Finite Group Databasebeta

← #13166997

Subgroups — #13166997

Order 1728.

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

normal subgroups only

This group has 660 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×#89042C211yesmaximal, characteristic
2C2×#88989C211yesmaximal, characteristic, complemented
3C3×#63900C211yesmaximal, characteristic, complemented
4#11722774C212yesmaximal, complemented
5#11722772C212yesmaximal, complemented
6#10593360—31nomaximal
7C2×C3×C4×C3⋊D3—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×#6296C2211yescharacteristic
10C2×#6295—21no—
11C2×#6295C2211yescharacteristic
12#89042—22no—
13#89042C414yescomplemented
14C2×C3×#624—21no—
15C2×C3⋊#622C2212yes—
16#88989—22no—
17C32×C4⋊Dic3—21no—
18C3×#3724C2212yes—
19C2×#3798—31no—
20C2×#3798—61no—
21C2×#3788—31no—
22C3×D3×C4⋊C4—61no—
23#63900D311yescharacteristic, complemented
24#63740—32no—
25#63739—32no—
26C22×C3×C3⋊D3—21no—
27C22×C3×C3⋊D3D411yescharacteristic
28C2×C33×C4D411yescharacteristic
29C2×C3×C3⋊Dic3—21no—
30C2×C3×C3⋊Dic3C2311yescharacteristic
31C3×C4×C3⋊D3—24no—
32C3×C4×C3⋊D3C2×C412yes—
33C3×C4×C3⋊D3D412yes—
34C2×C32×Dic3—22no—
35#6296—21no—
36#6296C2×C412yescomplemented
37#6295—24no—
38#6295—21no—
39#6295C2×C412yescomplemented
40C3×#624—42no—
41C3⋊#622—22no—
42#4966—91nomaximal
43C2×C4×C3⋊D3—21no—
44C2×C4×C3⋊D3C2×D311yescharacteristic
45C2×C3×C4×D3—61no—
46C2×C3×C4×D3—61no—
47C2×C3×C4×D3—61no—
48C2×C3×C4×D3—61no—
49C2×C3×C4×D3—62no—
50C2×C3×C4×D3—121no—
51C2×#627—31no—
52C2×#627—61no—
53C2×#626—61no—
54C2×#626—61no—
55C2×#626—61no—
56C2×#625—31no—
57C2×#625—61no—
58C2×#624—21no—
59#3798—34no—
60#3798—64no—
61#3798—64no—
62#3798—62no—
63C2×#622—32no—
64#3788—62no—
65C32×C4⋊C4—61no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3—41no—
69C3×Dic3⋊C4—62no—
70#3724C2×D312yes—
71C22×C33C2×D411yescharacteristic
72C2×C3×C3⋊D3—21no—
73C2×C3×C3⋊D3—21no—
74C2×C3×C3⋊D3—22no—
75C2×C3×C3⋊D3—41no—
76C2×C3×C3⋊D3—41no—
77C2×C3×C3⋊D3C22⋊C412yes—
78C33×C4C22⋊C412yes—
79C3×C3⋊Dic3—22no—
80C3×C3⋊Dic3C22×C411yescharacteristic
81C3×C3⋊Dic3C2×D411yescharacteristic
82C32×Dic3—42no—
83C2×#235—91no—
84C2×#235—181no—
85C2×C3×C4⋊C4—91no—
86C2×#208—91no—
87D3×C4⋊C4—61no—
88#810—92no—
89#809—92no—
90C22×C3⋊D3—21no—
91C22×C3⋊D3#6711yescharacteristic
92C22×C3×D3—61no—
93C22×C3×D3—61no—
94C22×C3×D3—61no—
95C22×C3×D3—61no—
96C22×C3×D3—121no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4—41no—
100C2×C32×C4—41no—
101C2×C32×C4—62no—
102C2×C32×C4#6711yescharacteristic
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3C22×D311yescharacteristic
105C4×C3⋊D3—24no—
106C4×C3⋊D3C4×D312yes—
107C4×C3⋊D3D1212yes—
108C2×C3×Dic3—22no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—42no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C2×C3×Dic3—61no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—121no—
