Finite Group Databasebeta

← #29577964

Subgroups — #29577964

Order 15616.

682 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 95 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#21454858C211yesmaximal, characteristic
2#21449178C211yesmaximal, characteristic
3#21445743C211yesmaximal, characteristic, complemented
4#21444155C211yesmaximal, characteristic
5#21443726C211yesmaximal, characteristic, complemented
6#21443724C211yesmaximal, characteristic, complemented
7C61×#1784C211yesmaximal, characteristic, complemented
8C2×#14519753—21no—
9C2×C4×C22⋊D61C2211yescharacteristic
10#16839838—22no—
11#16839837—22no—
12#16839812—21no—
13C2×#14519696—22no—
14#16839726C2211yescharacteristic
15#16839726C2211yescharacteristic
16#16839724—21no—
17#16839681—21no—
18#16839673—21no—
19#16839669—21no—
20#16839641C412yes—
21#16839635C412yes—
22#16839622C2211yescharacteristic
23#16839487—21no—
24#16839485—21no—
25C23×Q8×C61C2211yescharacteristic
26C2×#101×C61—22no—
27C61×#407C2211yescharacteristic
28C61×#407C2211yescharacteristic
29C22×C22⋊D61C2×C411yescharacteristic
30C22×C4×D61D411yescharacteristic
31C2×#11758194—21no—
32C2×#11758194D412yes—
33C2×#11758189—22no—
34#14519753—44no—
35C4×C22⋊D61—21no—
36C4×C22⋊D61—21no—
37C4×C22⋊D61—21no—
38C4×C22⋊D61—21no—
39C2×C4⋊Dic61—21no—
40C2×#11758196—22no—
41C2×#11758196D411yescharacteristic
42C2×C22⋊Dic61C2×C411yescharacteristic
43C2×C22⋊Dic61C2×C412yes—
44C2×#11758198—22no—
45C2×#11758198D411yescharacteristic
46C2×C4×Dic61D411yescharacteristic
47#14519706—22no—
48#14519696—44no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52C22⋊#92418—21no—
53#14519689—22no—
54C8×Dic61—22no—
55#14519684—21no—
56#14519684—22no—
57#14519684—22no—
58C22×Q8×C61—21no—
59C22×Q8×C61—21no—
60C22×Q8×C61—21no—
61C22×Q8×C61—21no—
62C22×Q8×C61—21no—
63C22×Q8×C61—21no—
64C22×Q8×C61—44no—
65C23×C4×C61C2×C412yes—
66C23×C4×C61C2311yescharacteristic
67C2×M16×C61—21no—
68C2×M16×C61D412yes—
69#101×C61—44no—
70C22⋊C8×C61—21no—
71C22⋊C8×C61—21no—
72C22⋊C8×C61—21no—
73C22⋊C8×C61—21no—
74C24×C61C22×C411yescharacteristic
75C23×D61C22⋊C411yescharacteristic
76C2×Q8×C61—24no—
77C2×Q8×C61—24no—
78C2×Q8×C61—24no—
79C2×Q8×C61—42no—
80C2×Q8×C61—42no—
81C2×Q8×C61—42no—
82C2×Q8×C61—44no—
83C2×Q8×C61—44no—
84C2×Q8×C61—44no—
85C2×Q8×C61—41no—
86C2×Q8×C61—41no—
87C2×Q8×C61—41no—
88C2×Q8×C61—41no—
89C22×C4×C61—22no—
90C22×C4×C61—22no—
91C22×C4×C61—22no—
92C22×C4×C61—44no—
93C22×C4×C61—42no—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61#4111yescharacteristic
99C22×C4×C61#4111yescharacteristic
100C2×C22⋊D61—21no—
101C2×C22⋊D61—21no—
102C2×C22⋊D61—22no—
103C2×C22⋊D61—41no—
104C22×Dic61—22no—
105C22×Dic61C22⋊C411yescharacteristic
106C22×Dic61C22⋊C411yescharacteristic
107C22×Dic61C22⋊C411yescharacteristic
108C2×C4×D61—21no—
109C2×C4×D61—21no—
110C2×C4×D61—42no—
111M16×C61—42no—
112M16×C61—42no—
113C2×C8×C61—22no—
114C2×C8×C61—41no—
115C22⋊Dic61—21no—
116C22⋊Dic61—21no—
117C22⋊Dic61—42no—
