Finite Group Databasebeta

← #29663972

Subgroups — #29663972

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
1#21455628C211yesmaximal, characteristic
2#21455302C211yesmaximal, characteristic
3C8×#11758222C211yesmaximal, characteristic
4#21445246C214yesmaximal, complemented
5#21445239C214yesmaximal, complemented
6#21444575C211yesmaximal, characteristic, complemented
7#21444504C211yesmaximal, characteristic, complemented
8#21444491C211yesmaximal, characteristic, complemented
9C2×C61×#487C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11C4×#11758222C2211yescharacteristic
12#16840054—21no—
13#16840049—21no—
14#16840044C2212yes—
15#16840039C2212yes—
16#16839969—21no—
17#16839969C2212yes—
18#16839964—21no—
19#16839964C2212yes—
20C8×Dic122C2212yes—
21C8×D244C2212yes—
22C8×C22⋊D61—22no—
23#16839815C2211yescharacteristic
24#16839813—21no—
25#16839802—21no—
26#16839786C2211yescharacteristic
27#16839786C2211yescharacteristic
28#16839771—21no—
29#16839761—21no—
30C4×C8×D61—21no—
31#16839513C2214yes—
32#16839508C2214yes—
33#16839508C2214yes—
34C2×C4⋊Q8×C61C2211yescharacteristic
35C2×C4⋊Q8×C61C2211yescharacteristic
36C61×#487—21no—
37C61×#487—21no—
38C61×#487C2214yes—
39C2×#103×C61—22no—
40C2×C4×C8×C61C2211yescharacteristic
41C2×#11758222D411yescharacteristic
42#14519808—44no—
43C2×#11758202—22no—
44C2×#11758201—22no—
45C2×#11758194—21no—
46C2×#11758194—21no—
47C2×C8×D61—22no—
48C4×Dic122D411yescharacteristic
49C4×Dic122C2311yescharacteristic
50C4×D244D411yescharacteristic
51C4×D244C2311yescharacteristic
52C4×C22⋊D61—21no—
53C4×C22⋊D61—21no—
54#14519736D411yescharacteristic
55#14519734—21no—
56C42×D61—21no—
57#14519710—22no—
58#14519708—22no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61C4⋊#92418C2312yes—
62C4⋊#92418C2312yes—
63C4⋊#92418C2312yes—
64#14519706—22no—
65#14519701—22no—
66#14519701—22no—
67#14519700—22no—
68#14519697—22no—
69C22⋊#92418—21no—
70#14519689—22no—
71C4×#92418—21no—
72C8×Dic61—22no—
73C22×Q8×C61—21no—
74C22×Q8×C61—21no—
75C22×C8×C61D412yes—
76C4⋊Q8×C61—21no—
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61—21no—
79C4⋊Q8×C61—21no—
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61C2312yes—
83C4⋊Q8×C61C2312yes—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C42×C61C2311yescharacteristic
89#103×C61—24no—
90#103×C61—24no—
91C4×C8×C61D411yescharacteristic
92C4×C8×C61D411yescharacteristic
93C4×C8×C61C2312yes—
94C2×Q8×C61—22no—
95C2×Q8×C61—22no—
96C2×Q8×C61—44no—
97C2×Q8×C61—41no—
98C2×Q8×C61—41no—
99C2×Q8×C61—42no—
100C2×Q8×C61—42no—
101C22×C4×C61—41no—
102C22×C4×C61—41no—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61#4111yescharacteristic
106C2×C22⋊D61—21no—
107#11758222—22no—
108#11758222SD1614yes—
109C2×D244C2×D411yescharacteristic
110C2×C4×D61—21no—
111C2×C4×D61—21no—
112C2×C4×D61—21no—
113C2×Dic122C2×D411yescharacteristic
114C2×C8×C61—42no—
115C2×C8×C61C2×D412yes—
116C2×C8×C61C2×D412yes—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—41no—
122C4⋊C4×C61—41no—
123C4⋊C4×C61—41no—
124C4⋊C4×C61—41no—
125C42×C61C2×D411yescharacteristic
126C42×C61C2×D411yescharacteristic
127C42×C61#4111yescharacteristic
128C42×C61C2411yescharacteristic
129C22⋊Dic61—21no—
130#11758202—44no—
131#11758201—44no—
132#11758198—21no—
133#11758198—21no—
134C4⋊Dic61C2×D411yescharacteristic
135C4⋊Dic61C2×D411yescharacteristic
136#11758196—21no—
137#11758196—21no—
138C4×Dic61—21no—
139C4×Dic61—21no—
140C4×Dic61—21no—
141#11758194—42no—
142#11758194—42no—
143C2×#92418—22no—
144C2×#92418—22no—
145C2×#92418—22no—
146C8×D61—44no—
147C23×C61#12111yescharacteristic
148C22×D61—21no—
149Q8×C61—42no—
150Q8×C61—42no—
151Q8×C61—44no—
152Q8×C61—82no—
153Q8×C61—82no—
154C2×C4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61—82no—
159C2×C4×C61—82no—
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61C2×SD1612yes—
164C2×C4×C61#13512yes—
165C2×C4×C61C22×D411yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C2×#4111yescharacteristic
168C22⋊D61—24no—
169C2×Dic61—21no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172D244—21no—
173D244C2×SD1612yes—
174C4×D61—24no—
175C4×D61—42no—
176C4×D61—42no—
177Dic122—21no—
178Dic122C2×SD1612yes—
179C8×C61—22no—
180C8×C61—22no—
181#92418—42no—
182#92418—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—82no—
195C4×C61—82no—
