Finite Group Databasebeta

← #29576205

Subgroups — #29576205

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840049C211yesmaximal, characteristic
2C2×#16839839C211yesmaximal, characteristic
3#21446781C211yesmaximal, characteristic, complemented
4#21445820C211yesmaximal, characteristic
5#21443703C211yesmaximal, characteristic, complemented
6#21443653C211yesmaximal, characteristic, complemented
7C2×C61×#333C211yesmaximal, characteristic, complemented
8C22×#11758189—21no—
9C2×C4×D244C2211yescharacteristic
10#16840049—22no—
11#16840049C414yescomplemented
12#16839839—22no—
13#16839839—22no—
14C2×#14519708—21no—
15C2×C4⋊#92418C2211yescharacteristic
16C2×#14519699—21no—
17C2×#14519697—21no—
18C2×#14519689—21no—
19C2×C4×#92418C2211yescharacteristic
20#16839632C2211yescharacteristic
21#16839525—21no—
22#16839524—21no—
23#16839524—21no—
24C2×C4⋊Q8×C61C2211yescharacteristic
25C2×C4⋊C8×C61C2211yescharacteristic
26C2×C4⋊C8×C61C2211yescharacteristic
27C61×#333—22no—
28C22×D244D411yescharacteristic
29C22×C4×D61—21no—
30C2×#11758189—24no—
31C2×#11758189—42no—
32C2×#11758189—42no—
33C22×#92418—21no—
34C22×#92418D411yescharacteristic
35C22×#92418D411yescharacteristic
36C4×D244—21no—
37C4×D244—21no—
38C4×D244C2×C412yes—
39C4×D244D412yes—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61D411yescharacteristic
44C2×#11758198—21no—
45#14519708—24no—
46C4⋊#92418—21no—
47C4⋊#92418C2×C412yes—
48#14519699—42no—
49#14519697—24no—
50#14519689—24no—
51C4×#92418—21no—
52C4×#92418—21no—
53C22×Q8×C61—21no—
54C22×C8×C61—21no—
55C22×C8×C61—21no—
56C4⋊Q8×C61—21no—
57C4⋊Q8×C61—21no—
58C4⋊Q8×C61—21no—
59C4⋊Q8×C61C2×C412yes—
60C2×C4⋊C4×C61—21no—
61C2×C4⋊C4×C61—21no—
62C2×C42×C61C2311yescharacteristic
63C4⋊C8×C61—21no—
64C4⋊C8×C61—21no—
65C4⋊C8×C61—21no—
66C4⋊C8×C61D412yes—
67C23×D61—21no—
68C2×Q8×C61—22no—
69C2×Q8×C61—41no—
70C2×Q8×C61—81no—
71C2×Q8×C61—81no—
72C22×C4×C61—41no—
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61#4111yescharacteristic
76C22×Dic61—21no—
77C22×Dic61—41no—
78C2×D244—21no—
79C2×D244—21no—
80C2×D244—21no—
81C2×D244—21no—
82C2×D244—21no—
83C2×D244C22⋊C412yes—
84C2×C4×D61—22no—
85C2×C4×D61—22no—
86C2×C4×D61—41no—
87C2×C4×D61—41no—
88C2×C4×D61—41no—
89C2×C4×D61—41no—
90C2×C8×C61—22no—
91C2×C8×C61—41no—
92C2×C8×C61—41no—
93C2×C8×C61—41no—
94C2×C8×C61—41no—
95C2×C8×C61—41no—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61—41no—
99C4⋊C4×C61—41no—
100C42×C61C22×C411yescharacteristic
101C42×C61C2×D411yescharacteristic
102C42×C61C2×D411yescharacteristic
103C42×C61#4111yescharacteristic
104#11758198—22no—
105#11758198—22no—
106C4⋊Dic61—21no—
107C4⋊Dic61—41no—
108C4⋊Dic61—41no—
109C4⋊Dic61—41no—
110C4⋊Dic61—41no—
111C4⋊Dic61—81no—
112C4⋊Dic61C22⋊C412yes—
113C2×#92418—21no—
114C2×#92418—21no—
115C2×#92418—22no—
116C2×#92418—21no—
117C2×#92418—21no—
118C2×#92418—21no—
119C2×#92418—41no—
120C2×#92418—41no—
121C2×#92418SD1612yes—
122#11758189—48no—
123C23×C61#12111yescharacteristic
124C22×D61—22no—
125C22×D61—41no—
126C22×D61—41no—
127C22×D61—41no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—41no—
131Q8×C61—82no—
132Q8×C61—82no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—42no—
136C2×C4×C61—81no—
137C2×C4×C61—81no—
138C2×C4×C61C2×C22⋊C411yescharacteristic
139C2×C4×C61C4×D411yescharacteristic
140C2×C4×C61C4×D411yescharacteristic
