Finite Group Databasebeta

← #29597550

Subgroups — #29597550

Order 15616.

702 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C22×#14519696C211yesmaximal, characteristic, complemented
2#21449473C211yesmaximal, characteristic
3C2×#16839558C212yesmaximal, complemented
4#21443945C212yesmaximal, complemented
5C61×#1784C211yesmaximal, characteristic, complemented
6C22×#11758194—21no—
7C2×#14519766C411yescharacteristic
8C2×#14519766C2211yescharacteristic
9C2×#14519751C411yescharacteristic
10C2×#14519696—24no—
11C2×#14519696—21no—
12C2×#14519696C2212yes—
13#16839673—21no—
14#16839673—21no—
15#16839669C411yescharacteristic
16#16839669C2211yescharacteristic
17#16839641—21no—
18#16839641—21no—
19C4×C22⋊Dic61C411yescharacteristic
20#16839558—24no—
21C23×Q8×C61C2211yescharacteristic
22C2×#101×C61—21no—
23C2×#101×C61C2212yes—
24C61×#407—21no—
25C22×D244C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C2×#11758194—24no—
28C2×#11758194—22no—
29C2×#11758194—41no—
30C2×#11758194—42no—
31C22×#92418—41no—
32#14519766—22no—
33#14519766—22no—
34#14519766—42no—
35#14519751—41no—
36#14519751—41no—
37C2×C4⋊Dic61C2×C411yescharacteristic
38C2×C22⋊Dic61—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×C4×Dic61C2×C411yescharacteristic
45C2×C4×Dic61C2×C411yescharacteristic
46#14519696—24no—
47#14519696—22no—
48#14519696D414yescomplemented
49#14519684—21no—
50#14519684—21no—
51#14519684—21no—
52#14519684—21no—
53#14519684—21no—
54C22×Q8×C61—22no—
55C22×Q8×C61—22no—
56C22×Q8×C61—41no—
57C22×Q8×C61—41no—
58C22×Q8×C61—42no—
59C22×Q8×C61C2×C411yescharacteristic
60C22×Q8×C61D411yescharacteristic
61C22×Q8×C61D411yescharacteristic
62C22×Q8×C61C2311yescharacteristic
63C23×C4×C61C2×C411yescharacteristic
64C23×C4×C61D411yescharacteristic
65C23×C4×C61D411yescharacteristic
66C2×M16×C61—22no—
67#101×C61—22no—
68#101×C61—22no—
69#101×C61—42no—
70C22⋊C8×C61—41no—
71C22⋊C8×C61—41no—
72C24×C61C22⋊C411yescharacteristic
73C23×D61—21no—
74C2×Q8×C61—22no—
75C2×Q8×C61—22no—
76C2×Q8×C61—22no—
77C2×Q8×C61—21no—
78C2×Q8×C61—22no—
79C2×Q8×C61—21no—
80C2×Q8×C61—41no—
81C2×Q8×C61—41no—
82C2×Q8×C61—41no—
83C2×Q8×C61—41no—
84C2×Q8×C61—41no—
85C2×Q8×C61—41no—
86C2×Q8×C61—44no—
87C2×Q8×C61—44no—
88C2×Q8×C61—44no—
89C2×Q8×C61—44no—
90C2×Q8×C61C2×D412yes—
91C2×Q8×C61#4112yes—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—22no—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—41no—
104C22×C4×C61—41no—
105C22×C4×C61C22⋊C411yescharacteristic
106C22×C4×C61C22⋊C411yescharacteristic
107C22×C4×C61C22⋊C411yescharacteristic
108C22×C4×C61C22×C411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C2×C22⋊D61—44no—
112C2×C22⋊D61—41no—
113C2×C22⋊D61—41no—
114C22×Dic61—21no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C2×D244—22no—
118C2×D244—41no—
119C2×D244—41no—
120M16×C61—42no—
121M16×C61—42no—
122C2×C8×C61—42no—
123C22⋊Dic61—42no—
124#11758198—42no—
125#11758198—42no—
126#11758198—42no—
127#11758198—41no—
128#11758198—42no—
129#11758198—41no—