196C4×C61#46512yes—
197C4×C61#47312yes—
198C4×C61C22×SD1611yescharacteristic
199C4×C61C2×#13511yescharacteristic
200C4×C61#57512yes—
201Dic61—22no—
202Dic61—41no—
203Dic61—41no—
204C2×#487D6111yescharacteristic, complemented
205#3079—611no—
206#3039—611no—
207C8×#41—611no—
208#1680—614no—
209#1679—614no—
210#1566—611no—
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⋊Q8C2×D6111yescharacteristic
220C2×C4⋊Q8C2×D6111yescharacteristic
221C4×#41—611no—
222#487—21no—
223#487—21no—
224#487C2×D6114yes—
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#423—611no—
240#423—611no—
241#419—1221no—
242#417—1221no—
243C2×#103—22no—
244C2×C4×C8—1221no—
245C2×C4×C8C2×D6111yescharacteristic
246#334—614no—
247#332—614no—
248#332—614no—
249C61#1927511yescharacteristic, complemented, Sylow
250C2×#41—611no—
251C22×Q8—21no—
252C22×Q8—21no—
253#137—2444no—
254C2×Q16—1222no—
255C2×SD16—1222no—
256C2×M16—1221no—
257C2×M16—1221no—
258C22×C8—1222no—
259C22×C8C22⋊D6112yes—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8C22×D6112yes—
267C4⋊Q8C22×D6112yes—
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—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C42—1221no—
281C2×C42C22×D6111yescharacteristic
282#107—1222no—
283#106—1222no—
284C4⋊C8—612no—
285C4⋊C8—612no—
286C4⋊C8—612no—
287C4⋊C8—1222no—
288C4⋊C8—1221no—
289C4⋊C8—1221no—
290#103—24no—
291#103—24no—
292#103—1222no—
293#103—1222no—
294#103—1222no—
295#102—1222no—
296C22⋊C8—1221no—
297C22⋊C8—1222no—
298C4×C8—1222no—
299C4×C8—1221no—
300C4×C8C22⋊D6111yescharacteristic
301C4×C8C22⋊D6111yescharacteristic
302C4×C8C22×D6112yes—
303#41—614no—
304#41—1222no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—44no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—611no—
313C2×D4—611no—
314C2×D4—1221no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—1221no—
318C22×C4—1221no—
319C22×C4—1221no—
320C22×C4D4×D6111yescharacteristic
321C22×C4#1175822411yescharacteristic
322C22×C4C2×C22⋊D6111yescharacteristic
323Q16—2444no—
324SD16—2444no—
325M16—2442no—
326M16—2442no—
327C2×C8—42no—
328C2×C8—1222no—
329C2×C8—1222no—
330C2×C8—1222no—
331C2×C8—2444no—
332C2×C8C2×C22⋊D6112yes—
333C2×C8C2×C22⋊D6112yes—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—611no—
343C4⋊C4—611no—
344C4⋊C4—1221no—
345C4⋊C4—1221no—
346C22⋊C4—1221no—
347C22⋊C4—1221no—
348C22⋊C4—1221no—
349C42—1221no—
350C42—1221no—
351C42—1221no—
352C42D4×D6111yescharacteristic
353C42#1175822411yescharacteristic
354C42C2×C22⋊D6111yescharacteristic
355C42C23×D6111yescharacteristic
356C23—1221no—
357C23#1451974811yescharacteristic
358Q8—42no—
359Q8—42no—
360Q8—44no—
361Q8—82no—
362Q8—82no—
363Q8—612no—
364Q8—1221no—
365D4—612no—
366D4—1224no—
367D4—1221no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—1221no—
375C2×C4—1224no—
376C2×C4—1221no—
377C2×C4—1221no—
378C2×C4—2442no—
379C2×C4—2442no—
380C2×C4#1451974811yescharacteristic
381C2×C4#1451974811yescharacteristic
382C2×C4#1451974811yescharacteristic
383C2×C4SD16×D6112yes—
384C2×C4#1451979812yes—
385C2×C4C2×D4×D6111yescharacteristic
386C2×C4C22×C22⋊D6111yescharacteristic
387C2×C4C2×#1175822411yescharacteristic
388C8—22no—
389C8—22no—
390C8—2442no—
391C8—2442no—
392C8—2442no—
393C22—21no—
394C22—21no—
395C22—1222no—
396C22—2441no—
397C22—2441no—
398C22#1683998812yes—
399C22C2×#1451974811yescharacteristic
400C4—21no—
401C4—21no—
402C4—82no—
403C4—82no—
404C4—1222no—
405C4—2441no—
406C4—2441no—
407C4#1683998812yes—
408C4#1684003412yes—
409C4C2×SD16×D6111yescharacteristic
410C4C2×#1451979811yescharacteristic
411C4#1684056012yes—
412C2—22no—
413C2—2442no—
414C2#2145530911yescharacteristic
415C2C2×#1683998811yescharacteristic
416C2C2×#1684003411yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#487C2
socleC22×C61C2×#121
prime radicalC2×#11758222D4
hypercenterC2×#487D61
hypocenterC61#19275
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
4. #21445246C2semidirect
5. #21445239C2semidirect
6. #21444575C2semidirect
7. #21444504C2semidirect
8. #21444491C2semidirect
9. C2×C61×#487C2semidirect
204. C2×#487D61semidirect
249. C61#19275semidirect

Maximal subgroups

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

subgroupindex
#214556282
#214553022
C8×#117582222
#214452462
#214452392
#214445752
#214445042
#214444912
C2×C61×#4872
#1927561