Finite Group Databasebeta

← #29658713

Subgroups — #29658713

Order 15616.

922 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456767C211yesmaximal, characteristic
2#21455858C211yesmaximal, characteristic
3#21455624C211yesmaximal, characteristic
4#21445040C214yesmaximal, complemented
5#21444959C214yesmaximal, complemented
6#21444500C211yesmaximal, characteristic
7#21444248C211yesmaximal, characteristic
8#21444214C211yesmaximal, characteristic
9C2×Q82×C61C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840141C2212yes—
12#16840112C2212yes—
13#16840097—21no—
14#16840069—22no—
15#16840057C2212yes—
16#16840057C2212yes—
17#16840047C2212yes—
18#16839972C2212yes—
19#16839969—22no—
20#16839918—22no—
21#16839918—21no—
22C4×#11758202—22no—
23#16839815C2211yescharacteristic
24#16839815C2211yescharacteristic
25#16839802—21no—
26#16839786C2211yescharacteristic
27#16839771—21no—
28#16839511C2214yes—
29Q8⋊#92418C2214yes—
30Q8⋊#92418C2214yes—
31Q82×C61—22no—
32Q82×C61—22no—
33Q82×C61—22no—
34Q82×C61C2214yes—
35C2×C4⋊Q8×C61—22no—
36C2×C4⋊Q8×C61—21no—
37C2×C4⋊Q8×C61—21no—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×C4×Q8×C61—21no—
40C2×C4×Q8×C61—21no—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4×Q8×C61C2211yescharacteristic
43C2×#11758202—22no—
44C2×#11758202—24no—
45C2×#11758194—21no—
46#14519777C2311yescharacteristic
47#14519777C2311yescharacteristic
48#14519769—21no—
49#14519756—21no—
50#14519756—21no—
51C4×Dic122—21no—
52C4×Dic122—21no—
53#14519736—21no—
54#14519710—22no—
55#14519710—22no—
56#14519708—22no—
57C4⋊#92418—21no—
58C4⋊#92418—21no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519701—24no—
62#14519701—22no—
63#14519701—22no—
64#14519701—22no—
65#14519700—22no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C4×#92418—21no—
69C4×#92418C2312yes—
70C22×Q8×C61—22no—
71C22×Q8×C61D411yescharacteristic
72C22×Q8×C61D411yescharacteristic
73C22×Q8×C61D412yes—
74C4⋊Q8×C61—28no—
75C4⋊Q8×C61—24no—
76C4⋊Q8×C61—24no—
77C4⋊Q8×C61—28no—
78C4⋊Q8×C61—24no—
79C4⋊Q8×C61—24no—
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61D412yes—
83C4⋊Q8×C61C2312yes—
84C4×Q8×C61—24no—
85C4×Q8×C61—24no—
86C4×Q8×C61—24no—
87C4×Q8×C61—24no—
88C4×Q8×C61—21no—
89C4×Q8×C61—21no—
90C4×Q8×C61—21no—
91C4×Q8×C61—21no—
92C4×Q8×C61D412yes—
93C4×Q8×C61D412yes—
94C4×Q8×C61C2312yes—
95C4×Q8×C61C2312yes—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C42×C61—22no—
105C2×C42×C61—21no—
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C2×Q8×C61—24no—
109C2×Q8×C61—28no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—82no—
115C2×Q8×C61—84no—
116C2×Q8×C61Q1614yescomplemented
117C2×Q8×C61C2×D412yes—
118C2×Q8×C61C2×D412yes—
119C2×Q8×C61C2×D414yes—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C2×Dic122—21no—
129C2×Dic122—21no—
130C4⋊C4×C61—28no—
131C4⋊C4×C61—28no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61C2×D412yes—
143C4⋊C4×C61C2×D412yes—
144C42×C61—28no—
145C42×C61—22no—
146C42×C61—22no—
147C42×C61—22no—
148C42×C61—22no—