130C4⋊Dic61—41no—
131C4×Dic61—22no—
132C4×Dic61—41no—
133#11758194—24no—
134#11758194—44no—
135#11758194—42no—
136C2×#92418—44no—
137C2×#92418—42no—
138C23×C61—22no—
139C23×C61—21no—
140C23×C61—21no—
141C23×C61—41no—
142C23×C61#9911yescharacteristic
143C23×C61#9911yescharacteristic
144C23×C61C2×C22⋊C411yescharacteristic
145C22×D61—42no—
146C22×D61—42no—
147C22×D61—41no—
148C22×D61—41no—
149C22×D61—41no—
150Q8×C61—42no—
151Q8×C61—42no—
152Q8×C61—44no—
153Q8×C61—44no—
154Q8×C61—42no—
155Q8×C61—42no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—42no—
178C2×C4×C61#9912yes—
179C2×C4×C61C4×D412yes—
180C2×C4×C61#12011yescharacteristic
181C2×C4×C61#12112yes—
182C2×C4×C61#12311yescharacteristic
183C22⋊D61—82no—
184C22⋊D61—82no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—42no—
190C2×Dic61—41no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193D244—82no—
194C8×C61—82no—
195#92418—44no—
196#13092—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—21no—
204C22×C61—42no—
205C22×C61—42no—
206C22×C61#35512yescomplemented
207C22×C61#38611yescharacteristic
208C22×C61C2×#9911yescharacteristic
209C22×C61C2×#9911yescharacteristic
210C2×D61—82no—
211C2×D61—82no—
212C2×D61—81no—
213C2×D61—81no—
214C2×D61—81no—
215C4×C61—21no—
216C4×C61—22no—
217C4×C61—22no—
218C4×C61—22no—
219C4×C61—21no—
220C4×C61—41no—
221C4×C61—41no—
222Dic61—81no—
223Dic61—81no—
224Dic61—81no—
225C22×#101—611no—
226#2436—611no—
227C2×#355—612no—
228#1784D6111yescharacteristic, complemented
229#1394—612no—
230C2×C61—21no—
231C2×C61—21no—
232C2×C61—22no—
233C2×C61—41no—
234C2×C61#189811yescharacteristic
235C2×C61C2×#35511yescharacteristic
236C2×C61#213311yescharacteristic
237D61—161no—
238C23×Q8C2×D6111yescharacteristic
239C22×M16—1221no—
240C2×#124—611no—
241C2×#124—611no—
242C2×#121—611no—
243C2×#101—21no—
244C2×#101—612no—
245C2×#101—1224no—
246C2×#101—1221no—
247C2×#101C2×D6112yes—
248#407—21no—
249#393—1221no—
250#393—1221no—
251#391—611no—
252#391—611no—
253#386—1221no—
254#386—1221no—
255C4×C22⋊C4—611no—
256#355—1224no—
257C61#1309211yescharacteristic, complemented, Sylow
258C22×Q8—22no—
259C22×Q8—22no—
260C22×Q8—41no—
261C22×Q8—41no—
262C22×Q8—42no—
263C22×Q8C4×D6111yescharacteristic
264C22×Q8D24411yescharacteristic
265C22×Q8C22⋊D6111yescharacteristic
266C22×Q8C22×D6111yescharacteristic
267C22×D4—611no—
268C22×D4—1221no—
269C23×C4C4×D6111yescharacteristic
270C23×C4D24411yescharacteristic
271C23×C4C22⋊D6111yescharacteristic
272C2×M16—22no—
273C2×M16—1224no—
274C2×M16—1222no—
275C2×M16—2441no—
276C2×M16—2442no—
277C22×C8—2441no—
278#124—1222no—
279#124—1222no—
280#124—2442no—
281#121—2441no—
282#121—2441no—
283C2×C4⋊C4—611no—
284C2×C22⋊C4—1221no—
285C2×C22⋊C4—1221no—
286C2×C22⋊C4—1221no—
287C2×C22⋊C4—1221no—
288C2×C22⋊C4—1221no—
289C2×C22⋊C4—1221no—
290C2×C42—611no—
291C2×C42—611no—
292#101—22no—
293#101—22no—
294#101—42no—
295#101—614no—
296#101—1224no—
297#101—1222no—
298C22⋊C8—41no—
299C22⋊C8—41no—
300#95—1221no—
301#95—1221no—
302#95—1221no—
