Finite Group Databasebeta

← #29580304

Subgroups — #29580304

Order 15616.

778 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21451626C211yesmaximal, characteristic
2C2×#16839697C211yesmaximal, characteristic
3#21445909C211yesmaximal, characteristic, complemented
4C2×#16839499C211yesmaximal, characteristic, complemented
5C2×#16839485C211yesmaximal, characteristic
6#21443699C211yesmaximal, characteristic, complemented
7C2×C61×#484C211yesmaximal, characteristic, complemented
8C22×#11758196—21no—
9C22×#11758198—21no—
10C22×#11758198C2211yescharacteristic
11C2×C22⋊#92418C2211yescharacteristic
12#16839697—22no—
13#16839697—22no—
14#16839697—21no—
15#16839684—22no—
16#16839673—21no—
17#16839634—21no—
18#16839622C2211yescharacteristic
19#16839617—21no—
20C2×#14519684C2211yescharacteristic
21#16839524—22no—
22#16839499—22no—
23#16839485—22no—
24C2×C22⋊Q8×C61C2211yescharacteristic
25C22×C4⋊C4×C61C2211yescharacteristic
26C61×#484—22no—
27C61×#484—22no—
28C2×#107×C61—21no—
29C2×#103×C61—21no—
30C2×C22⋊C8×C61C2211yescharacteristic
31C24×D61D411yescharacteristic
32C23×Dic61—21no—
33C23×Dic61D411yescharacteristic
34C22×#92418—21no—
35C2×C4⋊Dic61D412yes—
36C2×#11758196—24no—
37C2×#11758196—22no—
38C2×#11758196—44no—
39C2×C22⋊Dic61—21no—
40C2×#11758198—24no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198—22no—
48C2×#11758198—22no—
49C2×#11758198—44no—
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52#14519684—22no—
53#14519684—21no—
54#14519684—21no—
55#14519684D412yes—
56C22×Q8×C61D411yescharacteristic
57C23×C4×C61—21no—
58C23×C4×C61C2311yescharacteristic
59C22×C8×C61—21no—
60C22⋊Q8×C61—21no—
61C22⋊Q8×C61—21no—
62C22⋊Q8×C61—21no—
63C22⋊Q8×C61—21no—
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61D411yescharacteristic
71C2×C4⋊C4×C61D412yes—
72C2×C22⋊C4×C61—21no—
73#107×C61—24no—
74#103×C61—24no—
75C22⋊C8×C61—21no—
76C22⋊C8×C61D412yes—
77C24×C61C2×D411yescharacteristic
78C23×D61—21no—
79C23×D61—21no—
80C23×D61—21no—
81C23×D61—21no—
82C23×D61—21no—
83C23×D61—21no—
84C23×D61—42no—
85C23×D61—42no—
86C23×D61Q1612yes—
87C2×Q8×C61—21no—
88C2×Q8×C61—21no—
89C2×Q8×C61—42no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—22no—
93C22×C4×C61—42no—
94C22×C4×C61—42no—
95C22×C4×C61—42no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61#4111yescharacteristic
104C22×Dic61—24no—
105C22×Dic61—22no—
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61—22no—
109C22×Dic61—21no—
110C22×Dic61—42no—
111C22×Dic61—41no—
112C22×Dic61—41no—
113C22×Dic61—42no—
114C2×C8×C61—22no—
115C2×C8×C61—42no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—41no—
121C4⋊C4×C61—41no—
122C4⋊C4×C61—41no—
123C4⋊C4×C61—41no—
124C22⋊C4×C61—41no—
125C22⋊C4×C61—41no—
126C22⋊Dic61—41no—
127C22⋊Dic61—41no—
128#11758198—48no—
129#11758198—41no—
130#11758198—41no—
131#11758198—41no—
132#11758198—41no—
133#11758198—82no—
134#11758198—82no—
135C4⋊Dic61—42no—
136#11758196—48no—
137#11758196—82no—
138C2×#92418—42no—
139C2×#92418—41no—
140C23×C61—22no—
141C23×C61—22no—
142C23×C61#12011yescharacteristic
143C23×C61#12011yescharacteristic
144C23×C61#12111yescharacteristic
145C23×C61C2×Q1611yescharacteristic
146C23×C61#13511yescharacteristic
147C23×C61#13711yescharacteristic
148C23×C61#13711yescharacteristic
