Finite Group Databasebeta

← #13167283

Subgroups — #13167283

Order 1728.

688 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 87 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
3C2×#88815C211yesmaximal, characteristic, complemented
4#11722775C212yesmaximal, complemented
5#11722773C212yesmaximal, complemented
6#10593496—31nomaximal
7C22×C3×C3⋊Dic3—21no—
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×C32×#67—21no—
10C2×C3×#627C2211yescharacteristic
11C2×C3×#625C2211yescharacteristic
12C3×#3805—22no—
13C3×#3805C414yescomplemented
14C2×C3×D3×Dic3—21no—
15#88993—22no—
16C2×C3⋊#622C2212yes—
17#88815—21no—
18#88815C2212yes—
19C2×C3×#237—61no—
20C2×#3805D311yescharacteristic
21C2×#3792—31no—
22C2×#3724—31no—
23C2×Dic3⋊Dic3—61no—
24#63740—32no—
25#63739—32no—
26C23×C33D411yescharacteristic
27C22×C32×D3—21no—
28C2×C3×C3⋊Dic3—22no—
29C2×C3×C3⋊Dic3—42no—
30C2×C3×C3⋊Dic3C2×C412yes—
31C2×C3×C3⋊Dic3D411yescharacteristic
32C2×C3×C3⋊Dic3D412yes—
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C32×#67—24no—
35C2×C32×Dic3—21no—
36C3×#627—21no—
37C3×#627C2×C412yescomplemented
38C3×#625—21no—
39C3×#625C2×C412yescomplemented
40C3×D3×Dic3—24no—
41C3⋊#622—22no—
42#5232—91nomaximal
43C2×C32×D4—61no—
44C22×C3⋊Dic3—21no—
45C22×C3⋊Dic3C2×D311yescharacteristic
46C2×C3×#67—21no—
47C2×C3×#67—21no—
48C2×C3×#67—41no—
49C2×C3×#67—61no—
50C22×C3×Dic3—61no—
51C22×C3×Dic3—61no—
52C22×C3×Dic3—61no—
53C22×C3×Dic3—61no—
54C22×C3×Dic3—61no—
55C22×C3×Dic3—121no—
56C3×#237—64no—
57C3×#237—64no—
58C2×C3×C4×D3—61no—
59C2×C3×Dic6—61no—
60C2×#627C2×D311yescharacteristic
61C2×#625C2×D311yescharacteristic
62#3805—22no—
63#3805Dic314yescomplemented
64C2×D3×Dic3—21no—
65#3792—62no—
66C2×#622—32no—
67#3724—32no—
68#3724—61no—
69Dic3⋊Dic3—62no—
70Dic3⋊Dic3—62no—
71C22×C33—21no—
72C22×C33—21no—
73C22×C33C22⋊C412yes—
74C22×C33C2×D411yescharacteristic
75C2×C32×D3—22no—
76C2×C32×D3—41no—
77C2×C32×D3—41no—
78C3×C3⋊Dic3—42no—
79C3×C3⋊Dic3C22⋊C412yes—
80C3×C3⋊Dic3C22×C411yescharacteristic
81C3×C3⋊Dic3C2×D411yescharacteristic
82C32×Dic3—22no—
83C2×C3×#41—91no—
84C2×#237—61no—
85C2×C4⋊Dic3—91no—
86C2×Dic3⋊C4—181no—
87C2×#208—91no—
88#835—92no—
89#832—92no—
90C23×C32—21no—
91C23×C32—21no—
92C23×C32—41no—
93C23×C32—41no—
94C23×C32—61no—
95C23×C32#6711yescharacteristic
96C22×C3×D3—21no—
97C22×C3×D3—21no—
98C22×C3×D3—41no—
99C32×D4—64no—
100C2×C32×C4—61no—
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3—42no—
103C2×C3⋊Dic3C2×Dic312yes—
104C2×C3⋊Dic3#6711yescharacteristic
105C2×C3⋊Dic3#6712yes—
106C2×C3⋊Dic3C22×D311yescharacteristic
107C3×#67—24no—
108C3×#67—24no—
109C3×#67—44no—
110C3×#67—62no—
111C3×#67—62no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—41no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—64no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—122no—
