Finite Group Databasebeta

← #29577814

Subgroups — #29577814

Order 15616.

818 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 97 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#21449150C211yesmaximal, characteristic
2#21448174C211yesmaximal, characteristic, complemented
3#21445763C211yesmaximal, characteristic
4#21444148C211yesmaximal, characteristic, complemented
5#21443673C211yesmaximal, characteristic, complemented
6#21443672C211yesmaximal, characteristic, complemented
7C61×#1789C211yesmaximal, characteristic
8C2×#14519758—21no—
9C2×#14519733—21no—
10C22×#11758196C2211yescharacteristic
11C2×#14519704—22no—
12#16839726C2211yescharacteristic
13#16839681C412yescomplemented
14#16839676C412yescomplemented
15#16839642—21no—
16#16839635—22no—
17#16839625C2211yescharacteristic
18C4×#11758196—22no—
19#16839613—22no—
20C22⋊C4×Dic61—21no—
21C4×C22⋊Dic61C2211yescharacteristic
22#16839487—22no—
23C22×#41×C61C2211yescharacteristic
24C2×#104×C61—22no—
25C61×#407C2211yescharacteristic
26C4×C22⋊C4×C61C2211yescharacteristic
27C23×Dic61D411yescharacteristic
28C23×Dic61D411yescharacteristic
29C2×#11758194—21no—
30#14519758—44no—
31#14519733—28no—
32C2×#11758196—22no—
33C2×#11758196—21no—
34C2×#11758196—21no—
35C2×#11758196—21no—
36C2×#11758196—21no—
37C2×#11758196—22no—
38C2×#11758196C2×C412yes—
39C2×#11758196D412yes—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—22no—
43C2×C22⋊Dic61—22no—
44C2×C4×Dic61—22no—
45C2×C4×Dic61—22no—
46C2×C4×Dic61D412yes—
47#14519704—44no—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50#14519684—22no—
51#14519684—21no—
52#14519684—21no—
53#14519684—22no—
54#14519684C2×C412yes—
55C2×#41×C61—21no—
56C2×#41×C61—21no—
57C2×#41×C61—21no—
58C2×#41×C61—21no—
59C2×#41×C61—44no—
60C22×Q8×C61C2×C411yescharacteristic
61C22×D4×C61—21no—
62C22×D4×C61C2×C411yescharacteristic
63C23×C4×C61—21no—
64C23×C4×C61C2311yescharacteristic
65C2×M16×C61—21no—
66C2×C22⋊C4×C61—21no—
67C2×C42×C61D412yes—
68#104×C61—44no—
69C22⋊C8×C61—21no—
70C22⋊C8×C61—21no—
71#95×C61—21no—
72C24×C61—21no—
73C24×C61C2×D411yescharacteristic
74#41×C61—24no—
75#41×C61—24no—
76#41×C61—42no—
77#41×C61—42no—
78#41×C61—42no—
79#41×C61—42no—
80#41×C61—42no—
81#41×C61—42no—
82C2×Q8×C61—21no—
83C2×Q8×C61—21no—
84C2×Q8×C61—42no—
85C2×D4×C61—24no—
86C2×D4×C61—21no—
87C2×D4×C61—21no—
88C2×D4×C61—42no—
89C2×D4×C61—42no—
90C2×D4×C61—41no—
91C2×D4×C61—41no—
92C22×C4×C61—21no—
93C22×C4×C61—22no—
94C22×C4×C61—42no—
95C22×C4×C61—42no—
96C22×C4×C61—42no—
97C22×C4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61C22×C411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61#4111yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C22×Dic61—21no—
106C22×Dic61—21no—
107C22×Dic61—24no—
108C22×Dic61—21no—
109C22×Dic61—22no—
110C22×Dic61—21no—
111C22×Dic61—22no—
112C22×Dic61—42no—
113C22×Dic61C22⋊C412yes—
114C22×Dic61C22⋊C412yes—
115M16×C61—42no—
116C2×C8×C61—41no—
117C22⋊C4×C61—24no—
118C42×C61—42no—
119C22⋊Dic61—24no—
120C22⋊Dic61—44no—
121C22⋊Dic61—42no—
122C22⋊Dic61—44no—
123#11758196—28no—
124#11758196—22no—
125#11758196—22no—
126#11758196—22no—
127#11758196—22no—