118#11758198—21no—
119#11758198—21no—
120#11758198—44no—
121C4⋊Dic61—42no—
122#11758196—21no—
123#11758196—21no—
124#11758196—44no—
125C4×Dic61—21no—
126C4×Dic61—21no—
127#11758194—42no—
128#11758194—42no—
129C2×#92418—22no—
130C2×#92418—41no—
131#11758189—82no—
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—41no—
135C23×C61—41no—
136C23×C61C2×C22⋊C411yescharacteristic
137C23×C61C4×D411yescharacteristic
138C23×C61C4×D411yescharacteristic
139C22×D61—21no—
140C22×D61—21no—
141C22×D61—21no—
142C22×D61—41no—
143C22×D61—41no—
144Q8×C61—44no—
145Q8×C61—44no—
146Q8×C61—44no—
147Q8×C61—44no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—24no—
152C2×C4×C61—24no—
153C2×C4×C61—24no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61#12011yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12311yescharacteristic
165C22⋊D61—42no—
166C22⋊D61—42no—
167C2×Dic61—21no—
168C2×Dic61—21no—
169C2×Dic61—21no—
170C2×Dic61—21no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—42no—
174C2×Dic61—42no—
175C2×Dic61—42no—
176C2×Dic61#10112yescomplemented
177C4×D61—42no—
178C4×D61—81no—
179C8×C61—42no—
180C8×C61—81no—
181#92418—42no—
182#92418—81no—
183#9506—611nomaximal, Sylow
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—42no—
193C22×C61—42no—
194C22×C61#38611yescharacteristic
195C22×C61C2×#10111yescharacteristic
196C22×C61#41311yescharacteristic
197C2×D61—42no—
198C2×D61—41no—
199C2×D61—41no—
200C2×D61—41no—
201C2×D61—81no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—44no—
206C4×C61B(4,𝔽2)12yes—
207Dic61—22no—
208Dic61—41no—
209Dic61—41no—
210Dic61—82no—
211#2975—611no—
212#2409—611no—
213#1784—611no—
214#1784D6111yescharacteristic, complemented
215#1465—611no—
216#1305—611no—
217#1305—611no—
218C2×C61—21no—
219C2×C61—21no—
220C2×C61—41no—
221C2×C61—41no—
222C2×C61#188911yescharacteristic
223C2×C61#190111yescharacteristic
224C2×C61#191511yescharacteristic
225D61—81no—
226C23×Q8C2×D6111yescharacteristic
227C2×C22⋊Q8—1221no—
228C2×C4×D4—611no—
229#436—1222no—
230#436—1222no—
231#431—1221no—
232C2×#101—22no—
233C2×#101—1222no—
234#407—611no—
235#407—611no—
236#407—1221no—
237#407C2×D6111yescharacteristic
238#407C2×D6111yescharacteristic
239#395—1221no—
240#393—1221no—
241#391—1221no—
242#386—612no—
243#385—612no—
244#380—611no—
245#324—1221no—
246#323—1221no—
247C61#950611yescharacteristic, complemented, Sylow
248C22×Q8—21no—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—21no—
254C22×Q8—44no—
255C22×D4—611no—
256C23×C4—611no—
257C23×C4C4×D6112yes—
258C23×C4C22×D6111yescharacteristic
259C2×M16—21no—
260C2×M16—612no—
261C2×M16—1222no—
262C2×M16—1221no—
263C2×M16C22⋊D6112yes—
264C22⋊Q8—2444no—
265C4×D4—1221no—
266C4×D4—1221no—
267C4×D4—1221no—
268C4×D4—1221no—
269C2×C4⋊C4—611no—
270C2×C4⋊C4—1222no—
271C2×C4⋊C4—1221no—
272C2×C22⋊C4—611no—
273C2×C22⋊C4—611no—
274C2×C22⋊C4—612no—
275C2×C22⋊C4—1222no—
276C2×C42—611no—
277C4⋊C8—1222no—
278#101—44no—