127C2×C3×Dic3—122no—
128C2×C3×Dic3—122no—
129C2×C3×Dic3—124no—
130C3×C4×D3—64no—
131C3×Dic6—62no—
132C3×Dic6—62no—
133#627—21no—
134#627C2×Dic312yescomplemented
135#625—21no—
136#625C2×Dic312yescomplemented
137D3×Dic3—24no—
138#622—62no—
139#417—271noSylow
140C2×C33—22no—
141C2×C33C2×C22⋊C411yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C2×C33#13711yescharacteristic
144C32×D3—42no—
145C3×#41—94no—
146C3×#41—182no—
147C2×C3×Q8—91no—
148C2×C3×D4—61no—
149C2×C3×D4—61no—
150C2×C3×D4—91no—
151C2×C3×D4—121no—
152C22×C3×C4—91no—
153C22×C3×C4—181no—
154C22×C3×C4—181no—
155C2×#67—21no—
156C2×#67—61no—
157C22×Dic3—61no—
158C22×Dic3—61no—
159C22×Dic3—61no—
160C22×Dic3—61no—
161C22×Dic3—61no—
162C22×Dic3—121no—
163#237—64no—
164#237—64no—
165C2×C4×D3—61no—
166C2×Dic6—61no—
167C4⋊Dic3—92no—
168C4⋊Dic3—181no—
169Dic3⋊C4—182no—
170Dic3⋊C4—182no—
171#208—182no—
172C2×C3⋊C8—92no—
173C22×C32—22no—
174C22×C32—22no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—21no—
179C22×C32—22no—
180C22×C32—22no—
181C22×C32—42no—
182C22×C32—41no—
183C22×C32—41no—
184C22×C32—42no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32—42no—
188C22×C32—42no—
189C22×C32—62no—
190C22×C32—81no—
191C22×C32—81no—
192C22×C32—121no—
193C22×C32—121no—
194C22×C32#21712yes—
195C22×C32C2×#6711yescharacteristic
196C2×C3×D3—22no—
197C2×C3×D3—22no—
198C2×C3×D3—42no—
199C2×C3×D3—41no—
200C2×C3×D3—41no—
201C2×C3×D3—41no—
202C2×C3×D3—41no—
203C2×C3×D3—81no—
204C2×C3×D3—81no—
205C32×C4—62no—
206C3⋊Dic3—42no—
207C3⋊Dic3#21712yes—
208C3⋊Dic3C22×Dic311yescharacteristic
209C3⋊Dic3C2×#6711yescharacteristic
210C3×Dic3—22no—
211C3×Dic3—22no—
212C3×Dic3—42no—
213C3×Dic3—62no—
214C3×Dic3—62no—
215C3×Dic3—61no—
216C3×Dic3—61no—
217C3×Dic3—61no—
218C3×Dic3—61no—
219C3×Dic3—122no—
220C3×Dic3—124no—
221C3×Dic3—122no—
222C2×#41—91no—
223C2×M16—271no—
224C2×C4⋊C4—271no—
225#103—272no—
226#102—272no—
227C33#41711yescharacteristic, complemented, Sylow
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—61no—
233C23×C3—61no—
234C23×C3—121no—
235C23×C3GO+(4,𝔽2)11yescharacteristic
236C22×D3—21no—
237C3×Q8—92no—
238C3×Q8—181no—
239C3×D4—64no—
240C3×D4—64no—
241C3×D4—92no—
242C3×D4—124no—
243C3×D4—181no—
244C2×C3×C4—61no—
245C2×C3×C4—61no—
246C2×C3×C4—91no—
247C2×C3×C4—91no—
248C2×C3×C4—92no—
249C2×C3×C4—92no—
250C2×C3×C4—121no—
251C2×C3×C4—184no—
252C2×C3×C4—182no—
253C2×C3×C4—362no—
254#67—24no—
255#67—62no—
256#67—62no—
257C2×Dic3—21no—
258C2×Dic3—62no—
259C2×Dic3—62no—
260C2×Dic3—61no—
261C2×Dic3—61no—
262C2×Dic3—62no—
263C2×Dic3—62no—
264C2×Dic3—61no—
265C2×Dic3—61no—
266C2×Dic3—64no—
267C2×Dic3—62no—
268C2×Dic3—62no—
269C2×Dic3—122no—
270C2×Dic3—122no—
271C2×Dic3—122no—
272C2×Dic3—124no—
273C4×D3—64no—
274Dic6—62no—
275Dic6—62no—
276C3⋊C8—182no—
277C2×C32—24no—
278C2×C32—22no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—41no—
