Finite Group Databasebeta

← #29731851

Subgroups — #29731851

Order 15616.

838 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840043C211yesmaximal, characteristic
2#21450804C211yesmaximal, characteristic, complemented
3#21450737C211yesmaximal, characteristic
4#21447307C212yesmaximal, complemented
5#21447206C212yesmaximal
6#21446780C211yesmaximal, characteristic, complemented
7#21446578C211yesmaximal, characteristic, complemented
8#21446552C212yesmaximal, complemented
9#21446502C212yesmaximal
10C4×#14519699C211yesmaximal, characteristic
11C2×C8×Q8×C61C211yesmaximal, characteristic, complemented
12C2×#14519777C2211yescharacteristic
13C2×#14519775—21no—
14C2×#14519753—21no—
15C2×C4×D244C2211yescharacteristic
16#16840043—24no—
17C2×#14519711C2212yes—
18C2×#14519711C2212yes—
19C2×#14519699C2211yescharacteristic
20C2×#14519699C2212yes—
21C2×#14519699C2211yescharacteristic
22#16839692—22no—
23#16839684—22no—
24#16839667—21no—
25#16839664C2212yes—
26#16839660C2212yes—
27#16839660C2212yes—
28#16839660C2212yes—
29#16839632C2211yescharacteristic
30#16839619—21no—
31C4×C4⋊Dic61C2211yescharacteristic
32#16839525C2211yescharacteristic
33#16839525C2211yescharacteristic
34#16839525C2212yes—
35#16839525C2212yes—
36#16839525C2212yes—
37C2×C4×Q8×C61C2211yescharacteristic
38C8×Q8×C61—24no—
39C2×C4⋊C8×C61C2212yes—
40C2×C4⋊C8×C61C2211yescharacteristic
41C2×C4×C8×C61C2211yescharacteristic
42C2×C4×C8×C61C2212yes—
43C22×Dic122—21no—
44C22×D244C2311yescharacteristic
45C22×C4×D61—21no—
46#14519777—22no—
47#14519777—22no—
48#14519775—44no—
49#14519753—44no—
50C4×D244—22no—
51C4×D244—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×C4⋊Dic61C2312yes—
57C2×C4⋊Dic61C2312yes—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758196—22no—
61C2×#11758196—22no—
62C2×#11758196—22no—
63C2×#11758198—21no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×C4×Dic61—21no—
67#14519711—24no—
68#14519711—24no—
69#14519699—24no—
70#14519699—22no—
71#14519699Q814yescomplemented
72#14519684—22no—
73#14519684—21no—
74#14519684—22no—
75#14519684—22no—
76C22×Q8×C61D411yescharacteristic
77C22×C8×C61C2311yescharacteristic
78C22×C8×C61C2312yes—
79C22×C8×C61C2311yescharacteristic
80C4×Q8×C61—22no—
81C4×Q8×C61—22no—
82C2×C4⋊C4×C61D412yes—
83C2×C4⋊C4×C61D411yescharacteristic
84C2×C42×C61C2312yes—
85C2×C42×C61C2311yescharacteristic
86C4⋊C8×C61—24no—
87C4⋊C8×C61D414yes—
88C4×C8×C61—24no—
89C4×C8×C61—22no—
90C23×D61—21no—
91C2×Q8×C61—22no—
92C2×Q8×C61—22no—
93C2×Q8×C61D814yes—
94C22×C4×C61C2×D412yes—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2411yescharacteristic
99C22×Dic61—21no—
100C22×Dic61—21no—
101C22×Dic61—21no—
102C22×Dic61—22no—
103C22×Dic61—22no—
104C2×D244—21no—
105C2×D244—21no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244—21no—
109C2×D244C2×Q812yes—
110C2×C4×D61—42no—
111C2×C4×D61—42no—
112C2×C4×D61—42no—
113C2×Dic122—42no—
114C2×Dic122—42no—
115C2×Dic122—42no—
116C2×C8×C61—21no—
117C2×C8×C61—22no—
118C2×C8×C61—22no—
119C2×C8×C61—21no—
120C2×C8×C61—21no—
121C2×C8×C61—21no—
122C2×C8×C61—21no—
123C2×C8×C61C2×D414yes—
124C2×C8×C61C2×Q812yes—
125C4⋊C4×C61—24no—
126C4⋊C4×C61—22no—
127C42×C61—22no—
128C42×C61—22no—
129C42×C61C2×D412yes—
130C42×C61#4112yes—
131#11758198—42no—
132#11758198—44no—
133#11758198—82no—
134C4⋊Dic61—21no—
135C4⋊Dic61—21no—
136C4⋊Dic61—22no—
137C4⋊Dic61—21no—
138C4⋊Dic61—22no—
139C4⋊Dic61—21no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—42no—
143C4⋊Dic61—42no—
144C4⋊Dic61—81no—
145C4⋊Dic61C2×Q812yes—
146C4⋊Dic61#4112yes—
147#11758196—44no—
148#11758196—44no—
149#11758196—44no—
150C4×Dic61—42no—
151C23×C61C22×D411yescharacteristic