128#11758196—44no—
129C4×Dic61—28no—
130C4×Dic61—42no—
131C4×Dic61—82no—
132#11758194—42no—
133C2×#92418—41no—
134C23×C61—22no—
135C23×C61—22no—
136C23×C61—22no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—42no—
140C23×C61—41no—
141C23×C61—42no—
142C23×C61C2×C22⋊C411yescharacteristic
143C23×C61#12011yescharacteristic
144C23×C61#12311yescharacteristic
145Q8×C61—22no—
146Q8×C61—22no—
147Q8×C61—41no—
148Q8×C61—41no—
149D4×C61—22no—
150D4×C61—22no—
151D4×C61—41no—
152D4×C61—41no—
153D4×C61—44no—
154D4×C61—44no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—24no—
158C2×C4×C61—22no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—44no—
164C2×C4×C61—44no—
165C2×C4×C61—42no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—42no—
169C2×C4×C61C4×D411yescharacteristic
170C2×C4×C61C4×D411yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12111yescharacteristic
173C2×Dic61—22no—
174C2×Dic61—22no—
175C2×Dic61—24no—
176C2×Dic61—22no—
177C2×Dic61—22no—
178C2×Dic61—24no—
179C2×Dic61—22no—
180C2×Dic61—42no—
181C2×Dic61—42no—
182C2×Dic61—42no—
183C2×Dic61—42no—
184C2×Dic61—42no—
185C2×Dic61—42no—
186C2×Dic61—42no—
187C2×Dic61—81no—
188C2×Dic61#10414yescomplemented
189C8×C61—81no—
190#92418—81no—
191#9476—611nomaximal, Sylow
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—44no—
200C22×C61—44no—
201C22×C61—42no—
202C22×C61—82no—
203C22×C61—81no—
204C22×C61—81no—
205C22×C61#38611yescharacteristic
206C22×C61C2×#10412yes—
207C22×C61B(4,𝔽2)12yes—
208C4×C61—21no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—44no—
213C4×C61—42no—
214Dic61—24no—
215Dic61—44no—
216Dic61—42no—
217Dic61—82no—
218#2406—611no—
219#2274—611no—
220#1789—611no—
221#1789D6111yescharacteristic
222#1463—611no—
223#1284—611no—
224#1284—611no—
225C2×C61—22no—
226C2×C61—22no—
227C2×C61—41no—
228C2×C61—82no—
229C2×C61#188911yescharacteristic
230C2×C61#189611yescharacteristic
231C2×C61#191211yescharacteristic
232C22×#41C2×D6111yescharacteristic
233C2×#123—1221no—
234C2×#117—1221no—
235C22×C4⋊C4—611no—
236C2×#104—22no—
237C2×#104—1222no—
238#407—611no—
239#407C2×D6111yescharacteristic
240#395—612no—
241#394—612no—
242#386—1221no—
243#385—1222no—
244#381—611no—
245C4×C4⋊C4—1222no—
246C4×C4⋊C4—1222no—
247C4×C22⋊C4—611no—
248C4×C22⋊C4—1221no—
249C4×C22⋊C4C2×D6111yescharacteristic
250#324—1222no—
251C61#947611yescharacteristic, complemented, Sylow
252C2×#41—21no—
253C2×#41—21no—
254C2×#41—21no—
255C2×#41—21no—
256C2×#41—44no—
257C22×Q8C4×D6111yescharacteristic
258C22×D4—21no—
259C22×D4C4×D6111yescharacteristic
260C23×C4—21no—
261C23×C4—611no—
262C23×C4—611no—
263C23×C4C22×D6111yescharacteristic
264C2×M16—21no—
265C2×M16—1221no—
266#123—2444no—
267#117—1228no—
268C2×C4⋊C4—612no—
269C2×C4⋊C4—612no—
270C2×C4⋊C4—1222no—
271C2×C4⋊C4—1221no—
272C2×C4⋊C4—1221no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4—1221no—
275C2×C4⋊C4—1222no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—1221no—
278C2×C22⋊C4—1222no—
279C2×C22⋊C4—1222no—
280C2×C22⋊C4—1221no—
281C2×C42—612no—
282C2×C42—1222no—
283C2×C42—1222no—
284C2×C42C22⋊D6112yes—
285#104—44no—
286#104—2444no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—1221no—
