Finite Group Databasebeta

← #30600675

Subgroups — #30600675

Order 15616.

534 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 92 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
1#21462620C211yesmaximal, characteristic, complemented
2#21461961C211yesmaximal, characteristic
3#21461326C211yesmaximal, characteristic
4#21461149C211yesmaximal, characteristic, complemented
5#21460689C211yesmaximal, characteristic, complemented
6#21460664C211yesmaximal, characteristic, complemented
7C2×#16839961C211yesmaximal, characteristic
8#16840814—22no—
9#16840812—21no—
10#16840811—21no—
11#16840779C2211yescharacteristic
12#16840765—21no—
13#16840765C2211yescharacteristic
14#16840765C2211yescharacteristic
15C2×#14519837—21no—
16C2×#14519837—21no—
17C4×#11758217—21no—
18C4×#11758217C2211yescharacteristic
19#16840620—22no—
20#16840612—21no—
21#16840612—21no—
22C22×#11758190C411yescharacteristic
23C22×#11758200C411yescharacteristic
24C2×#14519773C2211yescharacteristic
25C2×#14519742C2211yescharacteristic
26#16839961—21no—
27#16839961—21no—
28#16839961—21no—
29#16839961—21no—
30C2×#14519706C2211yescharacteristic
31C2×#14519697C411yescharacteristic
32C2×#102×C61C411yescharacteristic
33C2×#11758217—21no—
34C2×#11758217—21no—
35C2×#11758217D411yescharacteristic
36C2×#11758217D411yescharacteristic
37#14519844—22no—
38#14519844—21no—
39#14519844—21no—
40#14519844D412yes—
41C4⋊#92420—22no—
42C4⋊#92420—21no—
43C4⋊#92420—21no—
44#14519837—42no—
45#14519837—42no—
46C22⋊#92420—21no—
47#14519832—22no—
48#14519832—21no—
49C4×#92420—22no—
50C4×#92420—21no—
51C4×#92420—21no—
52C22×Dic122C2×C411yescharacteristic
53C22×D244C2×C411yescharacteristic
54C22×C22⋊D61—21no—
55C22×C22⋊D61—21no—
56C2×#11758190—21no—
57C2×#11758190—21no—
58C2×#11758190—42no—
59C2×#11758200—21no—
60C2×#11758200—21no—
61C2×#11758200—42no—
62C22×#92418C2×C411yescharacteristic
63#14519773—21no—
64#14519773—21no—
65#14519773—21no—
66#14519773—21no—
67#14519742—21no—
68#14519742—21no—
69#14519742D411yescharacteristic
70#14519742D411yescharacteristic
71#14519742D411yescharacteristic
72#14519742D411yescharacteristic
73C2×#11758196—21no—
74C2×C4×Dic61—21no—
75C2×C4×Dic61C2311yescharacteristic
76#14519706—21no—
77#14519706—21no—
78#14519697—21no—
79#14519697—21no—
80C22×D4×C61C2×C411yescharacteristic
81C22×C8×C61C2×C411yescharacteristic
82C2×C4⋊C4×C61C2×C411yescharacteristic
83#102×C61—21no—
84#102×C61—21no—
85C24×C61—21no—
86C23×D61—21no—
87C2×D4×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—41no—
90C2×D4×C61—41no—
91C22×C4×C61—21no—
92C22×C4×C61C22×C411yescharacteristic
93C2×C22⋊D61—41no—
94C2×C22⋊D61—41no—
95C2×C22⋊D61—41no—
96C2×C22⋊D61—41no—
97C2×C22⋊D61—41no—
98C2×C22⋊D61—41no—
99C2×C22⋊D61—41no—
100C2×C22⋊D61—41no—
101C2×C22⋊D61—82no—
102C22×Dic61—21no—
103C22×Dic61C2×D411yescharacteristic
104C22×Dic61#4111yescharacteristic
105C2×D244—21no—
106C2×D244—21no—
107C2×D244—41no—
108C2×D244—41no—
109C2×Dic122—22no—
110C2×Dic122—21no—
111C2×Dic122—41no—
112C2×Dic122C22⋊C411yescharacteristic
113C2×Dic122C22⋊C411yescharacteristic
114#11758217—21no—
115#11758217—21no—
116#11758217—21no—
117#11758217—41no—
118#11758217—41no—
119#11758217—41no—
120#11758217—41no—
121#11758217SD1612yescomplemented
122C2×#92420—21no—
123C2×#92420—21no—
124C2×#92420—22no—
125C2×#92420—41no—
126C2×#92420—41no—
127C2×C8×C61—21no—
128C2×C8×C61—41no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61C22⋊C411yescharacteristic
131C4⋊C4×C61C22⋊C411yescharacteristic
132#11758200—42no—
133#11758200—42no—
