Finite Group Databasebeta

← #29663977

Subgroups — #29663977

Order 15616.

670 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840113C211yesmaximal, characteristic, complemented
2#21455635C211yesmaximal, characteristic
3#21455307C211yesmaximal, characteristic
4C8×#11758222C211yesmaximal, characteristic
5#21445250C214yesmaximal, complemented
6#21445243C214yesmaximal, complemented
7#21444505C211yesmaximal, characteristic, complemented
8#21444492C211yesmaximal, characteristic, complemented
9C61×#1566C211yesmaximal, characteristic, complemented
10C2×#14519810—21no—
11C2×#14519777C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13C4×#11758222C2211yescharacteristic
14#16840113—21no—
15#16840113—21no—
16#16840113C2214yes—
17#16840058—21no—
18#16840053—21no—
19#16840048C2212yes—
20#16840043C2212yes—
21#16839973—21no—
22#16839973C2212yes—
23#16839968—21no—
24#16839968C2212yes—
25C8×Dic122C2212yes—
26C8×D244C2212yes—
27C8×C22⋊D61—22no—
28#16839815C2211yescharacteristic
29#16839813—21no—
30#16839804—21no—
31#16839772—21no—
32#16839764—21no—
33C2×#14519702—22no—
34C4×C8×D61—21no—
35#16839509C2214yes—
36#16839507C2214yes—
37C61×#423C2211yescharacteristic
38C61×#423C2211yescharacteristic
39C2×C4×C8×C61C2211yescharacteristic
40C61×#334C2214yes—
41C2×#11758222D411yescharacteristic
42C22×Dic122—21no—
43C22×Dic122—21no—
44#14519810—44no—
45C2×Dic244—22no—
46C2×#11758190—22no—
47C2×C8×D61—22no—
48#14519777—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777C2312yes—
55#14519777C2312yes—
56C4×Dic122D411yescharacteristic
57C4×Dic122C2311yescharacteristic
58C4×D244D411yescharacteristic
59C4×D244C2311yescharacteristic
60C4×C22⋊D61—21no—
61C4×C22⋊D61—21no—
62#14519736D411yescharacteristic
63#14519734—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C42×D61—21no—
69#14519711—22no—
70#14519709—22no—
71C4⋊#92418C2312yes—
72#14519706—22no—
73#14519702—22no—
74#14519702—24no—
75#14519702—24no—
76#14519702—22no—
77#14519702—22no—
78#14519699—22no—
79C22⋊#92418—21no—
80#14519689—22no—
81C4×#92418—21no—
82C8×Dic61—22no—
83C2×M16×C61—21no—
84C2×M16×C61—21no—
85C22×C8×C61D412yes—
86C2×C42×C61C2311yescharacteristic
87C4⋊C8×C61—21no—
88C4⋊C8×C61—21no—
89C4⋊C8×C61C2312yes—
90C4⋊C8×C61C2312yes—
91C4×C8×C61D411yescharacteristic
92C4×C8×C61D411yescharacteristic
93C4×C8×C61C2312yes—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61#4111yescharacteristic
97C2×C22⋊D61—21no—
98C22×Dic61—41no—
99C22×Dic61—41no—
100#11758222—22no—
101#11758222SD1614yes—
102C2×D244C2×D411yescharacteristic
103C2×C4×D61—21no—
104C2×C4×D61—21no—
105C2×C4×D61—21no—
106C2×Dic122—22no—
107C2×Dic122—22no—
108C2×Dic122—44no—
109C2×Dic122—42no—
110C2×Dic122—42no—
111C2×Dic122—41no—
112C2×Dic122—41no—
113C2×Dic122C2×D411yescharacteristic
114M16×C61—42no—
115M16×C61—42no—
116C2×C8×C61—22no—
117C2×C8×C61—22no—
118C2×C8×C61—42no—
119C2×C8×C61C2×D412yes—
120C2×C8×C61C2×D412yes—
121C42×C61C2×D411yescharacteristic
122C42×C61C2×D411yescharacteristic
123C42×C61#4111yescharacteristic
124C42×C61C2411yescharacteristic
125C22⋊Dic61—21no—
126#11758198—21no—
127#11758198—21no—
128C4⋊Dic61—22no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131C4⋊Dic61—22no—
132C4⋊Dic61—41no—