290C22⋊C8—1221no—
291#95—21no—
292#95—612no—
293#95—1222no—
294#95—1222no—
295#95—1221no—
296#95—1221no—
297C24—21no—
298C24D4×D6111yescharacteristic
299#41—24no—
300#41—24no—
301#41—42no—
302#41—42no—
303#41—42no—
304#41—42no—
305#41—42no—
306#41—42no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—42no—
310C2×D4—24no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—41no—
316C2×D4—41no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—612no—
325C22×C4—612no—
326C22×C4—1224no—
327C22×C4—1221no—
328C22×C4—1221no—
329C22×C4—1221no—
330C22×C4—1222no—
331C22×C4—1221no—
332C22×C4—1222no—
333C22×C4—2442no—
334C22×C4C2×C4×D6111yescharacteristic
335C22×C4#1175822211yescharacteristic
336C22×C4D4×D6111yescharacteristic
337C22×C4D4×D6111yescharacteristic
338C22×C4#1175822411yescharacteristic
339C22×C4C2×C22⋊D6111yescharacteristic
340M16—42no—
341M16—2442no—
342C2×C8—41no—
343C2×C8—2441no—
344C4⋊C4—1228no—
345C4⋊C4—1222no—
346C4⋊C4—1222no—
347C4⋊C4—1222no—
348C4⋊C4—1222no—
349C4⋊C4—2444no—
350C22⋊C4—24no—
351C22⋊C4—1224no—
352C22⋊C4—2444no—
353C22⋊C4—2442no—
354C22⋊C4—2444no—
355C42—42no—
356C42—1228no—
357C42—4882no—
358C42—2442no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23—41no—
367C23C22⋊C4×D6111yescharacteristic
368C23#1451974511yescharacteristic
369C23#1451975911yescharacteristic
370Q8—22no—
371Q8—22no—
372Q8—41no—
373Q8—41no—
374D4—22no—
375D4—22no—
376D4—41no—
377D4—41no—
378D4—44no—
379D4—44no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—614no—
395C2×C4—1222no—
396C2×C4—1222no—
397C2×C4—1224no—
398C2×C4—1222no—
399C2×C4—1222no—
400C2×C4—1224no—
401C2×C4—1222no—
402C2×C4—4881no—
403C2×C4—2442no—
404C2×C4—2442no—
405C2×C4—2442no—
406C2×C4—2442no—
407C2×C4—2442no—
408C2×C4—2442no—
409C2×C4—2442no—
410C2×C4#1451973711yescharacteristic
411C2×C4C4×C22⋊D6111yescharacteristic
412C2×C4#1451974811yescharacteristic
413C2×C4#1451975011yescharacteristic
414C8—81no—
415C8—4881no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—44no—
424C22—44no—
425C22—42no—
426C22—82no—
427C22—81no—
428C22—81no—
429C22#1683963811yescharacteristic
430C22#104×D6112yes—
431C22#1683994812yes—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—44no—
437C4—42no—
438C4—1224no—
439C4—4882no—
440C4—2444no—
441C4—2442no—
442C2—22no—
443C2—22no—
444C2—41no—
445C2—82no—
446C2#2144622411yescharacteristic
447C2#2144625911yescharacteristic
448C2#2144635311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1789C2
socleC22×C61#386
prime radicalC22×#41×C61C22
cosocleC23×C4×C61C23
hypercenter#1789D61
hypocenterC61#9476
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1789

Complements

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

NH ≅ G/Ntype
2. #21448174C2semidirect
4. #21444148C2semidirect
5. #21443673C2semidirect
6. #21443672C2semidirect
13. #16839681C4semidirect
14. #16839676C4semidirect
188. C2×Dic61#104semidirect
251. C61#9476semidirect

Maximal subgroups

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

subgroupindex
#214491502
#214481742
#214457632
#214441482
#214436732
#214436722
C61×#17892
#947661