Finite Group Databasebeta

← #30624039

Subgroups — #30624039

Order 15616.

1176 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462431C212yesmaximal, complemented
2#21462123C212yesmaximal, complemented
3#21462116C214yesmaximal, complemented
4#21462112C212yesmaximal, complemented
5C2×#16840673C214yesmaximal, complemented
6#21459604C211yesmaximal, characteristic
7#16840840—22no—
8#16840749—24no—
9#16840748C2214yes—
10D61⋊#101—22no—
11C2×#14519838C2218yes—
12#16840673C2214yes—
13#16840673C2218yes—
14#16840671C2214yes—
15C2×#14519824C411yescharacteristic
16C2×#14519824C2211yescharacteristic
17Q8×D244—22no—
18#16840416—22no—
19#16840416—24no—
20#16840338C412yes—
21#16840338C411yescharacteristic
22#16840338C2211yescharacteristic
23#16840338C2212yes—
24#16840312C412yes—
25#16840312C2211yescharacteristic
26C2×#14519777C411yescharacteristic
27C2×#14519777C2212yes—
28C4×#41×C61C411yescharacteristic
29#14519864—48no—
30D61⋊M16—24no—
31C2×#11758217—28no—
32#14519838—28no—
33#14519838C2318yes—
34#14519837—24no—
35#14519824—44no—
36#14519824—44no—
37C2×#11758222C2×C412yes—
38C2×#11758222C2×C411yescharacteristic
39C2×#11758222C2×C412yes—
40C2×#11758222C2311yescharacteristic
41C2×#11758224—22no—
42C2×#11758224—21no—
43C2×#11758224D412yes—
44C2×#11758224D414yes—
45C22×Dic122C2×C411yescharacteristic
46C22×Dic122C2×C412yes—
47C22×Dic122C2×C411yescharacteristic
48C22×Dic122C2312yes—
49C2×Q8×D61—21no—
50C2×Q8×D61D412yes—
51#14519777—22no—
52#14519777—24no—
53#14519777C2×C412yes—
54#14519777C2×C412yes—
55#14519777C2×C412yes—
56#14519777C2311yescharacteristic
57#14519777C2312yes—
58#14519774C2×C411yescharacteristic
59#14519765C2×C412yes—
60#14519765C2×C412yes—
61#14519765C2×C411yescharacteristic
62#14519765C2311yescharacteristic
63#14519761—24no—
64#14519761—22no—
65#14519761—22no—
66#14519761—24no—
67#14519757—24no—
68#14519757—22no—
69#14519757—22no—
70#14519757—24no—
71#14519751—24no—
72#14519751—22no—
73C4×D244—24no—
74C4×D244—22no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61—24no—
77C2×#41×C61C2×C411yescharacteristic
78C4×Q8×C61—21no—
79C4×D4×C61—22no—
80C4×D4×C61—21no—
81#117×C61C2×C411yescharacteristic
82#117×C61C2×C412yes—
83C2×C42×C61C2×C411yescharacteristic
84C2×C42×C61C2×C412yes—
85#41×C61—24no—
86C2×Q8×C61C22⋊C411yescharacteristic
87C2×D4×C61C22⋊C412yes—
88C2×D4×C61C22⋊C411yescharacteristic
89C22×C4×C61—21no—
90C22×C4×C61—22no—
91C22×C4×C61C22×C411yescharacteristic
92C22×C4×C61C22×C412yes—
93C2×C22⋊D61—21no—
94C2×C22⋊D61—22no—
95C2×C22⋊D61C22⋊C414yes—
96C2×C22⋊D61C2×D412yes—
97C22×Dic61—21no—
98C22×Dic61—22no—
99C22×Dic61C22⋊C412yes—
100C22×Dic61C2×D414yes—
101Q8×D61—44no—
102Q8×D61—44no—
103#11758224—28no—
104#11758224—48no—
105#11758224—48no—
106#11758224—44no—
107#11758222—42no—
108#11758222—44no—
109#11758222—44no—
110#11758222—42no—
111C2×D244—22no—
112C2×D244C22×C411yescharacteristic
113C2×D244C22×C411yescharacteristic
114C2×C4×D61—22no—
115C2×C4×D61—21no—
116C2×C4×D61C22⋊C414yes—
117C2×C4×D61C2×D412yes—
118C2×Dic122—22no—
119C2×Dic122—21no—
120C2×Dic122—42no—
121C2×Dic122—44no—
122C2×Dic122—44no—
123C2×Dic122—42no—
124C2×Dic122C22⋊C412yes—
125C2×Dic122C22×C412yes—
126C2×Dic122C22×C411yescharacteristic
127C2×Dic122C22×C412yes—
128C2×Dic122C2×D414yes—
