Finite Group Databasebeta

← #10769249

Subgroups — #10769249

Order 768.

888 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85725C211yesmaximal, characteristic
2#82264C211yesmaximal, characteristic
3#79272C212yesmaximal, complemented
4C3×#1458C211yesmaximal, characteristic
5#70793C212yesmaximal, complemented
6#68141C214yesmaximal, complemented
7#68117C214yesmaximal
8#18311—31nomaximal, Sylow
9C2×C3×#117C2211yescharacteristic
10C2×#926—21no—
11#5519—22no—
12#5519—22no—
13#5517—24no—
14#5500—21no—
15#5500C2212yes—
16#5483—21no—
17#5483C2212yes—
18#5470—21no—
19#5470C2212yes—
20C2×#869C2211yescharacteristic
21C3×#431C2212yes—
22#5212—21no—
23#4986C2212yes—
24#4979—21no—
25#4949C2212yes—
26#4947—21no—
27#4936C2211yescharacteristic
28#4928—21no—
29#4850—24no—
30#4722C2218yes—
31#4682—22no—
32#4681—22no—
33C3×#324C2212yes—
34C3×#323C2212yes—
35#4467C2218yes—
36#3458—31no—
37#3230—32no—
38#2628—31no—
39#1617—34no—
40#1614—34no—
41#1518—32no—
42#1458D311yescharacteristic
43C23×C3×C4D411yescharacteristic
44C22×Dic6—21no—
45C3×#117—22no—
46C3×#117—22no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C4⋊C4C2311yescharacteristic
49C2×C3×C22⋊C4D411yescharacteristic
50C2×C3×C22⋊C4C2311yescharacteristic
51C2×C3×C42D411yescharacteristic
52C2×C3×C42C2311yescharacteristic
53C2×#217—22no—
54C2×#217—22no—
55#926—22no—
56#926—22no—
57#926—22no—
58#926—22no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3C2312yes—
64#924—22no—
65#924—22no—
66#924—22no—
67#924C2314yes—
68C2×Dic3⋊C4—22no—
69C2×C4×Dic3—22no—
70#890—24no—
71#890—24no—
72#888—24no—
73#888—24no—
74#878—24no—
75#878—24no—
76#878—24no—
77#878—24no—
78#870—22no—
79#870—22no—
80#869—21no—
81#869—21no—
82#869—22no—
83#869—22no—
84#869D412yes—
85#869D412yes—
86C4×Dic6—22no—
87C4×Dic6—22no—
88C4×Dic6—22no—
89C4×Dic6D414yes—
90C3×C4⋊C8—22no—
91C3×C22⋊C8C2314yes—
92C3×C4×C8—22no—
93C3⋊#95—22no—
94C3⋊#95—21no—
95C3⋊#95—22no—
96C3⋊#95C2312yes—
97#818—28no—
98#816—28no—
99#544—32no—
100#544—62no—
101#544—62no—
102#544—61no—
103#543—32no—
104#543—61no—
105#533—32no—
106#533—64no—
107#533—61no—
108C2×C4⋊Q8—31no—
109C2×#117—61no—
110C2×#117C2×D311yescharacteristic
111#500—62no—
112#487—62no—
113#484—38no—
114#477—64no—
115#431C2×D312yes—
116#400—32no—
117#399—32no—
118#395—61no—
119#393—61no—
120#389—61no—
121#382—31no—
122#380—61no—
123#328—38no—
124#324C2×D312yes—
125#323C2×D312yes—
126C24×C3C2×D411yescharacteristic
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4Q1614yes—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2411yescharacteristic
137C22×Dic3—22no—
138C22×Dic3—22no—
139C22×Dic3—22no—
140C22×Dic3—22no—
141C2×Dic6—22no—
142C2×Dic6—22no—
143C2×Dic6—22no—
144C2×Dic6—42no—
145C2×Dic6—82no—
146C2×Dic6C2×D414yes—
147C2×C3×C8—24no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—22no—
150C3×C22⋊C4—22no—
151C3×C22⋊C4—22no—
152C3×C42—21no—
153C3×C42—21no—
154C3×C42C2×D412yes—
155C3×C42C2×D412yes—
156#217—24no—
157#217—24no—
158#217—82no—
159C4⋊Dic3—22no—
160C4⋊Dic3—22no—
161C4⋊Dic3—22no—
162C4⋊Dic3—22no—
163C4⋊Dic3—22no—
164C4⋊Dic3—22no—
165C4⋊Dic3—22no—
166C4⋊Dic3—24no—
167C4⋊Dic3—41no—
168C4⋊Dic3—41no—
169C4⋊Dic3—42no—
170C4⋊Dic3C2×D414yes—
171Dic3⋊C4—24no—
172Dic3⋊C4—24no—
173Dic3⋊C4—24no—
174Dic3⋊C4—24no—
175Dic3⋊C4—82no—
176C4×Dic3—24no—
177C4×Dic3—24no—
178C22×Q8—61no—
179C23×C4D1211yescharacteristic
180C4⋊Q8—32no—
181C4⋊Q8—32no—
182C4⋊Q8—61no—
183C4⋊Q8—61no—
184C4⋊Q8—64no—
185C4⋊Q8—64no—
186C4⋊Q8—62no—
187C4⋊Q8—62no—
188#125—62no—
189#125—64no—
190#125—64no—
191#125—62no—
192C22⋊Q8—34no—
193C22⋊Q8—64no—
194C22⋊Q8—64no—
195C22⋊Q8—64no—
196C22⋊Q8—64no—
197C22⋊Q8—62no—
198C22⋊Q8—62no—
199C22⋊Q8—62no—
200C4×Q8—34no—
201C4×Q8—62no—
202C4×Q8—62no—
203C4×Q8—62no—
204#117—22no—
205#117—22no—
206#117—62no—
207#117—62no—
208#117—62no—
209#117—62no—
210C2×C4⋊C4—32no—
211C2×C4⋊C4—61no—
212C2×C4⋊C4—61no—
213C2×C4⋊C4—61no—
214C2×C4⋊C4—61no—
215C2×C4⋊C4—62no—