134#11758196—21no—
135#11758196—21no—
136#11758196—21no—
137#11758196—21no—
138C4×Dic61—21no—
139C4×Dic61—21no—
140C4×Dic61—21no—
141C4×Dic61—21no—
142C4×Dic61C2×D411yescharacteristic
143C4×Dic61C2×D411yescharacteristic
144C4×Dic61C2×D411yescharacteristic
145C4×Dic61#4111yescharacteristic
146C2×#92418—21no—
147C2×#92418—41no—
148#11758190—42no—
149#11758190—42no—
150C23×C61—41no—
151C23×C61—41no—
152C23×C61—41no—
153C23×C61—81no—
154C23×C61—81no—
155C23×C61C4×D411yescharacteristic
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61—81no—
160C22×D61—81no—
161D4×C61—41no—
162D4×C61—41no—
163D4×C61—81no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—81no—
169C2×C4×C61C2×C22⋊C411yescharacteristic
170C2×C4×C61C4×D411yescharacteristic
171C22⋊D61—82no—
172C22⋊D61—82no—
173C22⋊D61—81no—
174C22⋊D61—81no—
175C22⋊D61—81no—
176C22⋊D61—81no—
177C2×Dic61—21no—
178C2×Dic61—21no—
179C2×Dic61—41no—
180C2×Dic61—42no—
181C2×Dic61#12011yescharacteristic
182C2×Dic61#12111yescharacteristic
183C2×Dic61#12111yescharacteristic
184C2×Dic61#12311yescharacteristic
185C2×Dic61C2×SD1611yescharacteristic
186C2×Dic61#13511yescharacteristic
187C2×Dic61AGL(1,ℤ/8)11yescharacteristic
188C2×Dic61#13711yescharacteristic
189D244—41no—
190D244—41no—
191D244—81no—
192Dic122—21no—
193Dic122—21no—
194Dic122—21no—
195Dic122—42no—
196Dic122#10012yescomplemented
197#92420—22no—
198#92420—41no—
199#92420—42no—
200#92420—81no—
201#92420—81no—
202C8×C61—41no—
203#92418—41no—
204#9254—611nomaximal, Sylow
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—81no—
208C22×C61—81no—
209C22×C61—81no—
210C22×C61—161no—
211C22×C61—161no—
212C22×C61#38611yescharacteristic
213C22×C61C4×SD1611yescharacteristic
214C22×C61#44011yescharacteristic
215C2×D61—81no—
216C2×D61—81no—
217C2×D61—81no—
218C2×D61—161no—
219C2×D61—161no—
220C4×C61—21no—
221C4×C61—41no—
222C4×C61—41no—
223C4×C61C2×#10011yescharacteristic
224C4×C61#41311yescharacteristic
225Dic61—22no—
226Dic61—21no—
227Dic61—21no—
228Dic61—41no—
229Dic61#44811yescharacteristic
230Dic61#44911yescharacteristic
231Dic61#46011yescharacteristic
232Dic61#46411yescharacteristic
233C2×#460—611no—
234Q8×M16—611no—
235#1965—611no—
236#1545—611no—
237#1487—611no—
238#1292—611no—
239#1283—611no—
240C2×C61—21no—
241C2×C61—21no—
242C2×C61—161no—
243C2×C61#188011yescharacteristic
244C2×C61#190011yescharacteristic
245C2×C61#190311yescharacteristic
246D61—161no—
247C22×SD16—611no—
248C22×SD16—611no—
249C2×#127—611no—
250C2×C4×Q8—611no—
251#460—1221no—
252#460—1221no—
253#460—1221no—
254#460—1221no—
255#446—1222no—
256#446—1221no—
257C8×Q8—1221no—
258#431—611no—
259#423—611no—
260#423—611no—
261#423—1221no—
262C2×C4⋊C8—611no—
263C2×#102—611no—
264C2×#102#692311yescharacteristic
265C2×#100—1221no—
266C2×#100—1221no—
267C4×M16—611no—
268C4×M16—1221no—
269#331—1222no—
270Q8⋊C8—1221no—
271Q8⋊C8—1221no—
272C61#925411yescharacteristic, complemented, Sylow
273C22×Q8—611no—
274C22×D4—611no—
275C22×D4—1221no—
276C22×D4—1221no—
277C22×D4C2×#692311yescharacteristic
278C2×SD16—1221no—
279C2×SD16—1221no—
280C2×SD16—1221no—
281C2×SD16—1221no—
282C2×SD16—2442no—
283C2×SD16—2442no—
284C2×M16—611no—
285C2×M16—611no—
286C2×M16—1221no—
287C2×M16—1221no—
288C22×C8—611no—
289C22×C8C2×#692311yescharacteristic
290#127—1221no—
291#127—1221no—
292#127—1221no—
293#127—1221no—
294C4×Q8—611no—
295C4×Q8—611no—
296C4×Q8—611no—
297C4×Q8—611no—
298C4×Q8—1221no—
299C4×Q8—1221no—
300C2×C4⋊C4—1221no—
301C2×C4⋊C4C2×#692311yescharacteristic
302C2×C42—611no—
