Finite Group Databasebeta

← #13167300

Subgroups — #13167300

Order 1728.

636 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 79 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×C3×#3805C211yesmaximal, characteristic
2C2×#88993C211yesmaximal, characteristic, complemented
3#11723715C211yesmaximal, characteristic, complemented
4#11722777C214yesmaximal, complemented
5#10593506—31nomaximal
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C2×C32×#67—21no—
8C2×C3×#627C411yescharacteristic, complemented
9C2×C3×#625C411yescharacteristic, complemented
10C3×#3805—21no—
11C3×#3805—21no—
12C3×#3805C412yescomplemented
13C3×#3805C2212yes—
14C2×C3×D3×Dic3—21no—
15#88993—21no—
16#88993C2212yescomplemented
17C2×C3⋊#622—21no—
18C3×#3757—21no—
19#88815—21no—
20#88813C2212yescomplemented
21C2×C3×#237—61no—
22C2×#3805D311yescharacteristic
23C2×#3792—31no—
24#63957—31no—
25#63955—61no—
26#63741—34no—
27C23×C33D411yescharacteristic
28C22×C32×D3—21no—
29C2×C3×C3⋊Dic3—22no—
30C2×C3×C3⋊Dic3C2×C411yescharacteristic
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C2×C3×C3⋊Dic3D411yescharacteristic
33C2×C3×C3⋊Dic3D411yescharacteristic
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3C2311yescharacteristic
36C32×#67—22no—
37C32×#67—22no—
38C2×C32×Dic3—21no—
39C3×#627—21no—
40C3×#627C2×C412yes—
41C3×#625—21no—
42C3×#625C2×C412yes—
43C3×D3×Dic3—22no—
44C3×D3×Dic3—22no—
45C3⋊#622—22no—
46#5236—91nomaximal
47C2×C32×D4—61no—
48C22×C3⋊Dic3C2×D311yescharacteristic
49C2×C3×#67—21no—
50C2×C3×#67—21no—
51C2×C3×#67—41no—
52C2×C3×#67—61no—
53C22×C3×Dic3—61no—
54C22×C3×Dic3—61no—
55C22×C3×Dic3—61no—
56C3×#237—62no—
57C3×#237—62no—
58C3×#237—62no—
59C3×#237—62no—
60C2×C3×C4×D3—61no—
61C2×C3×Dic6—61no—
62C2×#627Dic311yescharacteristic, complemented
63C2×#625Dic311yescharacteristic, complemented
64#3805—21no—
65#3805—21no—
66#3805Dic312yescomplemented
67#3805C2×D312yes—
68C2×D3×Dic3—21no—
69#3792—32no—
70#3792—61no—
71C2×#622—61no—
72#3757—21no—
73C3×#217—61no—
74C3×#217—61no—
75C3×#217—121no—
76#3724—61no—
77Dic3⋊Dic3—62no—
78Dic32—32no—
79Dic32—62no—
80C22×C33—41no—
81C22×C33C22⋊C411yescharacteristic
82C22×C33C22⋊C411yescharacteristic
83C22×C33C2×D411yescharacteristic
84C2×C32×D3—22no—
85C2×C32×D3—42no—
86C3×C3⋊Dic3—42no—
87C3×C3⋊Dic3C22⋊C411yescharacteristic
88C3×C3⋊Dic3C22⋊C411yescharacteristic
89C3×C3⋊Dic3C22×C411yescharacteristic
90C3×C3⋊Dic3C2×D411yescharacteristic
91C32×Dic3—22no—
92C2×C3×#41—91no—
93C2×#237—61no—
94#926—91no—
95C2×#208—91no—
96#877—181no—
97#837—94no—
98C23×C32—21no—
99C23×C32—21no—
100C23×C32—41no—
101C23×C32—41no—
102C23×C32—61no—
103C23×C32#6711yescharacteristic
104C22×C3×D3—21no—
105C22×C3×D3—21no—
106C22×C3×D3—41no—
107C32×D4—62no—
108C32×D4—62no—
109C2×C32×C4—61no—
110C2×C3⋊Dic3—22no—
111C2×C3⋊Dic3C2×Dic311yescharacteristic