149C22×D61—22no—
150C22×D61—22no—
151C22×D61—22no—
152C22×D61—44no—
153C22×D61—42no—
154C22×D61—44no—
155C22×D61—42no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61—42no—
160C22×D61—42no—
161C22×D61—41no—
162C22×D61—82no—
163C22×D61—82no—
164C22×D61—81no—
165C22×D61—81no—
166Q8×C61—42no—
167Q8×C61—81no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—82no—
182C2×C4×C61—82no—
183C2×C4×C61#12011yescharacteristic
184C2×C4×C61#12111yescharacteristic
185C2×C4×C61#12111yescharacteristic
186C2×C4×C61#12711yescharacteristic
187C2×Dic61—24no—
188C2×Dic61—44no—
189C2×Dic61—44no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—81no—
199C8×C61—42no—
200#92418—81no—
201#9928—611nomaximal, Sylow
202C22×C61—22no—
203C22×C61—22no—
204C22×C61—22no—
205C22×C61—22no—
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61#39211yescharacteristic
211C22×C61#45111yescharacteristic
212C22×C61#45111yescharacteristic
213C22×C61#45211yescharacteristic
214C22×C61#45211yescharacteristic
215C22×C61#45312yes—
216C22×C61#45812yes—
217C22×C61#46711yescharacteristic
218C22×C61#47011yescharacteristic
219C2×D61—44no—
220C2×D61—44no—
221C2×D61—42no—
222C2×D61—82no—
223C2×D61—82no—
224C2×D61—82no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—82no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—81no—
233C4×C61—82no—
234C4×C61—81no—
235Dic61—44no—
236Dic61—82no—
237Dic61—81no—
238C2×#484D6111yescharacteristic, complemented
239#2626—611no—
240C2×#399—611no—
241#1824—611no—
242C2×#328—611no—
243C2×#323—611no—
244#1294—611no—
245C2×C61—21no—
246C2×C61—21no—
247C2×C61—21no—
248C2×C61—21no—
249C2×C61#161412yes—
250C2×C61#161812yes—
251C2×C61#200111yescharacteristic
252C2×C61#200211yescharacteristic
253C2×C61#202211yescharacteristic
254D61—82no—
255C2×C22⋊Q8C2×D6111yescharacteristic
256C22×C4⋊C4—1221no—
257C22×C4⋊C4C2×D6111yescharacteristic
258C22×C22⋊C4—611no—
259C22×C22⋊C4—1221no—
260#484—22no—
261#484—22no—
262C2×#107—21no—
263C2×#103—21no—
264C2×C22⋊C8—611no—
265C2×C22⋊C8C2×D6111yescharacteristic
266#399—1222no—
267#399—1221no—
268#399—1222no—
269#396—1222no—
270#393—1221no—
271#385—1221no—
272#380—611no—
273#379—1221no—
274C2×#95—611no—
275#340—1222no—
276#328—1222no—
277#323—1222no—
278C61#992811yescharacteristic, complemented, Sylow
279C25—611no—
280C22×Q8C22⋊D6111yescharacteristic
281C23×C4—21no—
282C23×C4—611no—
283C23×C4—1221no—
284C23×C4C22×D6111yescharacteristic
285C22×C8—21no—
286C22×C8—1221no—
287C22⋊Q8—21no—
288C22⋊Q8—21no—
289C22⋊Q8—21no—
290C22⋊Q8—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—612no—
298C2×C4⋊C4—1222no—
299C2×C4⋊C4—1224no—
300C2×C4⋊C4—2444no—
301C2×C4⋊C4C22⋊D6112yes—
302C2×C4⋊C4C22⋊D6111yescharacteristic
303C2×C22⋊C4—21no—
304C2×C22⋊C4—1222no—
305C2×C22⋊C4—1222no—
306C2×C22⋊C4—1224no—
307C2×C22⋊C4—1221no—
308C2×C22⋊C4—1221no—
309C2×C22⋊C4—1221no—
310C2×C22⋊C4—1221no—
311C2×C22⋊C4—1221no—
312C2×C22⋊C4—1221no—
313C2×C22⋊C4—1221no—
314C2×C22⋊C4—2444no—
315#107—24no—
316#103—24no—
317C22⋊C8—21no—
318C22⋊C8—1221no—
319C22⋊C8—1221no—
320C22⋊C8C22⋊D6112yes—
321#95—612no—
322#95—1222no—
323#95—1221no—
324#95—1221no—
325C24—612no—
326C24—1221no—
327C24—1221no—
328C24—1221no—
329C24—1221no—
330C24—1221no—
331C24—1221no—
332C24—2442no—
333C24—2442no—
