
454C3—21no—
455C3—41no—
456C3—41no—
457C3#1059350611yescharacteristic, complemented
458C2—21no—
459C2—21no—
460C2—21no—
461C2—362no—
462C2C2×#8893211yescharacteristic
subgroupG/H
centerC2C2×#88932
commutatorC3×C3⋊Dic3C22×C4
FrattiniC22C2×D32⋊D3
FittingC23×C33D4
socleC2×C33C2×C22⋊C4
prime radicalC2×#89046C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23D32⋊D3
hypocenterC33#421
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88993C2semidirect
3. C3×#63957C2semidirect
4. #11722776C2semidirect
8. C2×#6296C4semidirect
9. C2×#6295C4semidirect
12. #89046C4semidirect
16. #88993C22semidirect
20. C3×Dic32C22semidirect
25. #63957D3semidirect
79. Dic32C2×D3semidirect
247. C33#421semidirect
397. C32#5005semidirect
457. C3#10593506semidirect

Maximal subgroups

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

subgroupindex
C2×#890462
C2×#889932
C3×#639572
#117227762
#105935063
#50059