116C3×C4×D3—62no—
117C3×C4×D3—62no—
118C3×C4×D3—62no—
119C3×C4×D3—62no—
120C3×C4×D3—64no—
121C3×C4×D3—64no—
122C3×C4×D3—124no—
123C3×C4×D3—124no—
124#627—32no—
125#627—62no—
126#627—62no—
127#627—61no—
128#626—62no—
129#626—62no—
130#626—64no—
131#626—64no—
132#625—32no—
133#625—64no—
134#625—61no—
135#624—42no—
136#622—62no—
137#417—271noSylow
138C2×C33C2×C22⋊C411yescharacteristic
139C2×C33AGL(1,ℤ/8)11yescharacteristic
140C2×C33#13711yescharacteristic
141C3×C3⋊D3—22no—
142C3×C3⋊D3—42no—
143C22×C3×C4—91no—
144C22×C3×C4—181no—
145C22×C3×C4—181no—
146C2×#67—181no—
147C2×#67—181no—
148C2×#67—181no—
149#235—94no—
150#235—184no—
151#235—182no—
152#235—184no—
153C2×D12—91no—
154C2×D12—181no—
155C2×C4×D3—61no—
156C2×C4×D3—61no—
157C2×C4×D3—61no—
158C2×C4×D3—61no—
159C2×C4×D3—62no—
160C2×C4×D3—121no—
161C2×Dic6—91no—
162C2×Dic6—181no—
163C3×C4⋊C4—61no—
164C3×C4⋊C4—61no—
165C3×C4⋊C4—92no—
166C3×C4⋊C4—121no—
167C4⋊Dic3—21no—
168Dic3⋊C4—62no—
169#208—182no—
170C2×C3⋊C8—92no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—41no—
174C22×C32—41no—
175C22×C32C2×#6711yescharacteristic
176C2×C3⋊D3—21no—
177C2×C3⋊D3—21no—
178C2×C3⋊D3—22no—
179C2×C3⋊D3—41no—
180C2×C3⋊D3—41no—
181C2×C3⋊D3(C2×D3)⋊C412yes—
182C2×C3×D3—62no—
183C2×C3×D3—62no—
184C2×C3×D3—62no—
185C2×C3×D3—62no—
186C2×C3×D3—62no—
187C2×C3×D3—62no—
188C2×C3×D3—121no—
189C2×C3×D3—121no—
190C2×C3×D3—122no—
191C2×C3×D3—121no—
192C2×C3×D3—121no—
193C2×C3×D3—122no—
194C2×C3×D3—122no—
195C2×C3×D3—121no—
196C2×C3×D3—121no—
197C32×C4—22no—
198C32×C4—22no—
199C32×C4—42no—
200C32×C4—42no—
201C32×C4—122no—
202C32×C4(C2×D3)⋊C412yes—
203C3⋊Dic3—22no—
204C3⋊Dic3C2×C4×D311yescharacteristic
205C3⋊Dic3C2×D1211yescharacteristic
206C3×Dic3—42no—
207C3×Dic3—42no—
208C3×Dic3—61no—
209C3×Dic3—61no—
210C3×Dic3—61no—
211C3×Dic3—61no—
212C3×Dic3—62no—
213C3×Dic3—62no—
214C3×Dic3—82no—
215C3×Dic3—122no—
216C2×#41—271no—
217C2×M16—271no—
218C2×C4⋊C4—91no—
219#103—272no—
220#102—272no—
221C33#41711yescharacteristic, complemented, Sylow
222C23×C3—91no—
223C23×C3—181no—
224C22×D3—61no—
225C22×D3—61no—
226C22×D3—61no—
227C22×D3—61no—
228C22×D3—121no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—62no—
234C2×C3×C4—62no—
235C2×C3×C4—91no—
236C2×C3×C4—92no—
237C2×C3×C4—92no—
238C2×C3×C4—122no—
239C2×C3×C4—184no—
240C2×C3×C4—181no—
241C2×C3×C4—362no—
242C2×C3×C4GO+(4,𝔽2)11yescharacteristic
243#67—182no—
244#67—184no—
245#67—182no—
246#67—184no—
247C2×Dic3—22no—
248C2×Dic3—61no—
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—121no—
253D12—92no—
254D12—181no—
255D12—184no—
256C4×D3—62no—
257C4×D3—62no—
258C4×D3—62no—
259C4×D3—62no—
260C4×D3—64no—
261C4×D3—64no—
262C4×D3—124no—
263C4×D3—124no—
264Dic6—92no—
265Dic6—182no—
266Dic6—181no—
267Dic6—182no—
268C3⋊C8—182no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—41no—
276C2×C32—41no—