141C2×C4×C61#12011yescharacteristic
142C2×C4×C61#12111yescharacteristic
143C2×C4×C61#12311yescharacteristic
144C2×C4×C61C2×SD1611yescharacteristic
145C2×C4×C61#13511yescharacteristic
146C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C61#13711yescharacteristic
149C2×C4×C61#13711yescharacteristic
150C2×C4×C61#13711yescharacteristic
151C2×C4×C61#13711yescharacteristic
152C2×Dic61—22no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—81no—
156C2×Dic61—81no—
157C2×Dic61—81no—
158D244—22no—
159D244—22no—
160D244—41no—
161D244—41no—
162C4×D61—42no—
163C4×D61—42no—
164C4×D61—42no—
165C4×D61—42no—
166C8×C61—42no—
167C8×C61—81no—
168#92418—22no—
169#92418—42no—
170#92418—41no—
171#9223—611nomaximal, Sylow
172C22×C61#38611yescharacteristic
173C22×C61#46111yescharacteristic
174C22×C61#46311yescharacteristic
175C22×C61#46811yescharacteristic
176C22×C61#47011yescharacteristic
177C22×C61#47111yescharacteristic
178C22×C61#47211yescharacteristic
179C2×D61—42no—
180C2×D61—42no—
181C2×D61—42no—
182C2×D61—81no—
183C2×D61—81no—
184C2×D61—81no—
185C2×D61—81no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—82no—
189C4×C61#41711yescharacteristic
190C4×C61#41911yescharacteristic
191C4×C61C4×SD1611yescharacteristic
192C4×C61#44011yescharacteristic
193C4×C61#45011yescharacteristic
194C4×C61#45211yescharacteristic
195C4×C61#46811yescharacteristic
196C4×C61#47011yescharacteristic
197Dic61—42no—
198Dic61—161no—
199C2×#476—611no—
200C2×#436—611no—
201#1990—611no—
202#1802—611no—
203C2×#333D6111yescharacteristic, complemented
204#1296—611no—
205#1277—611no—
206C2×C61#169111yescharacteristic
207C2×C61#169511yescharacteristic
208C2×C61#169711yescharacteristic
209C2×C61#170311yescharacteristic
210C2×C61#187711yescharacteristic
211C2×C61#189411yescharacteristic
212C2×C61#190711yescharacteristic
213D61—82no—
214C22×M16—1221no—
215C2×C4⋊Q8C2×D6111yescharacteristic
216C2×C4×D4—611no—
217#476—614no—
218#476—1222no—
219#436—1222no—
220#436—1222no—
221C2×#106—1221no—
222C2×C4⋊C8—611no—
223C2×C4⋊C8C2×D6111yescharacteristic
224C2×C4⋊C8C2×D6111yescharacteristic
225C2×#102—1221no—
226C2×#102—1221no—
227C2×C22⋊C8—1221no—
228C2×C4×C8—611no—
229#384—611no—
230#340—1221no—
231#340—1221no—
232#340—1221no—
233#333—22no—
234C61#922311yescharacteristic, complemented, Sylow
235C22×Q8—21no—
236C22×D4—611no—
237C23×C4—1221no—
238C2×M16—1224no—
239C2×M16—2442no—
240C2×M16—2442no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—611no—
244C22×C8—611no—
245C22×C8—1221no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8C4×D6112yes—
250C4×D4—612no—
251C4×D4—612no—
252C4×D4—1221no—
253C4×D4—1221no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—611no—
257C2×C4⋊C4—1221no—
258C2×C4⋊C4—1221no—
259C2×C4⋊C4—1221no—
260C2×C22⋊C4—1221no—
261C2×C42C22×D6111yescharacteristic
262#106—1224no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—612no—
267C4⋊C8—1221no—
268C4⋊C8C22⋊D6112yes—
269#102—1224no—
270#102—2442no—
271C22⋊C8—1224no—
272C4×C8—1221no—
273C4×C8—1221no—
274C24—1221no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—81no—
278C2×Q8—81no—
279C2×D4—612no—
280C2×D4—1221no—
281C2×D4—1221no—
282C2×D4—1221no—
283C2×D4—1221no—
284C2×D4—1221no—
285C22×C4—41no—
286C22×C4—1221no—
287C22×C4—1222no—
288C22×C4—1222no—
289C22×C4—2441no—
290C22×C4—2441no—