279#101—2444no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—1222no—
285C22⋊C8—1221no—
286C22⋊C8—1221no—
287C22⋊C8—1221no—
288C22⋊C8—1221no—
289C4×C8—1222no—
290#95—1222no—
291#95—1222no—
292#95—1221no—
293C24—611no—
294C24C2×C4×D6111yescharacteristic
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×D4—1222no—
309C2×D4—1221no—
310C2×D4—1221no—
311C2×D4—2441no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—44no—
316C22×C4—42no—
317C22×C4—611no—
318C22×C4—611no—
319C22×C4—611no—
320C22×C4—1221no—
321C22×C4—1221no—
322C22×C4—1222no—
323C22×C4—2442no—
324C22×C4#1175822211yescharacteristic
325C22×C4D4×D6111yescharacteristic
326C22×C4D4×D6111yescharacteristic
327C22×C4D4×D6111yescharacteristic
328C22×C4#1175822411yescharacteristic
329C22×C4C2×C22⋊D6111yescharacteristic
330M16—42no—
331M16—42no—
332M16—4882no—
333M16—2442no—
334M16—2442no—
335C2×C8—22no—
336C2×C8—41no—
337C2×C8—1222no—
338C2×C8—2441no—
339C4⋊C4—1221no—
340C4⋊C4—1221no—
341C4⋊C4—2444no—
342C4⋊C4—2442no—
343C22⋊C4—1221no—
344C22⋊C4—1221no—
345C22⋊C4—1221no—
346C22⋊C4—1221no—
347C22⋊C4—2444no—
348C22⋊C4—2442no—
349C42—1221no—
350C42—1221no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23—1221no—
356C23—1221no—
357C23—1221no—
358C23—2441no—
359C23—2441no—
360C23C22⋊C4×D6111yescharacteristic
361C23#1451973711yescharacteristic
362C23C4×C22⋊D6111yescharacteristic
363Q8—44no—
364Q8—44no—
365Q8—44no—
366Q8—44no—
367D4—2442no—
368D4—2442no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—612no—
383C2×C4—1221no—
384C2×C4—1221no—
385C2×C4—1221no—
386C2×C4—1221no—
387C2×C4—4881no—
388C2×C4—2442no—
389C2×C4—2442no—
390C2×C4—2442no—
391C2×C4—2442no—
392C2×C4—2441no—
393C2×C4—2441no—
394C2×C4#1451974511yescharacteristic
395C2×C4#1451974811yescharacteristic
396C2×C4#1451975011yescharacteristic
397C2×C4#1451975911yescharacteristic
398C8—42no—
399C8—81no—
400C8—4881no—
401C8—2442no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—42no—
411C22—42no—
412C22—4881no—
413C22—2442no—
414C22—2441no—
415C22—2441no—
416C22—2441no—
417C22#1683963811yescharacteristic
418C22#101×D6111yescharacteristic
419C22#1683974511yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—44no—
424C4—1222no—
425C4—4882no—
426C4—2441no—
427C4—2441no—
428C4#1683994812yes—
429C2—21no—
430C2—21no—
431C2—41no—
432C2—41no—
433C2—4881no—
434C2#2144622411yescharacteristic
435C2#2144628411yescharacteristic
436C2#2144636811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1784C2
socleC22×C61#386
prime radicalC22×C22⋊D61C2×C4
cosocleC23×C4×C61C23
hypercenter#1784D61
hypocenterC61#9506
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
3. #21445743C2semidirect
5. #21443726C2semidirect
6. #21443724C2semidirect
7. C61×#1784C2semidirect
176. C2×Dic61#101semidirect
214. #1784D61semidirect
247. C61#9506semidirect

Maximal subgroups

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

subgroupindex
#214548582
#214491782
#214457432
#214441552
#214437262
#214437242
C61×#17842
#950661