152C22×D61—22no—
153C22×D61—41no—
154C22×D61—41no—
155C22×D61—41no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—41no—
159Q8×C61—28no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61#12112yes—
168C2×C4×C61#12112yes—
169C2×C4×C61C2×D814yes—
170C2×C4×C61C2×D812yes—
171C2×C4×C61#13512yes—
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C22×Q811yescharacteristic
174C2×C4×C61C2×#4111yescharacteristic
175C2×C4×C61C2×#4111yescharacteristic
176C2×C4×C612−1+411yescharacteristic
177C2×C4×C612−1+411yescharacteristic
178C2×Dic61—22no—
179C2×Dic61—22no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—42no—
185C2×Dic61—41no—
186C2×Dic61—42no—
187C2×Dic61—42no—
188C2×Dic61—41no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61—41no—
192C2×Dic61—42no—
193D244—22no—
194D244—22no—
195D244—41no—
196D244—41no—
197C4×D61—84no—
198Dic122—84no—
199C8×C61—24no—
200C8×C61—22no—
201C8×C61—41no—
202#25469—611nomaximal, Sylow
203C22×C61C2×#12111yescharacteristic
204C22×C61#54811yescharacteristic
205C22×C61D4×Q811yescharacteristic
206C22×C61C22×D811yescharacteristic
207C22×C61C2×#13511yescharacteristic
208C22×C61#57711yescharacteristic
209C22×C61#57811yescharacteristic
210C2×D61—42no—
211C2×D61—42no—
212C2×D61—42no—
213C2×D61—81no—
214C2×D61—81no—
215C2×D61—81no—
216C2×D61—81no—
217C4×C61—24no—
218C4×C61—22no—
219C4×C61—42no—
220C4×C61#46614yes—
221C4×C61#53311yescharacteristic
222C4×C61#54111yescharacteristic
223C4×C61#54711yescharacteristic
224C4×C61D4×Q811yescharacteristic
225Dic61—42no—
226Dic61—42no—
227Dic61—82no—
228Dic61—82no—
229Dic61—81no—
230C2×#474—611no—
231C2×C8×Q8D6111yescharacteristic, complemented
232#2546—611no—
233#2538—611no—
234#2086—612no—
235#2067—612no—
236#1990—611no—
237#1947—611no—
238#1942—612no—
239#1934—612no—
240C4×#102—611no—
241C2×C61#242911yescharacteristic
242C2×C61C2×#46611yescharacteristic
243C2×C61#306311yescharacteristic
244C2×C61#335811yescharacteristic
245C2×C61#336411yescharacteristic
246C2×C61Q8×D811yescharacteristic
247C2×C61#338111yescharacteristic
248D61—82no—
249C2×C4⋊Q8—611no—
250C2×C4⋊Q8—1221no—
251C2×C22⋊Q8—1221no—
252C2×C4×Q8C2×D6111yescharacteristic
253C2×C4×D4—611no—
254#474—1224no—
255C8×Q8—24no—
256C2×#107—612no—
257C2×#107—612no—
258C2×C4⋊C8C2×D6112yes—
259C2×C4⋊C8C2×D6111yescharacteristic
260C2×#102—611no—
261C2×#102—611no—
262C2×#102—612no—
263C2×C4×C8C2×D6112yes—
264C2×C4×C8C2×D6111yescharacteristic
265#398—1222no—
266#396—1222no—
267#391—1221no—
268#390—612no—
269#389—612no—
270#389—612no—
271#389—612no—
272#384—611no—
273#380—1221no—
274C4×C4⋊C4—611no—
275#340—612no—
276#340—611no—
277#340—612no—
278#340—612no—
279#340—611no—
280C61#2546911yescharacteristic, complemented, Sylow
281C22×Q8—1221no—
282C22×Q8D24411yescharacteristic
283C22×D4—611no—
284C23×C4—1221no—
285C22×C8C22×D6111yescharacteristic
286C22×C8C22×D6111yescharacteristic
287C22×C8C22×D6112yes—
288C4⋊Q8—1222no—
289C4⋊Q8—1222no—
290C4⋊Q8—2444no—
291C22⋊Q8—2444no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×D4—1222no—
295C4×D4—1222no—
296C2×C4⋊C4—611no—
297C2×C4⋊C4—611no—
298C2×C4⋊C4—611no—
299C2×C4⋊C4—612no—
300C2×C4⋊C4—612no—
301C2×C4⋊C4—1221no—
302C2×C4⋊C4—1221no—
303C2×C4⋊C4—1222no—
304C2×C4⋊C4—1221no—
305C2×C4⋊C4—1222no—
306C2×C4⋊C4—1222no—
307C2×C4⋊C4D24411yescharacteristic
308C2×C4⋊C4D24412yes—
309C2×C22⋊C4—1222no—
310C2×C22⋊C4—1221no—
311C2×C22⋊C4—1222no—
312C2×C42—1221no—
313C2×C42C22×D6112yes—
314C2×C42C22×D6111yescharacteristic
315#107—1224no—
316#107—1224no—
317C4⋊C8—24no—
318C4⋊C8C22⋊D6114yes—
319#102—614no—
320#102—1222no—
321#102—1224no—
322C4×C8—24no—
323C4×C8—22no—
324#95—1221no—
325#95—1222no—
326#95—1222no—