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61C2411yescharacteristic
153C22⋊Dic61—41no—
154#11758202—44no—
155#11758202—84no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160#11758196—41no—
161#11758196—41no—
162C4×Dic61—41no—
163#11758194—42no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—22no—
167C23×C61#12011yescharacteristic
168Q8×C61—24no—
169Q8×C61—22no—
170Q8×C61—22no—
171Q8×C61—24no—
172Q8×C61—44no—
173Q8×C61—82no—
174Q8×C61—88no—
175Q8×C61C2×Q1614yes—
176Q8×C61#13714yes—
177C2×C4×C61—24no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—24no—
181C2×C4×C61—22no—
182C2×C4×C61—24no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—41no—
186C2×C4×C61—44no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61#12011yescharacteristic
190C2×C4×C61#12011yescharacteristic
191C2×C4×C61#12011yescharacteristic
192C2×C4×C61C2×Q1612yes—
193C2×C4×C61#13712yes—
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×Dic61—41no—
198C2×Dic61—41no—
199Dic122—42no—
200Dic122—42no—
201#92418—42no—
202#92418—42no—
203#92418—42no—
204#18769—611nomaximal, Sylow
205C22×C61—21no—
206C22×C61—21no—
207C22×C61#45112yes—
208C22×C61C2×#12011yescharacteristic
209C4×C61—24no—
210C4×C61—24no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—44no—
215C4×C61—42no—
216C4×C61—42no—
217C4×C61#45112yes—
218C4×C61C22×Q1611yescharacteristic
219C4×C61C2×#13711yescharacteristic
220C4×C61C2×#13712yes—
221Dic61—81no—
222Dic61—81no—
223C2×Q82D6111yescharacteristic
224#3209—611no—
225#3104—611no—
226#3078—611no—
227#1645—614no—
228#1633—614no—
229#1548—611no—
230#1487—611no—
231#1479—611no—
232C2×C61—22no—
233C2×C61C2×#45111yescharacteristic
234C2×C61#300411yescharacteristic
235C2×C61#301911yescharacteristic
236Q82—22no—
237Q82—22no—
238Q82—22no—
239Q82C2×D6114yes—
240#533—611no—
241C2×C4⋊Q8—22no—
242C2×C4⋊Q8—21no—
243C2×C4⋊Q8—21no—
244C2×C4⋊Q8C2×D6111yescharacteristic
245C2×C4×Q8—21no—
246C2×C4×Q8—21no—
247C2×C4×Q8C2×D6111yescharacteristic
248C2×C4×Q8C2×D6111yescharacteristic
249#494—612no—
250#487—612no—
251#484—1221no—
252#479—1222no—
253#477—612no—
254#477—612no—
255#475—612no—
256#462—612no—
257#462—1222no—
258#451—1221no—
259#451—1222no—
260C4×Q16—1222no—
261#431—611no—
262#431—611no—
263#428—1221no—
264#423—611no—
265#419—1221no—
266#333—614no—
267Q8⋊C8—614no—
268Q8⋊C8—614no—
269C61#1876911yescharacteristic, complemented, Sylow
270C22×Q8—22no—
271C22×Q8C22⋊D6111yescharacteristic
272C22×Q8C22⋊D6111yescharacteristic
273C22×Q8C22⋊D6112yes—
274C2×Q16—1224no—
275C2×Q16—1222no—
276C2×M16—1221no—
277C4⋊Q8—28no—
278C4⋊Q8—24no—
279C4⋊Q8—24no—
280C4⋊Q8—28no—
281C4⋊Q8—24no—
282C4⋊Q8—24no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—611no—
286C4⋊Q8—611no—
287C4⋊Q8C22⋊D6112yes—
288C4⋊Q8C22×D6112yes—
289#125—1221no—
290C22⋊Q8—1221no—
291C22⋊Q8—1221no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8—1221no—
301C4×Q8—1221no—
302C4×Q8C22⋊D6112yes—
303C4×Q8C22⋊D6112yes—
304C4×Q8C22×D6112yes—
305C4×Q8C22×D6112yes—