129C2×Dic122C2411yescharacteristic
130#11758217—28no—
131#11758217—48no—
132C2×#92420—48no—
133#11758213—44no—
134D61⋊C8—44no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—21no—
137C22⋊C4×C61—21no—
138C22⋊C4×C61—22no—
139C42×C61—22no—
140C42×C61—21no—
141C42×C61C22×C411yescharacteristic
142C42×C61C22×C412yes—
143C42×C61C22×C411yescharacteristic
144#11758198—24no—
145#11758198—22no—
146#11758198—24no—
147#11758198—22no—
148C4⋊Dic61—24no—
149C4⋊Dic61—24no—
150C4⋊Dic61—24no—
151C4⋊Dic61—22no—
152C4⋊Dic61—24no—
153C23×C61C2×C22⋊C411yescharacteristic
154C23×C61C2×C22⋊C412yes—
155C22×D61—21no—
156C22×D61C2×C22⋊C412yes—
157Q8×C61—22no—
158D4×C61—24no—
159D4×C61#9914yes—
160C2×C4×C61—21no—
161C2×C4×C61—24no—
162C2×C4×C61—22no—
163C2×C4×C61—21no—
164C2×C4×C61—22no—
165C2×C4×C61—44no—
166C2×C4×C61—42no—
167C2×C4×C61C2×C22⋊C412yes—
168C2×C4×C61C2×C22⋊C411yescharacteristic
169C2×C4×C61C23×C411yescharacteristic
170C22⋊D61—28no—
171C22⋊D61—42no—
172C22⋊D61—42no—
173C22⋊D61—44no—
174C2×Dic61—28no—
175C2×Dic61—22no—
176C2×Dic61—21no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—44no—
180C2×Dic61C2×C22⋊C414yes—
181C2×Dic61C22×D412yes—
182D244—41no—
183D244—41no—
184D244—44no—
185C4×D61—24no—
186C4×D61—44no—
187C4×D61—44no—
188C4×D61—42no—
189Dic122—24no—
190Dic122—42no—
191Dic122—41no—
192Dic122—44no—
193Dic122—44no—
194Dic122—42no—
195Dic122—41no—
196Dic122—42no—
197#92420—48no—
198#33372—611nomaximal, Sylow
199C22×C61—21no—
200C22×C61—42no—
201C22×C61—41no—
202C22×C61C2×#9914yes—
203C22×C61C22×C22⋊C411yescharacteristic
204C2×D61—22no—
205C2×D61—42no—
206C2×D61—42no—
207C2×D61—41no—
208C4×C61—22no—
209C4×C61—21no—
210C4×C61—22no—
211C4×C61—41no—
212C4×C61—42no—
213C4×C61C2×#9912yes—
214Dic61—22no—
215Dic61—24no—
216Dic61—41no—
217Dic61—42no—
218#3499—611no—
219#2889—612no—
220#2135—612no—
221#2133—614no—
222#2132—612no—
223C2×#356—614no—
224C2×C61—21no—
225C2×C61—24no—
226C2×C61—41no—
227C2×C61C22×#9911yescharacteristic
228D61—42no—
229D61—81no—
230C2×2−1+4—611no—
231C2×2−1+4—611no—
232#568—1222no—
233D4×Q8—1222no—
234#547—1224no—
235#547—1222no—
236#536—612no—
237#536—611no—
238#536—612no—
239#536—611no—
240#532—611no—
241#532—612no—
242C2×C4⋊Q8—612no—
243C2×C4⋊Q8—611no—
244C4×#41#692311yescharacteristic
245#413—614no—
246#413—1224no—
247C2×#101—618no—
248C2×#101—1222no—
249#356—614no—
250#356—618no—
251#355—614no—
252C61#3337211yescharacteristic, complemented, Sylow
2532−1+4—2444no—
2542−1+4—2444no—
255C2×#41—614no—
256C2×#41—612no—
257C2×#41—612no—
258C2×#41—611no—
259C2×#41—612no—
260C2×#41—611no—
261C2×#41—1222no—
262C2×#41—1221no—
263C2×#41C2×#692311yescharacteristic
264C22×Q8—612no—
265C22×Q8—611no—
266C22×Q8—612no—
267C22×Q8—611no—
268C22×Q8—612no—
269C22×Q8—1221no—
270#131—2448no—
271C2×M16—1228no—
272C2×M16—1224no—
273C4⋊Q8—611no—
274C4⋊Q8—612no—
275C4⋊Q8—612no—
276C4⋊Q8—612no—
277C4⋊Q8—612no—
278C4⋊Q8—1222no—
279C4⋊Q8—1224no—
280#127—611no—
281#124—611no—
282#124—612no—
283#124—611no—
284#124—612no—
285#123—1224no—
286#123—1222no—
287#123—1224no—
288#123—1222no—
289C22⋊Q8—1222no—
290C22⋊Q8—1224no—
291C22⋊Q8—1222no—
292C22⋊Q8—1224no—
293#121—1224no—
294#121—1222no—
295C4×Q8—21no—
296C4×D4—22no—
297C4×D4—21no—
298C4×D4—1222no—
299C4×D4—1224no—
