Finite Group Databasebeta

← #29661325

Subgroups — #29661325

Order 15616.

896 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 171 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#21456834C212yesmaximal, complemented
2#21456195C211yesmaximal, characteristic, complemented
3#21445121C212yesmaximal, complemented
4#21445117C214yesmaximal, complemented
5#21445112C212yesmaximal, complemented
6#21444457C211yesmaximal, characteristic, complemented
7#21444426C212yesmaximal, complemented
8C61×#3531C211yesmaximal, characteristic, complemented
9C2×#14519774C2211yescharacteristic
10#16840152C2214yes—
11#16840128C2212yes—
12#16840128C2214yes—
13#16840128C2212yes—
14#16840110—22no—
15#16839962—22no—
16#16839957—22no—
17#16839912—22no—
18#16839895—22no—
19#16839819C2212yes—
20#16839766—22no—
21C4×#11758194C2211yescharacteristic
22#16839504C2214yescomplemented
23#16839504C2212yescomplemented
24#16839504C2212yescomplemented
25#16839502C2214yescomplemented
26C61×#564C2211yescharacteristic, complemented
27C61×#563—22no—
28C61×#563—22no—
29C61×#563C2212yescomplemented
30C61×#562—22no—
31C61×#562C2211yescharacteristic, complemented
32C61×#533—22no—
33C61×#533C2211yescharacteristic
34C61×#529—22no—
35C61×#529—22no—
36C61×#529C2212yes—
37C22×D244—21no—
38C22×D244—22no—
39C2×#11758201—24no—
40C2×#11758199—24no—
41C2×#11758194—21no—
42#14519774—22no—
43#14519774—21no—
44#14519774C2312yes—
45C4⋊#92418—22no—
46#14519697—24no—
47#14519697—24no—
48#14519697—24no—
49#14519697—24no—
50C22⋊#92418—22no—
51#14519688C2312yes—
52#14519688C2314yes—
53C4×#92418—21no—
54C4⋊Q8×C61—22no—
55C4⋊Q8×C61—22no—
56C4⋊Q8×C61D412yes—
57#126×C61—22no—
58#126×C61—22no—
59#126×C61—22no—
60#126×C61—22no—
61#126×C61—22no—
62#125×C61—22no—
63#125×C61—22no—
64#125×C61—22no—
65#125×C61—22no—
66#125×C61C2311yescharacteristic
67#125×C61C2311yescharacteristic
68#125×C61C2312yes—
69#124×C61—22no—
70#124×C61—22no—
71#124×C61C2312yes—
72#123×C61—22no—
73#123×C61—22no—
74#123×C61—22no—
75#123×C61—22no—
76#123×C61—22no—
77C22⋊Q8×C61—22no—
78C22⋊Q8×C61—22no—
79C22⋊Q8×C61—22no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61D412yes—
85#121×C61—22no—
86#121×C61—22no—
87#121×C61D412yes—
88C4×Q8×C61—22no—
89C4×Q8×C61—22no—
90C4×Q8×C61D412yes—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C4×D4×C61D412yes—
94#117×C61—22no—
95#117×C61—22no—
96#117×C61D412yes—
97C2×C42×C61—22no—
98C2×C42×C61C2311yescharacteristic
99C23×D61—41no—
100C2×Q8×C61—22no—
101C2×Q8×C61—22no—
102C2×Q8×C61C2×D412yes—
103C2×D4×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61C2×D412yes—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D412yes—
111C2×D244—22no—
112C2×D244—24no—
113C2×D244—44no—
114C2×D244—42no—
115C2×D244—41no—
116C2×D244—84no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61C2×D412yes—
131C4⋊C4×C61C2×D412yes—
132C4⋊C4×C61C2×D412yes—
133C4⋊C4×C61C2×D412yes—
134C22⋊C4×C61—22no—
135C22⋊C4×C61—22no—
136C22⋊C4×C61—22no—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—22no—
140C22⋊C4×C61—22no—
141C22⋊C4×C61—22no—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—22no—
144C42×C61—22no—
145C42×C61—22no—
146C42×C61—22no—
147C42×C61—22no—
148C42×C61C2×D412yes—
149C42×C61C2×D411yescharacteristic
150C42×C61C2411yescharacteristic
151#11758201—84no—
152#11758199—84no—
153#11758194—42no—
154C2×#92418—22no—
155C2×#92418—24no—
156C23×C61—22no—
157C23×C61#12011yescharacteristic
158C22×D61—42no—
159C22×D61—82no—
160C22×D61—84no—
161Q8×C61—42no—
162Q8×C61—82no—
163Q8×C61—82no—
164D4×C61—42no—
165D4×C61—82no—
166D4×C61—82no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—42no—
174C2×C4×C61—44no—
175C2×C4×C61—82no—
176C2×C4×C61—82no—
177C2×C4×C61#12012yes—
178C2×C4×C61#12011yescharacteristic
179C2×C4×C61C22×D411yescharacteristic
180C2×C4×C61C22×D412yes—
181D244—44no—
182D244—44no—
183D244—44no—
184D244—84no—
185D244—82no—
186#92418—42no—
187#92418—44no—
188#19004—611nomaximal, Sylow
189C22×C61—21no—
190C22×C61—22no—
191C22×C61—42no—
192C22×C61—42no—
193C22×C61—42no—
194C22×C61C2×#12011yescharacteristic
195C2×D61—82no—
196C2×D61—84no—
197C2×D61—164no—
198C4×C61—21no—
199C4×C61—22no—
200C4×C61—22no—
201C4×C61—42no—
202C4×C61—42no—
203C4×C61—42no—
204C4×C61—42no—
205C4×C61—42no—
206C4×C61#45412yes—
207#3531D6111yescharacteristic, complemented
208#3217—612no—
209#3138—611no—
210#1658—612no—
211#1657—614no—