286C2×C32—41no—
287C2×C32—41no—
288C2×C32—41no—
289C2×C32—41no—
290C2×C32—41no—
291C2×C32—42no—
292C2×C32—44no—
293C2×C32—82no—
294C2×C32—122no—
295C2×C32#94911yescharacteristic
296C2×C32#95111yescharacteristic
297C2×C32C2×#21711yescharacteristic
298C3×D3—42no—
299C3×D3—42no—
300C3×D3—82no—
301#41—94no—
302#41—182no—
303C2×Q8—91no—
304C2×D4—61no—
305C2×D4—91no—
306C22×C4—91no—
307C22×C4—181no—
308C22×C4—181no—
309M16—542no—
310C2×C8—272no—
311C4⋊C4—272no—
312C4⋊C4—541no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3—62no—
330C22×C3—62no—
331C22×C3—81no—
332C22×C3—81no—
333C22×C3—122no—
334C22×C3—121no—
335C22×C3—121no—
336C22×C3—121no—
337C22×C3—121no—
338C22×C3—241no—
339C22×C3—241no—
340C22×C3D32⋊C412yes—
341C22×C3C2×GO+(4,𝔽2)11yescharacteristic
342C2×D3—22no—
343C2×D3—41no—
344C2×D3—41no—
345C3×C4—62no—
346C3×C4—62no—
347C3×C4—91no—
348C3×C4—92no—
349C3×C4—91no—
350C3×C4—122no—
351C3×C4—362no—
352Dic3—22no—
353Dic3—61no—
354Dic3—61no—
355Dic3—62no—
356Dic3—62no—
357Dic3—61no—
358Dic3—61no—
359Dic3—122no—
360Dic3—124no—
361Dic3—122no—
362C32—21no—
363C32—21no—
364C32—41no—
365C32—41no—
366C32#523211yescharacteristic, complemented
367C23—61no—
368C23D32⋊D311yescharacteristic
369Q8—92no—
370Q8—181no—
371D4—64no—
372D4—92no—
373D4—181no—
374C2×C4—61no—
375C2×C4—91no—
376C2×C4—91no—
377C2×C4—92no—
378C2×C4—92no—
379C2×C4—184no—
380C2×C4—182no—
381C2×C4—362no—
382C8—542no—
383C2×C3—24no—
384C2×C3—22no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—42no—
392C2×C3—44no—
393C2×C3—41no—
394C2×C3—41no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—82no—
400C2×C3—122no—
401C2×C3—122no—
402C2×C3—242no—
403C2×C3C2×D32⋊C411yescharacteristic
404C2×C3#6423611yescharacteristic
405C2×C3#6423911yescharacteristic
406D3—42no—
407C22—21no—
408C22—21no—
409C22—62no—
410C22—121no—
411C22—121no—
412C22D32⋊Dic312yes—
413C22C2×D32⋊D311yescharacteristic
414C4—62no—
415C4—91no—
416C4—92no—
417C4—91no—
418C4—362no—
419C3—21no—
420C3—21no—
421C3—41no—
422C3—41no—
423C3#1059349611yescharacteristic, complemented
424C2—22no—
425C2—122no—
426C2C2×D32⋊Dic311yescharacteristic
427C2#1172432211yescharacteristic
428C2#1172432811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC2×C3×#3805C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC23D32⋊D3
hypocenterC33#417
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88993C2semidirect
3. C2×#88815C2semidirect
4. #11722775C2semidirect
5. #11722773C2semidirect
13. C3×#3805C4semidirect
37. C3×#627C2×C4semidirect
39. C3×#625C2×C4semidirect
63. #3805Dic3semidirect
134. #627C2×Dic3semidirect
136. #625C2×Dic3semidirect
227. C33#417semidirect
366. C32#5232semidirect
423. C3#10593496semidirect

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
C2×#888152
#117227752
#117227732
#105934963
#52329