112C2×C3⋊Dic3C2×Dic311yescharacteristic
113C2×C3⋊Dic3#6711yescharacteristic
114C2×C3⋊Dic3#6711yescharacteristic
115C2×C3⋊Dic3#6711yescharacteristic
116C2×C3⋊Dic3C22×D311yescharacteristic
117C3×#67—22no—
118C3×#67—22no—
119C3×#67—22no—
120C3×#67—22no—
121C3×#67—42no—
122C3×#67—42no—
123C3×#67—62no—
124C3×#67—62no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—21no—
127C2×C3×Dic3—41no—
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—62no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—122no—
145C3×C4×D3—62no—
146C3×C4×D3—62no—
147C3×Dic6—62no—
148C3×Dic6—62no—
149#627—21no—
150#627C2×Dic312yes—
151#625—21no—
152#625C2×Dic312yes—
153D3×Dic3—22no—
154D3×Dic3—22no—
155#622—62no—
156#421—271noSylow
157C2×C33—21no—
158C2×C33—21no—
159C2×C33—21no—
160C2×C33C2×C22⋊C411yescharacteristic
161C32×D3—42no—
162C3×#41—92no—
163C3×#41—92no—
164C3×#41—181no—
165C3×#41—181no—
166C2×C3×Q8—91no—
167C2×C3×D4—61no—
168C2×C3×D4—61no—
169C2×C3×D4—91no—
170C2×C3×D4—121no—
171C22×C3×C4—91no—
172C22×C3×C4—181no—
173C2×#67—21no—
174C2×#67—61no—
175C22×Dic3—61no—
176C22×Dic3—61no—
177C22×Dic3—61no—
178#237—62no—
179#237—62no—
180#237—62no—
181#237—62no—
182C2×C4×D3—61no—
183C2×Dic6—61no—
184C3×C22⋊C4—181no—
185#217—61no—
186#217—61no—
187#217—121no—
188C4⋊Dic3—181no—
189Dic3⋊C4—182no—
190C4×Dic3—92no—
191C4×Dic3—182no—
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—62no—
211C22×C32—82no—
212C22×C32—82no—
213C22×C32—122no—
214C22×C32#21711yescharacteristic
215C22×C32#21711yescharacteristic
216C22×C32C2×#6711yescharacteristic
217C2×C3×D3—22no—
218C2×C3×D3—22no—
219C2×C3×D3—42no—
220C2×C3×D3—42no—
221C2×C3×D3—42no—
222C2×C3×D3—82no—
223C32×C4—62no—
224C3⋊Dic3—42no—
225C3⋊Dic3#21711yescharacteristic
226C3⋊Dic3#21711yescharacteristic
227C3⋊Dic3C22×Dic311yescharacteristic
228C3⋊Dic3C2×#6711yescharacteristic
229C3×Dic3—22no—
230C3×Dic3—22no—
231C3×Dic3—42no—
232C3×Dic3—61no—
233C3×Dic3—61no—
234C3×Dic3—61no—
235C3×Dic3—61no—
236C3×Dic3—61no—
237C3×Dic3—61no—
238C3×Dic3—61no—
239C3×Dic3—61no—
240C3×Dic3—122no—
241C3×Dic3—122no—
242C3×Dic3—242no—
243C2×#41—91no—
244C2×M16—271no—
245#117—271no—
246#104—274no—
247C33#42111yescharacteristic, complemented, Sylow
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—61no—
253C23×C3—61no—
254C23×C3—121no—
255C23×C3GO+(4,𝔽2)11yescharacteristic
256C22×D3—21no—
257C3×Q8—92no—
258C3×Q8—181no—
259C3×D4—62no—
260C3×D4—62no—
261C3×D4—62no—
262C3×D4—62no—
263C3×D4—92no—
264C3×D4—122no—
265C3×D4—122no—
266C3×D4—181no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—91no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272C2×C3×C4—91no—
273C2×C3×C4—91no—
274C2×C3×C4—91no—
275C2×C3×C4—121no—
276C2×C3×C4—182no—
277C2×C3×C4—182no—
278C2×C3×C4—182no—
279#67—22no—
280#67—22no—
281#67—62no—
282#67—62no—
283C2×Dic3—21no—