133C4⋊Dic61—41no—
134C4⋊Dic61—41no—
135C4⋊Dic61—41no—
136C4⋊Dic61C2×D411yescharacteristic
137C4⋊Dic61C2×D411yescharacteristic
138#11758196—21no—
139#11758196—21no—
140C4×Dic61—21no—
141C4×Dic61—21no—
142C4×Dic61—21no—
143C2×#92418—22no—
144Dic244—44no—
145#11758190—44no—
146C8×D61—44no—
147C23×C61#12111yescharacteristic
148C22×D61—21no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61#12111yescharacteristic
152C2×C4×C61#12111yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61C2×SD1612yes—
155C2×C4×C61#13512yes—
156C2×C4×C61C22×D411yescharacteristic
157C2×C4×C61C22×D411yescharacteristic
158C2×C4×C61C2×#4111yescharacteristic
159C22⋊D61—24no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61—21no—
163C2×Dic61—42no—
164C2×Dic61—42no—
165C2×Dic61—82no—
166C2×Dic61—82no—
167D244—21no—
168D244C2×SD1612yes—
169C4×D61—24no—
170C4×D61—42no—
171C4×D61—42no—
172Dic122—21no—
173Dic122—42no—
174Dic122—42no—
175Dic122—44no—
176Dic122—82no—
177Dic122—82no—
178Dic122C2×SD1612yes—
179C8×C61—22no—
180C8×C61—22no—
181C8×C61—42no—
182C8×C61—42no—
183#92418—42no—
184#19275—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61#46512yes—
188C22×C61C2×#12111yescharacteristic
189C2×D61—22no—
190C2×D61—41no—
191C2×D61—41no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61#46512yes—
195C4×C61#47312yes—
196C4×C61C22×SD1611yescharacteristic
197C4×C61C2×#13511yescharacteristic
198C4×C61#57512yes—
199Dic61—22no—
200Dic61—41no—
201Dic61—41no—
202Dic61—82no—
203Dic61—82no—
204C2×#487—611no—
205#3079—611no—
206#3039—611no—
207C8×#41—611no—
208#1680—614no—
209#1679—614no—
210#1566D6111yescharacteristic, complemented
211#1549—611no—
212#1546—611no—
213C2×C61—22no—
214C2×C61#304011yescharacteristic
215C2×C61C2×#46511yescharacteristic
216C2×C61C2×#47311yescharacteristic
217D61—42no—
218C2×#137—1221no—
219C2×C4⋊Q8—611no—
220C2×C4⋊Q8—611no—
221C4×#41—611no—
222#487—614no—
223#487—1221no—
224#487—1221no—
225#477—1221no—
226#476—1221no—
227#475—612no—
228#474—612no—
229#462—612no—
230#462—1221no—
231#461—612no—
232#461—1221no—
233C8×Q8—612no—
234C8×D4—612no—
235C8×D4—1222no—
236#431—611no—
237#431—1221no—
238#428—1221no—
239#423C2×D6111yescharacteristic
240#423C2×D6111yescharacteristic
241#419—1221no—
242#417—1221no—
243C2×#103—1222no—
244C2×C4×C8—1221no—
245C2×C4×C8C2×D6111yescharacteristic
246#334C2×D6114yes—
247#332—614no—
248#332—614no—
249C61#1927511yescharacteristic, complemented, Sylow
250C2×#41—611no—
251C22×Q8—1221no—
252C22×Q8—1221no—
253#137—2444no—
254C2×Q16—1222no—
255C2×SD16—1222no—
256C2×M16—21no—
257C2×M16—21no—
258C22×C8—1222no—
259C22×C8D24412yes—
260C4⋊Q8—612no—
261C4⋊Q8—612no—
262C4⋊Q8—1221no—
263C4⋊Q8—1221no—
264C4⋊Q8—1221no—
265C4⋊Q8—1221no—
266C4⋊Q8—1221no—
267C4⋊Q8—1221no—
268C4×Q8—611no—
269C4×Q8—611no—
270C4×D4—611no—
271C4×D4—611no—
272C4×D4—1221no—
273C4×D4—1221no—
274#117—611no—
275#117—1221no—
276C2×C4⋊C4—1221no—
277C2×C4⋊C4—1221no—
278C2×C4⋊C4—1221no—
279C2×C4⋊C4—1221no—
280C2×C42—1221no—
281C2×C42C22×D6111yescharacteristic
282#107—1222no—
283#106—1222no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—612no—
287C4⋊C8—1222no—
288C4⋊C8C22×D6112yes—
289C4⋊C8C22×D6112yes—
290#103—1222no—