303C2×C42—1221no—
304C4⋊C8—612no—
305C4⋊C8—1221no—
306C4⋊C8—1221no—
307C4⋊C8—1221no—
308C4⋊C8—1221no—
309C4⋊C8—1221no—
310C4⋊C8—1221no—
311C4⋊C8—1222no—
312C4⋊C8—1222no—
313#102—21no—
314#102—21no—
315#102—1221no—
316#102—1221no—
317#100—2442no—
318#100—2442no—
319C22⋊C8—1221no—
320#97—1221no—
321#97—1222no—
322C4×C8—1221no—
323C4×C8—1221no—
324C4×C8—1222no—
325C24—21no—
326C24—1221no—
327C2×Q8—611no—
328C2×Q8—611no—
329C2×Q8—1222no—
330C2×Q8—1221no—
331C2×Q8—2441no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—1221no—
337C2×D4—1221no—
338C2×D4—4882no—
339C2×D4—2441no—
340C2×D4—2441no—
341C2×D4—2441no—
342C2×D4—2441no—
343C2×D4—2441no—
344C2×D4—2441no—
345C2×D4—2441no—
346C2×D4—2441no—
347C2×D4—2441no—
348C2×D4—2441no—
349C22×C4—21no—
350C22×C4—611no—
351C22×C4—611no—
352C22×C4—1221no—
353C22×C4C22×#692311yescharacteristic
354SD16—2442no—
355SD16—2442no—
356SD16—2442no—
357SD16—2442no—
358M16—612no—
359M16—1221no—
360M16—1221no—
361M16—1221no—
362M16—2441no—
363M16—2441no—
364M16—2441no—
365M16—2441no—
366C2×C8—21no—
367C2×C8—41no—
368C2×C8—1221no—
369C2×C8—1222no—
370C2×C8—1221no—
371C2×C8—1221no—
372C2×C8—2441no—
373C2×C8—2441no—
374C2×C8—2441no—
375C4⋊C4—21no—
376C4⋊C4—1221no—
377C4⋊C4—1221no—
378C4⋊C4—1221no—
379C4⋊C4—1221no—
380C4⋊C4C22⋊#692311yescharacteristic
381C4⋊C4C22⋊#692311yescharacteristic
382C42—611no—
383C42—611no—
384C42—611no—
385C42—611no—
386C42—1221no—
387C42—1221no—
388C42—1221no—
389C42—1221no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—81no—
394C23—81no—
395C23—4881no—
396C23—4881no—
397C23—2441no—
398C23—2441no—
399C23—2441no—
400C23D4×#692311yescharacteristic
401Q8—612no—
402Q8—1221no—
403Q8—1221no—
404Q8—1221no—
405Q8—2442no—
406D4—41no—
407D4—41no—
408D4—81no—
409D4—4881no—
410D4—4881no—
411D4—4881no—
412D4—4881no—
413D4—4881no—
414D4—4882no—
415D4—4882no—
416D4—2441no—
417D4—2441no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—81no—
423C2×C4—611no—
424C2×C4—611no—
425C2×C4—611no—
426C2×C4—611no—
427C2×C4—611no—
428C2×C4—611no—
429C2×C4—611no—
430C2×C4—611no—
431C2×C4—1221no—
432C2×C4—1221no—
433C2×C4—2441no—
434C2×C4—2442no—
435C2×C4C2×C22⋊#692311yescharacteristic
436C2×C4D4×#692311yescharacteristic
437C8—41no—
438C8—1222no—
439C8—4881no—
440C8—4881no—
441C8—2441no—
442C8—2442no—
443C8—2441no—
444C22—21no—
445C22—21no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—161no—
450C22—161no—
451C22—9761no—
452C22—9761no—
453C22—4881no—
454C22—4881no—
455C22—4881no—
456C22#1684071011yescharacteristic
457C22SD16×#692311yescharacteristic
458C22#1684080111yescharacteristic
459C4—21no—
460C4—41no—
461C4—41no—
462C4—611no—
463C4—611no—
464C4—611no—
465C4—611no—
466C4—1222no—
467C4—1221no—
468C4—1221no—
469C4—2441no—
470C4D61⋊#10011yescharacteristic
471C4#1684074811yescharacteristic
472C2—21no—
473C2—21no—
474C2—161no—
475C2—9761no—
476C2#2146181511yescharacteristic
477C2#2146184511yescharacteristic
478C2#2146185011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#102×C61C4
socleC22×C61#386
prime radicalC2×#14519773C22
cosocleC2×C4×Dic61C23
hypercenterC2×#102#6923
hypocenterC61#9254
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #21462620C2semidirect
4. #21461149C2semidirect
5. #21460689C2semidirect
6. #21460664C2semidirect
121. #11758217SD16semidirect
196. Dic122#100semidirect
272. C61#9254semidirect

Maximal subgroups

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

subgroupindex
#214626202
#214619612
#214613262
#214611492
#214606892
#214606642
C2×#168399612
#925461