212#1656—612no—
213#1537—611no—
214#1529—612no—
215C2×C61—41no—
216C2×C61—82no—
217C2×C61C2×#45411yescharacteristic
218C2×C61#302112yes—
219D61—162no—
220#564C2×D6111yescharacteristic, complemented
221#563—22no—
222#563—22no—
223#563C2×D6112yescomplemented
224#562—22no—
225#562C2×D6111yescharacteristic, complemented
226#533—22no—
227#533C2×D6111yescharacteristic
228C2×#127—611no—
229#529—22no—
230#529—22no—
231#529C2×D6112yes—
232#496—614no—
233#490—612no—
234#490—612no—
235#490—614no—
236#486—1222no—
237#460—1222no—
238#459—1222no—
239#450—1222no—
240#447—1222no—
241#432—612no—
242#418—1222no—
243C4×M16—611no—
244#330—612no—
245#330—612no—
246#330—614no—
247#329—614no—
248C61#1900411yescharacteristic, complemented, Sylow
249C22×D4—1222no—
250C22×D4—1221no—
251C2×SD16—1224no—
252C2×D8—1224no—
253C2×M16—1221no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8C22⋊D6112yes—
257#127—612no—
258#127—1222no—
259#127—1221no—
260#126—22no—
261#126—22no—
262#126—22no—
263#126—22no—
264#126—22no—
265#125—22no—
266#125—22no—
267#125—22no—
268#125—22no—
269#125C22×D6112yes—
270#125C22×D6111yescharacteristic
271#125C22×D6111yescharacteristic
272#124—22no—
273#124—22no—
274#124C22×D6112yes—
275#123—22no—
276#123—22no—
277#123—22no—
278#123—22no—
279#123—22no—
280C22⋊Q8—22no—
281C22⋊Q8—22no—
282C22⋊Q8—22no—
283C22⋊Q8—22no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—22no—
287C22⋊Q8C22⋊D6112yes—
288#121—22no—
289#121—22no—
290#121C22⋊D6112yes—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×Q8C22⋊D6112yes—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4C22⋊D6112yes—
297#117—22no—
298#117—22no—
299#117C22⋊D6112yes—
300C2×C42—22no—
301C2×C42C22×D6111yescharacteristic
302C4⋊C8—1222no—
303#102—1224no—
304#102—1224no—
305#102—1224no—
306#102—1224no—
307C22⋊C8—1222no—
308#97—612no—
309#97—614no—
310C4×C8—1221no—
311C24—2441no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8C2×C22⋊D6112yes—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—1222no—
318C2×D4—1224no—
319C2×D4—4884no—
320C2×D4—2444no—
321C2×D4—2442no—
322C2×D4—2441no—
323C2×D4C2×C22⋊D6112yes—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4C2×C22⋊D6111yescharacteristic
328C22×C4C2×C22⋊D6112yes—
329SD16—4884no—
330D8—4884no—
331M16—2442no—
332C2×C8—1222no—
333C2×C8—1224no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4C2×C22⋊D6112yes—
348C4⋊C4C2×C22⋊D6112yes—
349C4⋊C4C2×C22⋊D6112yes—
350C4⋊C4C2×C22⋊D6112yes—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42C2×C22⋊D6111yescharacteristic
366C42C2×C22⋊D6112yes—
367C42C23×D6111yescharacteristic
368C23—22no—
369C23—4884no—
370C23—4882no—
371C23—2442no—
372C23#1451974611yescharacteristic
373Q8—42no—
374Q8—82no—
375Q8—82no—
376D4—42no—
377D4—82no—
378D4—82no—
379D4—4884no—
380D4—4882no—
381D4—2444no—
382D4—2444no—
383D4—2444no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4#1451974612yes—
395C2×C4#1451974611yescharacteristic
396C2×C4C22×C22⋊D6111yescharacteristic
397C2×C4C22×C22⋊D6112yes—
398C8—2442no—
399C8—2444no—
400C22—21no—
401C22—22no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—9764no—
406C22—4882no—
407C22—4884no—
408C22C2×#1451974611yescharacteristic
409C4—21no—
410C4—22no—
411C4—22no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4#1683993512yes—
418C2—41no—
419C2—82no—
420C2—9762no—
421C2C2×#1683993511yescharacteristic
422C2#2145519112yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3531C2
socleC22×C61C2×#120
hypercenter#3531D61
hypocenterC61#19004
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3531

Complements

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

NH ≅ G/Ntype
1. #21456834C2semidirect
2. #21456195C2semidirect
3. #21445121C2semidirect
4. #21445117C2semidirect
5. #21445112C2semidirect
6. #21444457C2semidirect
7. #21444426C2semidirect
8. C61×#3531C2semidirect
22. #16839504C22semidirect
23. #16839504C22semidirect
24. #16839504C22semidirect
25. #16839502C22semidirect
26. C61×#564C22semidirect
29. C61×#563C22semidirect
31. C61×#562C22semidirect
207. #3531D61semidirect
220. #564C2×D61semidirect
223. #563C2×D61semidirect
225. #562C2×D61semidirect
248. C61#19004semidirect

Maximal subgroups

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

subgroupindex
#214568342
#214561952
#214451212
#214451172
#214451122
#214444572
#214444262
C61×#35312
#1900461