334C24D4×D6111yescharacteristic
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—42no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—1224no—
347C22×C4—1222no—
348C22×C4—1221no—
349C22×C4—1221no—
350C22×C4—1222no—
351C22×C4—1221no—
352C22×C4—2442no—
353C22×C4—2442no—
354C22×C4—2441no—
355C22×C4—2441no—
356C22×C4D4×D6111yescharacteristic
357C22×C4D4×D6111yescharacteristic
358C22×C4#1175822411yescharacteristic
359C22×C4C2×C22⋊D6111yescharacteristic
360C22×C4C2×C22⋊D6111yescharacteristic
361C22×C4C2×C22⋊D6111yescharacteristic
362C2×C8—22no—
363C2×C8—42no—
364C2×C8—2442no—
365C2×C8—2441no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—4882no—
375C4⋊C4—2442no—
376C4⋊C4—2448no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—4882no—
380C22⋊C4—4882no—
381C22⋊C4—2448no—
382C22⋊C4—2441no—
383C22⋊C4—2441no—
384C22⋊C4—2441no—
385C22⋊C4—2441no—
386C22⋊C4—2441no—
387C22⋊C4—2441no—
388C23—22no—
389C23—22no—
390C23—1222no—
391C23—1222no—
392C23—1222no—
393C23—4882no—
394C23—4881no—
395C23—4881no—
396C23—4882no—
397C23—2442no—
398C23—2444no—
399C23—2442no—
400C23—2444no—
401C23—2442no—
402C23—2441no—
403C23—2441no—
404C23—2441no—
405C23—2441no—
406C23—2442no—
407C23#1451974511yescharacteristic
408C23#1451974511yescharacteristic
409C23#1451974811yescharacteristic
410C23Q16×D6111yescharacteristic
411C23#1451980011yescharacteristic
412C23#1451980911yescharacteristic
413C23#1451980911yescharacteristic
414Q8—42no—
415Q8—81no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—1224no—
432C2×C4—4881no—
433C2×C4—2442no—
434C2×C4—2442no—
435C2×C4—2442no—
436C2×C4—2441no—
437C2×C4—2442no—
438C2×C4—2442no—
439C2×C4—2444no—
440C2×C4—2442no—
441C2×C4—2444no—
442C2×C4—2442no—
443C2×C4#1451974611yescharacteristic
444C2×C4#1451974911yescharacteristic
445C2×C4#1451974911yescharacteristic
446C2×C4#1451977311yescharacteristic
447C8—42no—
448C8—4881no—
449C22—22no—
450C22—22no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—22no—
456C22—22no—
457C22—4882no—
458C22—4882no—
459C22—4882no—
460C22—4882no—
461C22—4881no—
462C22—4881no—
463C22—2444no—
464C22—2442no—
465C22—2444no—
466C22#1683967011yescharacteristic
467C22#1683991611yescharacteristic
468C22#1683991611yescharacteristic
469C22#1683992311yescharacteristic
470C22#1683992311yescharacteristic
471C22#1683992812yes—
472C22#1683995112yes—
473C22#1684000011yescharacteristic
474C22#1684001711yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—81no—
480C4—81no—
481C4—82no—
482C4—4882no—
483C4—4881no—
484C4—2444no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—4882no—
490C2#2144484212yes—
491C2#2144486612yes—
492C2#2144682611yescharacteristic
493C2#2144683111yescharacteristic
494C2#2144694711yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#484C2
socleC23×C61#120
prime radicalC24×D61D4
hypercenterC2×#484D61
hypocenterC61#9928
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
3. #21445909C2semidirect
4. C2×#16839499C2semidirect
6. #21443699C2semidirect
7. C2×C61×#484C2semidirect
238. C2×#484D61semidirect
278. C61#9928semidirect

Maximal subgroups

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

subgroupindex
#214516262
C2×#168396972
#214459092
C2×#168394992
C2×#168394852
#214436992
C2×C61×#4842
#992861