300#117C2×#692312yes—
301#117C2×#692311yescharacteristic
302C2×C4⋊C4—1224no—
303C2×C4⋊C4—1222no—
304C2×C42C2×#692312yes—
305C2×C42C2×#692311yescharacteristic
306#101—618no—
307#101—1228no—
308#100—1224no—
309#41—24no—
310#41—1228no—
311#41—2442no—
312#41—2444no—
313#41—2444no—
314#41—2442no—
315#41—2448no—
316#41—2448no—
317#41—2444no—
318C2×Q8—612no—
319C2×Q8—614no—
320C2×Q8—611no—
321C2×Q8—612no—
322C2×Q8—611no—
323C2×Q8—612no—
324C2×Q8—1222no—
325C2×Q8—1221no—
326C2×Q8—2444no—
327C2×Q8—2442no—
328C2×Q8—2442no—
329C2×Q8—2444no—
330C2×Q8—2444no—
331C2×Q8—2444no—
332C2×Q8C22⋊#692311yescharacteristic
333C2×D4—614no—
334C2×D4—612no—
335C2×D4—611no—
336C2×D4—611no—
337C2×D4—1222no—
338C2×D4—1221no—
339C2×D4—1222no—
340C2×D4C22⋊#692312yes—
341C2×D4C22⋊#692311yescharacteristic
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—614no—
345C22×C4—612no—
346C22×C4—612no—
347C22×C4—614no—
348C22×C4—1221no—
349C22×C4—1222no—
350C22×C4—1222no—
351C22×C4—1221no—
352C22×C4C22×#692312yes—
353C22×C4C22×#692311yescharacteristic
354M16—1228no—
355M16—2444no—
356M16—2448no—
357C2×C8—2444no—
358C2×C8—2448no—
359C4⋊C4—22no—
360C4⋊C4—21no—
361C4⋊C4—1224no—
362C4⋊C4—1222no—
363C4⋊C4—1224no—
364C4⋊C4—1224no—
365C4⋊C4—1224no—
366C22⋊C4—21no—
367C22⋊C4—22no—
368C22⋊C4—1224no—
369C22⋊C4—1222no—
370C22⋊C4—1222no—
371C22⋊C4—1224no—
372C42—22no—
373C42—21no—
374C42C22×#692311yescharacteristic
375C42C22×#692311yescharacteristic
376C42C22×#692312yes—
377C23—612no—
378C23—1221no—
379C23C2×C22⋊#692311yescharacteristic
380C23C2×C22⋊#692312yes—
381Q8—22no—
382Q8—1224no—
383Q8—2441no—
384Q8—2442no—
385Q8—2444no—
386Q8—2442no—
387Q8—2441no—
388Q8—2444no—
389Q8—2442no—
390D4—24no—
391D4—1228no—
392D4—2441no—
393D4—2441no—
394D4—2444no—
395D4—2442no—
396D4—2442no—
397D4—2444no—
398D4#1451983614yes—
399C2×C4—24no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—614no—
407C2×C4—612no—
408C2×C4—1228no—
409C2×C4—1224no—
410C2×C4—1222no—
411C2×C4—1221no—
412C2×C4—2444no—
413C2×C4—2444no—
414C2×C4—2442no—
415C2×C4—2442no—
416C2×C4—2444no—
417C2×C4—2442no—
418C2×C4C2×C22⋊#692312yes—
419C2×C4C2×C22⋊#692311yescharacteristic
420C2×C4C23×#692311yescharacteristic
421C8—2448no—
422C22—21no—
423C22—42no—
424C22—41no—
425C22—1222no—
426C22—2442no—
427C22—2442no—
428C22—2441no—
429C22C2×#1451983614yes—
430C22C22×C22⋊#692311yescharacteristic
431C4—22no—
432C4—21no—
433C4—22no—
434C4—42no—
435C4—41no—
436C4—1222no—
437C4—1224no—
438C4—2442no—
439C4—2441no—
440C4C2×#1451983612yes—
441C2—21no—
442C2—24no—
443C2—41no—
444C2—4881no—
445C2—2442no—
446C2C22×#1451983611yescharacteristic
subgroupG/H
centerC2C22×#14519836
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC4×#41×C61C4
socleC2×C61C22×#99
prime radical#21459604C2
cosocleC2×Dic122C24
hypercenterC4×#41#6923
hypocenterC61#33372
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×D4 ◁ C2×#41 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
1. #21462431C2semidirect
2. #21462123C2semidirect
3. #21462116C2semidirect
4. #21462112C2semidirect
5. C2×#16840673C2semidirect
252. C61#33372semidirect

Maximal subgroups

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

subgroupindex
#214624312
#214621232
#214621162
#214621122
C2×#168406732
#214596042
#3337261