306#117—1221no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4C22⋊D6111yescharacteristic
314C2×C4⋊C4C22⋊D6111yescharacteristic
315C2×C42—22no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42C22×D6111yescharacteristic
319#107—1222no—
320#107—1222no—
321#106—1222no—
322C4⋊C8—612no—
323C4⋊C8—612no—
324C4⋊C8—1221no—
325C4⋊C8—1221no—
326#103—1222no—
327#103—1222no—
328#103—1222no—
329#103—1224no—
330#103—1222no—
331C22⋊C8—1221no—
332C22⋊C8—1221no—
333C4×C8—612no—
334C4×C8—1221no—
335C2×Q8—24no—
336C2×Q8—28no—
337C2×Q8—22no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—22no—
341C2×Q8—82no—
342C2×Q8—84no—
343C2×Q8—1221no—
344C2×Q8—1221no—
345C2×Q8#1175820214yescomplemented
346C2×Q8C2×C22⋊D6112yes—
347C2×Q8C2×C22⋊D6112yes—
348C2×Q8C2×C22⋊D6114yes—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4C2×C22⋊D6111yescharacteristic
355C22×C4C2×C22⋊D6111yescharacteristic
356C22×C4C2×C22⋊D6111yescharacteristic
357Q16—4884no—
358Q16—2444no—
359M16—2442no—
360C2×C8—1222no—
361C2×C8—1222no—
362C2×C8—1222no—
363C4⋊C4—28no—
364C4⋊C4—28no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—1221no—
376C4⋊C4—1221no—
377C4⋊C4—1221no—
378C4⋊C4—1221no—
379C4⋊C4—2441no—
380C4⋊C4—2441no—
381C4⋊C4C2×C22⋊D6112yes—
382C4⋊C4C2×C22⋊D6112yes—
383C22⋊C4—2441no—
384C42—28no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—22no—
389C42—2441no—
390C42C2×C22⋊D6111yescharacteristic
391C42C2×C22⋊D6111yescharacteristic
392C42C2×C22⋊D6111yescharacteristic
393C42C23×D6111yescharacteristic
394C23#1451974611yescharacteristic
395Q8—24no—
396Q8—22no—
397Q8—22no—
398Q8—24no—
399Q8—44no—
400Q8—82no—
401Q8—88no—
402Q8—2442no—
403Q8—2442no—
404Q8C2×#1175820214yes—
405Q8#1451980814yes—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—2441no—
419C2×C4—2441no—
420C2×C4#1451974611yescharacteristic
421C2×C4#1451974611yescharacteristic
422C2×C4#1451974611yescharacteristic
423C2×C4C2×#1175820212yes—
424C2×C4#1451980812yes—
425C2×C4C22×C22⋊D6111yescharacteristic
426C2×C4C22×C22⋊D6111yescharacteristic
427C2×C4C22×C22⋊D6111yescharacteristic
428C8—2442no—
429C8—2442no—
430C8—2442no—
431C22—21no—
432C22—21no—
433C22#1683991812yes—
434C22C2×#1451974611yescharacteristic
435C4—21no—
436C4—21no—
437C4—24no—
438C4—24no—
439C4—21no—
440C4—44no—
441C4—42no—
442C4—42no—
443C4—4881no—
444C4—4881no—
445C4#1683991812yes—
446C4C22×#1175820211yescharacteristic
447C4C2×#1451980811yescharacteristic
448C4C2×#1451980812yes—
449C2—22no—
450C2C2×#1683991811yescharacteristic
451C2#2145505311yescharacteristic
452C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×Q82×C61C2
socleC22×C61C2×#120
prime radicalC23×C61#120
hypercenterC2×Q82D61
hypocenterC61#18769
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
4. #21445040C2semidirect
5. #21444959C2semidirect
116. C2×Q8×C61Q16semidirect
269. C61#18769semidirect
345. C2×Q8#11758202semidirect

Maximal subgroups

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

subgroupindex
#214567672
#214558582
#214556242
#214450402
#214449592
#214445002
#214442482
#214442142
C2×Q82×C612
#1876961