303#95—1221no—
304#95—1221no—
305C24—1221no—
306C24#1175819811yescharacteristic
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—21no—
311C2×Q8—22no—
312C2×Q8—21no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8D4×D6112yes—
324C2×Q8#1175822412yes—
325C2×D4—1222no—
326C2×D4—2441no—
327C2×D4—2441no—
328C2×D4—2444no—
329C2×D4—2441no—
330C2×D4—2441no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—1221no—
345C22×C4—1221no—
346C22×C4—1221no—
347C22×C4#1175819811yescharacteristic
348C22×C4#1175819811yescharacteristic
349C22×C4#1175819811yescharacteristic
350C22×C4C2×C4×D6111yescharacteristic
351C22×C4C2×D24411yescharacteristic
352C22×C4C2×C22⋊D6111yescharacteristic
353M16—42no—
354M16—42no—
355M16—1224no—
356M16—2444no—
357M16—2442no—
358C2×C8—42no—
359C2×C8—2444no—
360C2×C8—2442no—
361C4⋊C4—2441no—
362C22⋊C4—2442no—
363C22⋊C4—2442no—
364C22⋊C4—2442no—
365C22⋊C4—2442no—
366C22⋊C4—2441no—
367C22⋊C4—2442no—
368C22⋊C4—2441no—
369C42—1222no—
370C42—2441no—
371C23—21no—
372C23—22no—
373C23—21no—
374C23—41no—
375C23—2442no—
376C23—2442no—
377C23—2441no—
378C23—2441no—
379C23—2441no—
380C23#1451969111yescharacteristic
381C23#1451969111yescharacteristic
382C23C2×#1175819811yescharacteristic
383Q8—42no—
384Q8—42no—
385Q8—44no—
386Q8—44no—
387Q8—42no—
388Q8—42no—
389D4—4882no—
390D4—4882no—
391D4—4882no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—2441no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4—2441no—
418C2×C4—2442no—
419C2×C4—2441no—
420C2×C4—2441no—
421C2×C4—2441no—
422C2×C4#1451969112yes—
423C2×C4#1451973712yes—
424C2×C4#1451974711yescharacteristic
425C2×C4#1451974912yes—
426C2×C4#1451976111yescharacteristic
427C8—82no—
428C8—2444no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—42no—
437C22—42no—
438C22—4882no—
439C22—4882no—
440C22—4881no—
441C22—4881no—
442C22—4881no—
443C22#1683955812yescomplemented
444C22#1683963911yescharacteristic
445C22C2×#1451969111yescharacteristic
446C22C2×#1451969111yescharacteristic
447C4—22no—
448C4—21no—
449C4—22no—
450C4—21no—
451C4—22no—
452C4—41no—
453C4—41no—
454C4—4881no—
455C4—4881no—
456C4—4881no—
457C2—21no—
458C2—21no—
459C2—22no—
460C2—41no—
461C2—9761no—
462C2#2144627011yescharacteristic
463C2C2×#1683955811yescharacteristic
464C2#2144750611yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#1784C2
socleC22×C61C2×#99
prime radicalC2×#14519751C4
cosocleC22×Q8×C61C23
hypercenter#1784D61
hypocenterC61#13092
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ #1784

Complements

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

NH ≅ G/Ntype
1. C22×#14519696C2semidirect
3. C2×#16839558C2semidirect
4. #21443945C2semidirect
5. C61×#1784C2semidirect
48. #14519696D4semidirect
206. C22×C61#355semidirect
228. #1784D61semidirect
257. C61#13092semidirect
443. C22#16839558semidirect

Maximal subgroups

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

subgroupindex
C22×#145196962
#214494732
C2×#168395582
#214439452
C61×#17842
#1309261