291#103—1222no—
292#103—1222no—
293#103—1224no—
294#103—1224no—
295#102—1222no—
296C22⋊C8—1221no—
297C22⋊C8—1222no—
298C4×C8—1222no—
299C4×C8—1221no—
300C4×C8D24411yescharacteristic
301C4×C8D24411yescharacteristic
302C4×C8C22×D6112yes—
303#41—614no—
304#41—1222no—
305C2×Q8—611no—
306C2×Q8—1222no—
307C2×Q8—1222no—
308C2×Q8—2444no—
309C2×Q8—2442no—
310C2×Q8—2442no—
311C2×Q8—2441no—
312C2×Q8—2441no—
313C2×D4—611no—
314C2×D4—1221no—
315C22×C4—1221no—
316C22×C4—1221no—
317C22×C4—1221no—
318C22×C4—2441no—
319C22×C4—2441no—
320C22×C4C2×D24411yescharacteristic
321C22×C4D4×D6111yescharacteristic
322C22×C4#1175822611yescharacteristic
323Q16—2444no—
324SD16—2444no—
325M16—42no—
326M16—42no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—42no—
330C2×C8—1222no—
331C2×C8—2444no—
332C2×C8C2×D24412yes—
333C2×C8C2×D24412yes—
334C4⋊C4—611no—
335C4⋊C4—611no—
336C4⋊C4—1221no—
337C4⋊C4—1221no—
338C4⋊C4—1222no—
339C4⋊C4—1222no—
340C4⋊C4—1222no—
341C4⋊C4—1222no—
342C4⋊C4—2441no—
343C4⋊C4—2441no—
344C4⋊C4—2441no—
345C4⋊C4—2441no—
346C22⋊C4—1221no—
347C22⋊C4—1221no—
348C22⋊C4—1221no—
349C42—1221no—
350C42—1221no—
351C42—1221no—
352C42C2×D24411yescharacteristic
353C42D4×D6111yescharacteristic
354C42#1175822611yescharacteristic
355C42C23×D6111yescharacteristic
356C23—1221no—
357C23#1451975211yescharacteristic
358Q8—612no—
359Q8—1221no—
360Q8—4882no—
361Q8—4882no—
362Q8—2442no—
363Q8—2442no—
364Q8—2444no—
365D4—612no—
366D4—1224no—
367D4—1221no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—1221no—
371C2×C4—1224no—
372C2×C4—1221no—
373C2×C4—1221no—
374C2×C4—4882no—
375C2×C4—4882no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4—2442no—
379C2×C4—2442no—
380C2×C4#1451975211yescharacteristic
381C2×C4#1451975211yescharacteristic
382C2×C4#1451975211yescharacteristic
383C2×C4SD16×D6112yes—
384C2×C4#1451979812yes—
385C2×C4C2×D4×D6111yescharacteristic
386C2×C4C22×D24411yescharacteristic
387C2×C4C2×#1175822611yescharacteristic
388C8—22no—
389C8—22no—
390C8—42no—
391C8—42no—
392C8—2442no—
393C22—21no—
394C22—21no—
395C22—1222no—
396C22—2441no—
397C22—2441no—
398C22#1683999212yes—
399C22C2×#1451975211yescharacteristic
400C4—21no—
401C4—21no—
402C4—1222no—
403C4—4882no—
404C4—4882no—
405C4—2441no—
406C4—2441no—
407C4#1683999212yes—
408C4#1684003812yes—
409C4C2×SD16×D6111yescharacteristic
410C4C2×#1451979811yescharacteristic
411C4#1684056212yes—
412C2—22no—
413C2—2442no—
414C2#2145531411yescharacteristic
415C2C2×#1683999211yescharacteristic
416C2C2×#1684003811yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1566C2
socleC22×C61C2×#121
prime radicalC2×#11758222D4
hypercenter#1566D61
hypocenterC61#19275
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1566

Complements

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

NH ≅ G/Ntype
1. C2×#16840113C2semidirect
5. #21445250C2semidirect
6. #21445243C2semidirect
7. #21444505C2semidirect
8. #21444492C2semidirect
9. C61×#1566C2semidirect
210. #1566D61semidirect
249. C61#19275semidirect

Maximal subgroups

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

subgroupindex
C2×#168401132
#214556352
#214553072
C8×#117582222
#214452502
#214452432
#214445052
#214444922
C61×#15662
#1927561