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—61no—
293C2×Dic3—61no—
294C2×Dic3—61no—
295C2×Dic3—61no—
296C2×Dic3—62no—
297C2×Dic3—62no—
298C2×Dic3—62no—
299C2×Dic3—62no—
300C2×Dic3—122no—
301C4×D3—62no—
302C4×D3—62no—
303Dic6—62no—
304Dic6—62no—
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×C32—122no—
326C2×C32C2×#21711yescharacteristic
327C3×D3—42no—
328C3×D3—42no—
329C3×D3—82no—
330#41—92no—
331#41—92no—
332#41—181no—
333#41—181no—
334C2×Q8—91no—
335C2×D4—61no—
336C2×D4—91no—
337C22×C4—91no—
338C22×C4—181no—
339M16—272no—
340M16—541no—
341C2×C8—541no—
342C4⋊C4—541no—
343C22⋊C4—181no—
344C42—272no—
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—62no—
361C22×C3—62no—
362C22×C3—82no—
363C22×C3—82no—
364C22×C3—122no—
365C22×C3—122no—
366C22×C3—122no—
367C22×C3—242no—
368C22×C3D32⋊C411yescharacteristic
369C22×C3D32⋊C411yescharacteristic
370C22×C3C2×GO+(4,𝔽2)11yescharacteristic
371C2×D3—22no—
372C2×D3—42no—
373C3×C4—62no—
374C3×C4—62no—
375C3×C4—91no—
376C3×C4—91no—
377C3×C4—91no—
378C3×C4—91no—
379C3×C4—122no—
380C3×C4—362no—
381Dic3—22no—
382Dic3—61no—
383Dic3—61no—
384Dic3—61no—
385Dic3—61no—
386Dic3—61no—
387Dic3—61no—
388Dic3—61no—
389Dic3—61no—
390Dic3—122no—
391Dic3—122no—
392Dic3—242no—
393C32—21no—
394C32—21no—
395C32—41no—
396C32—41no—
397C32#523611yescharacteristic, complemented
398C23—61no—
399C23D32⋊D311yescharacteristic
400Q8—92no—
401Q8—181no—
402D4—62no—
403D4—62no—
404D4—92no—
405D4—181no—
406C2×C4—61no—
407C2×C4—91no—
408C2×C4—91no—
409C2×C4—91no—
410C2×C4—91no—
411C2×C4—91no—
412C2×C4—91no—
413C2×C4—182no—
414C2×C4—182no—
415C2×C4—182no—
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—122no—
437C2×C3—122no—
438C2×C3—242no—
439C2×C3C2×D32⋊C411yescharacteristic
440D3—42no—
441C22—41no—
442C22—62no—
443C22—122no—
444C22D32⋊Dic311yescharacteristic
445C22D32⋊Dic311yescharacteristic
446C22C2×D32⋊D311yescharacteristic
447C4—62no—
448C4—91no—
449C4—91no—
450C4—91no—
451C4—91no—
452C4—362no—
453C3—21no—
454C3—21no—
455C3—41no—
456C3—41no—
457C3#1059350611yescharacteristic, complemented
458C2—21no—
459C2—21no—
460C2—21no—
461C2—122no—
462C2C2×D32⋊Dic311yescharacteristic
subgroupG/H
centerC2C2×D32⋊Dic3
commutatorC3×C3⋊Dic3C22×C4
FrattiniC22C2×D32⋊D3
FittingC23×C33D4
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3805C2
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. #11723715C2semidirect
4. #11722777C2semidirect
8. C2×C3×#627C4semidirect
9. C2×C3×#625C4semidirect
12. C3×#3805C4semidirect
16. #88993C22semidirect
20. #88813C22semidirect
62. C2×#627Dic3semidirect
63. C2×#625Dic3semidirect
66. #3805Dic3semidirect
247. C33#421semidirect
397. C32#5236semidirect
457. C3#10593506semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38052
C2×#889932
#117237152
#117227772
#105935063
#52369