216C2×C4⋊C4D1211yescharacteristic
217C2×C4⋊C4C22×D311yescharacteristic
218C2×C22⋊C4—62no—
219C2×C22⋊C4—62no—
220C2×C22⋊C4D1211yescharacteristic
221C2×C22⋊C4C22×D311yescharacteristic
222C2×C42—62no—
223C2×C42D1211yescharacteristic
224C2×C42C22×D311yescharacteristic
225#107—68no—
226C4⋊C8—22no—
227#103—68no—
228C22⋊C8C22×D314yes—
229C4×C8—22no—
230#95—32no—
231#95—62no—
232#95—61no—
233#95—62no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3C2×Q1612yes—
239C23×C3C22×D411yescharacteristic
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4#12012yes—
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12011yescharacteristic
253C2×C3×C4C2×Q1612yes—
254C2×C3×C4C2×Q1612yes—
255C2×C3×C4#13712yes—
256C2×C3×C4C22×D412yes—
257C2×Dic3—24no—
258C2×Dic3—24no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—82no—
266C2×Dic3—82no—
267Dic6—44no—
268Dic6—82no—
269Dic6—82no—
270Dic6—82no—
271Dic6—84no—
272C3×C8—44no—
273C24C2×D1211yescharacteristic
274C2×Q8—34no—
275C2×Q8—62no—
276C2×Q8—62no—
277C2×Q8—62no—
278C2×Q8—122no—
279C2×Q8—242no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—62no—
284C22×C4—62no—
285C22×C4—62no—
286C22×C4—62no—
287C22×C4Dic1214yes—
288C22×C4C2×D1211yescharacteristic
289C22×C4C2×D1211yescharacteristic
290C22×C4C2×D1211yescharacteristic
291C22×C4C2×D1211yescharacteristic
292C22×C4C2×D1211yescharacteristic
293C22×C4C23×D311yescharacteristic
294C2×C8—24no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—34no—
298C4⋊C4—62no—
299C4⋊C4—62no—
300C4⋊C4—64no—
301C4⋊C4—64no—
302C4⋊C4—62no—
303C4⋊C4—62no—
304C4⋊C4—62no—
305C4⋊C4—62no—
306C4⋊C4—64no—
307C4⋊C4—62no—
308C4⋊C4—64no—
309C4⋊C4—64no—
310C4⋊C4—121no—
311C4⋊C4—121no—
312C4⋊C4—122no—
313C4⋊C4—242no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—64no—
317C22⋊C4—64no—
318C22⋊C4—242no—
319C42—21no—
320C42—21no—
321C42—64no—
322C42—64no—
323C42D3×D412yes—
324C42D3×D412yes—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—22no—
328C22×C3—42no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—81no—
332C22×C3C2×#12011yescharacteristic
333C22×C3C22×Q1611yescharacteristic
334C22×C3C2×#13711yescharacteristic
335C3×C4—22no—
336C3×C4—22no—
337C3×C4—22no—
338C3×C4—41no—
339C3×C4—41no—
340C3×C4#45112yes—
341C3×C4#45112yes—
342Dic3—44no—
343Dic3—44no—
344Dic3—82no—
345Dic3—82no—
346C23—21no—
347C23—21no—
348C23—41no—
349C23—41no—
350C23C2×Dic1212yes—
351C23C22×D1211yescharacteristic
352Q8—124no—
353Q8—242no—
354Q8—244no—
355Q8—242no—
356Q8—242no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—64no—
368C2×C4—64no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—242no—
376C2×C4—242no—
377C2×C4#88212yes—
378C2×C4#88211yescharacteristic
379C2×C4#88211yescharacteristic
380C2×C4C2×Dic1212yes—
381C2×C4C2×Dic1212yes—
382C2×C4#90912yes—
383C2×C4C2×D3×D412yes—
384C8—44no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—42no—
388C2×C3C2×#45111yescharacteristic
389C2×C3#302111yescharacteristic
390C2×C3#302811yescharacteristic
391C22—21no—
392C22—21no—
393C22—22no—
394C22—42no—
395C22—41no—
396C22—41no—
397C22—81no—
398C22C2×#88211yescharacteristic
399C22C22×Dic1211yescharacteristic
400C22C2×#90911yescharacteristic
401C4—22no—
402C4—22no—
403C4—22no—
404C4—41no—
405C4—41no—
406C4—124no—
407C4—124no—
408C4—242no—
409C4—242no—
410C4#473912yes—
411C4#473912yes—
412C3#1831111yescharacteristic, complemented, Sylow
413C2—21no—
414C2—21no—
415C2—42no—
416C2C2×#473911yescharacteristic
417C2#7869211yescharacteristic
418C2#7871711yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1458C2
socleC22×C3C2×#120
prime radicalC24×C3C2×D4
hypercenter#1458D3
hypocenterC3#18311
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #79272C2semidirect
5. #70793C2semidirect
6. #68141C2semidirect
412. C3#18311semidirect

Maximal subgroups

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

subgroupindex
#857252
#822642
#792722
C3×#14582
#707932
#681412
#681172
#183113