327#95—1222no—
328C24—1221no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—2442no—
332C2×Q8—2442no—
333C2×Q8—2442no—
334C2×Q8D48814yes—
335C2×D4—612no—
336C2×D4—1221no—
337C2×D4—1221no—
338C2×D4—1221no—
339C2×D4—1221no—
340C2×D4—1221no—
341C22×C4—1221no—
342C22×C4—1221no—
343C22×C4—1221no—
344C22×C4—1222no—
345C22×C4—1222no—
346C22×C4—2442no—
347C22×C4—2442no—
348C22×C4—2442no—
349C22×C4C2×D24412yes—
350C22×C4C2×D24411yescharacteristic
351C22×C4C2×D24412yes—
352C22×C4C2×D24411yescharacteristic
353C22×C4C23×D6111yescharacteristic
354C2×C8—21no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8Q8×D6112yes—
362C2×C8C2×C22⋊D6114yes—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—612no—
366C4⋊C4—612no—
367C4⋊C4—1221no—
368C4⋊C4—1221no—
369C4⋊C4—1222no—
370C4⋊C4—1221no—
371C4⋊C4—1222no—
372C4⋊C4—1221no—
373C4⋊C4—1222no—
374C4⋊C4—1222no—
375C4⋊C4—4881no—
376C4⋊C4—2442no—
377C4⋊C4—2444no—
378C4⋊C4—2444no—
379C4⋊C4—2444no—
380C4⋊C4—2442no—
381C22⋊C4—4882no—
382C22⋊C4—2442no—
383C22⋊C4—2444no—
384C42—22no—
385C42—22no—
386C42—2442no—
387C42#1175822212yes—
388C42C2×C22⋊D6112yes—
389C23—1222no—
390C23—2441no—
391C23—2441no—
392C23—2441no—
393C23—2441no—
394C23—2441no—
395C23—2441no—
396C23C22×D24411yescharacteristic
397Q8—28no—
398Q8—4884no—
399D4—1222no—
400D4—1222no—
401D4—2441no—
402D4—2441no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—1222no—
411C2×C4—1222no—
412C2×C4—4884no—
413C2×C4—2441no—
414C2×C4—2442no—
415C2×C4—2442no—
416C2×C4—2441no—
417C2×C4—2442no—
418C2×C4—2441no—
419C2×C4—2442no—
420C2×C4—2442no—
421C2×C4—2442no—
422C2×C4—2441no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4—2441no—
426C2×C4#1451975112yes—
427C2×C4#1451975112yes—
428C2×C4C2×D48814yes—
429C2×C4C2×D48812yes—
430C2×C4#1451979912yes—
431C2×C4C22×C22⋊D6111yescharacteristic
432C2×C4C2×Q8×D6111yescharacteristic
433C2×C4#41×D6111yescharacteristic
434C2×C4C2×#1175822211yescharacteristic
435C2×C4#1451982411yescharacteristic
436C2×C4#1451982511yescharacteristic
437C8—24no—
438C8—22no—
439C8—41no—
440C22—4881no—
441C22—4881no—
442C22—4881no—
443C22—4881no—
444C22—2442no—
445C22—2442no—
446C22—2442no—
447C22C2×#1451975111yescharacteristic
448C22#1684042411yescharacteristic
449C22Q8×D24411yescharacteristic
450C22C22×D48811yescharacteristic
451C22C2×#1451979911yescharacteristic
452C22#1684057411yescharacteristic
453C22#1684057911yescharacteristic
454C4—24no—
455C4—22no—
456C4—42no—
457C4—4882no—
458C4—4881no—
459C4—4882no—
460C4—2442no—
461C4—2442no—
462C4#1683999814yes—
463C4#1684031611yescharacteristic
464C4#1684037011yescharacteristic
465C4#1684041111yescharacteristic
466C4Q8×C22⋊D6111yescharacteristic
467C2—4882no—
468C2#2144941211yescharacteristic
469C2C2×#1683999811yescharacteristic
470C2#2145551011yescharacteristic
471C2#2145845511yescharacteristic
472C2#2145850611yescharacteristic
473C2Q8×D48811yescharacteristic
474C2#2145869611yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C8×Q8×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×C8×Q8D61
hypocenterC61#25469
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8

Complements

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

NH ≅ G/Ntype
2. #21450804C2semidirect
4. #21447307C2semidirect
6. #21446780C2semidirect
7. #21446578C2semidirect
8. #21446552C2semidirect
11. C2×C8×Q8×C61C2semidirect
71. #14519699Q8semidirect
231. C2×C8×Q8D61semidirect
280. C61#25469semidirect

Maximal subgroups

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

subgroupindex
C2×#168400432
#214508042
#214507372
#214473072
#214472062
#214467802
#214465782
#214465522
#214465022
C4×#145196992
C2×C8×Q8×C612
#2546961