291C22×C4—2441no—
292C22×C4—2441no—
293C22×C4—2441no—
294C22×C4#1175822211yescharacteristic
295C22×C4D4×D6111yescharacteristic
296C22×C4D4×D6111yescharacteristic
297M16—2448no—
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—612no—
305C2×C8—1221no—
306C2×C8—1221no—
307C2×C8—1221no—
308C2×C8—1222no—
309C2×C8—1221no—
310C2×C8—1221no—
311C2×C8—2441no—
312C2×C8—2441no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—612no—
318C4⋊C4—1221no—
319C4⋊C4—4881no—
320C4⋊C4—2441no—
321C4⋊C4—2441no—
322C4⋊C4—2441no—
323C4⋊C4—2441no—
324C22⋊C4—1222no—
325C22⋊C4—1222no—
326C42C2×C4×D6111yescharacteristic
327C42D4×D6111yescharacteristic
328C42#1175822411yescharacteristic
329C42C2×C22⋊D6111yescharacteristic
330C23—1222no—
331C23—2441no—
332C23—2441no—
333C23—2441no—
334C23—2441no—
335C23—2441no—
336C23—2441no—
337C23#1451975011yescharacteristic
338Q8—82no—
339Q8—82no—
340D4—1222no—
341D4—1222no—
342D4—2441no—
343D4—2441no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—1222no—
350C2×C4—4881no—
351C2×C4—4881no—
352C2×C4—4881no—
353C2×C4—2441no—
354C2×C4—2441no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4—2442no—
358C2×C4—2442no—
359C2×C4C22⋊C4×D6111yescharacteristic
360C2×C4#1451973711yescharacteristic
361C2×C4C4×C22⋊D6111yescharacteristic
362C2×C4#1451974511yescharacteristic
363C2×C4#1451974811yescharacteristic
364C2×C4#1451975911yescharacteristic
365C2×C4SD16×D6111yescharacteristic
366C2×C4#1451980011yescharacteristic
367C2×C4#1451980411yescharacteristic
368C2×C4#1451980411yescharacteristic
369C2×C4#1451980611yescharacteristic
370C2×C4#1451980911yescharacteristic
371C2×C4#1451980911yescharacteristic
372C2×C4#1451980911yescharacteristic
373C8—42no—
374C8—81no—
375C8—1222no—
376C8—2442no—
377C8—2441no—
378C22—4881no—
379C22—4881no—
380C22—4881no—
381C22—4881no—
382C22—2442no—
383C22—2442no—
384C22—2442no—
385C22#1683963811yescharacteristic
386C22#1683996611yescharacteristic
387C22#1683998011yescharacteristic
388C22#1684001111yescharacteristic
389C22#1684001911yescharacteristic
390C22#1684002411yescharacteristic
391C22#1684003111yescharacteristic
392C4—21no—
393C4—21no—
394C4—82no—
395C4—9761no—
396C4—2442no—
397C4#1683976211yescharacteristic
398C4#1683976911yescharacteristic
399C4#1683984711yescharacteristic
400C4#1683986511yescharacteristic
401C4#1683991111yescharacteristic
402C4#1683992311yescharacteristic
403C4#1684000511yescharacteristic
404C4#1684001711yescharacteristic
405C2—4882no—
406C2#2144532111yescharacteristic
407C2#2144534111yescharacteristic
408C2#2144535511yescharacteristic
409C2#2144539711yescharacteristic
410C2#2144615011yescharacteristic
411C2#2144624911yescharacteristic
412C2#2144632411yescharacteristic
subgroupG/H
centerC23#14519750
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC2×C61×#333C2
socleC23×C61#121
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenterC2×#333D61
hypocenterC61#9223
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
3. #21446781C2semidirect
5. #21443703C2semidirect
6. #21443653C2semidirect
7. C2×C61×#333C2semidirect
11. #16840049C4semidirect
203. C2×#333D61semidirect
234. C61#9223semidirect

Maximal subgroups

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

subgroupindex
C2×#168400492
C2×#168398392
#214467812
#214458202
#214437032
#214436532
C2×C61×#3332
#922361