277C2×C32—41no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32—41no—
281C2×C32C2×(C2×D3)⋊C411yescharacteristic
282C2×C32#93211yescharacteristic
283C2×C32#94211yescharacteristic
284C3⋊D3—22no—
285C3⋊D3—42no—
286C3×D3—64no—
287C3×D3—122no—
288C3×D3—122no—
289C3×D3—124no—
290C3×D3—122no—
291#41—274no—
292#41—542no—
293C2×Q8—271no—
294C2×D4—271no—
295C2×D4—541no—
296C22×C4—91no—
297C22×C4—181no—
298C22×C4—181no—
299M16—542no—
300C2×C8—272no—
301C4⋊C4—61no—
302C4⋊C4—92no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—92no—
308C22×C3—182no—
309C22×C3—181no—
310C22×C3—181no—
311C22×C3—361no—
312C22×C3—361no—
313C22×C3C2×GO+(4,𝔽2)11yescharacteristic
314C2×D3—62no—
315C2×D3—62no—
316C2×D3—62no—
317C2×D3—62no—
318C2×D3—62no—
319C2×D3—62no—
320C2×D3—121no—
321C2×D3—121no—
322C2×D3—122no—
323C2×D3—121no—
324C2×D3—121no—
325C2×D3—122no—
326C2×D3—122no—
327C2×D3—121no—
328C2×D3—121no—
329C3×C4—22no—
330C3×C4—22no—
331C3×C4—42no—
332C3×C4—42no—
333C3×C4—91no—
334C3×C4—91no—
335C3×C4—122no—
336C3×C4—122no—
337C3×C4—182no—
338C3×C4—242no—
339C3×C4D32⋊C412yes—
340Dic3—42no—
341Dic3—61no—
342Dic3—61no—
343Dic3—61no—
344Dic3—61no—
345Dic3—62no—
346Dic3—62no—
347Dic3—122no—
348C32—21no—
349C32—21no—
350C32—41no—
351C32—41no—
352C32#496611yescharacteristic, complemented
353C23—91no—
354C23—181no—
355Q8—272no—
356Q8—541no—
357D4—272no—
358D4—541no—
359D4—544no—
360C2×C4—62no—
361C2×C4—91no—
362C2×C4—92no—
363C2×C4—92no—
364C2×C4—181no—
365C2×C4—184no—
366C2×C4—362no—
367C2×C4D32⋊D311yescharacteristic
368C8—542no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3—182no—
382C2×C3—362no—
383C2×C3#6422411yescharacteristic
384C2×C3#6422611yescharacteristic
385C2×C3C2×D32⋊C411yescharacteristic
386D3—64no—
387D3—122no—
388D3—122no—
389D3—124no—
390D3—122no—
391C22—92no—
392C22—181no—
393C22—181no—
394C22—182no—
395C22—361no—
396C22—361no—
397C22C2×D32⋊D311yescharacteristic
398C4—91no—
399C4—91no—
400C4—122no—
401C4—182no—
402C4#8893212yes—
403C3—21no—
404C3—21no—
405C3—41no—
406C3—41no—
407C3#1059336011yescharacteristic, complemented
408C2—182no—
409C2—362no—
410C2#1172430411yescharacteristic
411C2#1172430711yescharacteristic
412C2C2×#8893211yescharacteristic
subgroupG/H
center, FrattiniC22C2×D32⋊D3
commutatorC3×C3⋊Dic3C22×C4
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×#89042C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D32⋊D3
hypocenterC33#417
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88989C2semidirect
3. C3×#63900C2semidirect
4. #11722774C2semidirect
5. #11722772C2semidirect
13. #89042C4semidirect
23. #63900D3semidirect
36. #6296C2×C4semidirect
39. #6295C2×C4semidirect
221. C33#417semidirect
352. C32#4966semidirect
407. C3#10593360semidirect

Maximal subgroups

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

subgroupindex
C2×#890422
C2×#889892
C3×#639002
#117227742
#117227722
#105933603
#49669