Finite Group Databasebeta

← #29653925

Subgroups — #29653925

Order 15616.

1220 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 187 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
1C2×#16840343C211yesmaximal, characteristic
2#21456577C212yesmaximal, complemented
3#21451632C211yesmaximal, characteristic, complemented
4#21444858C214yesmaximal, complemented
5#21444790C214yesmaximal, complemented
6#21444388C212yesmaximal, complemented
7C2×C61×#324C211yesmaximal, characteristic, complemented
8C22×#11758222—21no—
9#16840343—24no—
10#16840343—22no—
11#16840343C2214yes—
12C2×#14519757—22no—
13C2×#14519756C2211yescharacteristic
14C2×#14519752—22no—
15C2×#14519751—21no—
16C2×#14519751C2212yes—
17#16840093—22no—
18#16840081C2214yescomplemented
19#16839991—24no—
20#16839915—22no—
21#16839903C2214yescomplemented
22#16839701C2212yescomplemented
23#16839695—22no—
24#16839682C2212yescomplemented
25#16839635—21no—
26#16839625—21no—
27#16839622C2211yescharacteristic
28#16839497C2218yescomplemented
29C22×C4⋊C4×C61C2211yescharacteristic
30C2×C22⋊C8×C61C2212yes—
31C61×#324C2214yescomplemented
32C2×#11758222—24no—
33C2×#11758222—24no—
34C2×#11758222—44no—
35C2×#11758222—44no—
36C22×Dic122—21no—
37C22×Dic122C2311yescharacteristic
38C22×D244—21no—
39C22×D244—21no—
40C22×D244D412yes—
41C22×C22⋊D61—22no—
42C22×C4×D61—22no—
43C2×#11758190—24no—
44#14519757—28no—
45#14519757—28no—
46#14519756—22no—
47#14519756—22no—
48#14519752—28no—
49#14519752—28no—
50#14519751—24no—
51#14519751—24no—
52#14519751—22no—
53#14519751—22no—
54#14519751D414yes—
55#14519751C2314yes—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61D412yes—
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758196—21no—
61C2×#11758196—22no—
62C2×C22⋊Dic61—21no—
63C2×C22⋊Dic61—21no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66#14519709—24no—
67#14519702—24no—
68#14519699—24no—
69#14519684—22no—
70#14519684—22no—
71#14519684—22no—
72#14519684C2312yes—
73C23×C4×C61D411yescharacteristic
74C23×C4×C61D411yescharacteristic
75C23×C4×C61C2311yescharacteristic
76C22×C8×C61—22no—
77C2×C4⋊C4×C61—22no—
78C2×C4⋊C4×C61—22no—
79C2×C4⋊C4×C61D412yes—
80C2×C4⋊C4×C61C2312yes—
81C22⋊C8×C61—22no—
82C22⋊C8×C61C2314yes—
83C24×C61C2×D411yescharacteristic
84C23×D61—22no—
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—22no—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—81no—
91C22×C4×C61—81no—
92C22×C4×C61SD1614yescomplemented
93C22×C4×C61C2×D412yes—
94C22×C4×C61C2×D412yes—
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2×D412yes—
98C22×C4×C61C2411yescharacteristic
99C2×C22⋊D61—24no—
100C2×C22⋊D61—24no—
101C2×C22⋊D61—44no—
102C2×C22⋊D61—44no—
103C22×Dic61—22no—
104C22×Dic61—21no—
105C22×Dic61—21no—
106C22×Dic61—22no—
107#11758222—48no—
108#11758222—416no—
109#11758222—48no—
110C2×D244—24no—
111C2×D244—24no—
112C2×D244—22no—
113C2×D244—44no—
114C2×D244—82no—
115C2×D244—82no—
116C2×D244C2×D414yes—
117C2×C4×D61—28no—
118C2×C4×D61—48no—
119C2×Dic122—24no—
120C2×Dic122—22no—
121C2×Dic122—44no—
122C2×Dic122—81no—
123C2×C8×C61—24no—
124C2×C8×C61—82no—
125C4⋊C4×C61—24no—
126C4⋊C4×C61—24no—
127C22⋊Dic61—41no—
128C22⋊Dic61—41no—
129C22⋊Dic61—42no—
130#11758198—28no—
131#11758198—24no—
132#11758198—24no—
133C4⋊Dic61—24no—
134C4⋊Dic61—22no—
135C4⋊Dic61—22no—
136C4⋊Dic61—81no—
137C4⋊Dic61C2×D414yes—
138#11758196—42no—
139#11758196—82no—
140#11758190—84no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61#12011yescharacteristic
144C23×C61#12011yescharacteristic
145C23×C61C2×SD1612yes—
146C23×C61AGL(1,ℤ/8)12yes—
147C23×C61C22×D411yescharacteristic
148C22×D61—24no—
149C22×D61—44no—
150C22×D61—44no—
151C22×D61—44no—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—24no—
155C2×C4×C61—24no—
156C2×C4×C61—22no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—42no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61—81no—
164C2×C4×C61#12012yes—
165C2×C4×C61C2×SD1614yes—
166C2×C4×C61C22×D412yes—
167C22⋊D61—44no—
168C22⋊D61—44no—
169C22⋊D61—48no—
170C2×Dic61—24no—
171C2×Dic61—44no—
172C2×Dic61—42no—
173C2×Dic61—42no—
174C2×Dic61—42no—
175C2×Dic61—42no—
176C2×Dic61—41no—
177C2×Dic61—41no—
178C2×Dic61—42no—
179D244—44no—
180D244—48no—
181D244—82no—
182D244—84no—
183C4×D61—416no—
184Dic122—48no—
185Dic122—44no—
186C8×C61—44no—
187#18293—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—42no—
195C22×C61—42no—
196C22×C61#45012yes—
197C22×C61#45012yes—
198C22×C61C2×#12011yescharacteristic
199C22×C61C22×SD1611yescharacteristic
200C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
201C2×D61—44no—
202C2×D61—44no—
203C2×D61—44no—
204C2×D61—88no—
205C4×C61—24no—
206C4×C61—22no—
207C4×C61—22no—
208C4×C61—42no—
209Dic61—44no—
210Dic61—82no—
211Dic61—81no—
212Dic61—81no—
213C2×#537—611no—
214#3187—612no—
215#2627—611no—
216#1616—614no—
217#1604—614no—
218#1519—612no—
219C2×#324D6111yescharacteristic, complemented
220C2×C61—21no—
221C2×C61—22no—
222C2×C61—21no—
223C2×C61—22no—
224C2×C61C2×#45011yescharacteristic
225C2×C61#302011yescharacteristic
226C2×C61#302611yescharacteristic
227D61—84no—
228C22×#41—1221no—
229#537—614no—
230#537—1224no—
231#537—1222no—
232C2×C22⋊Q8—611no—
233C2×C22⋊Q8—1222no—
234C2×#121—612no—
235C2×#121—1222no—
236C2×#121—1221no—
237C22×C4⋊C4C2×D6111yescharacteristic
238#483—1222no—
239#481—614no—
240#465—1224no—
241#450—1222no—
242#448—614no—
243C2×C22⋊C8C2×D6112yes—
244#400—612no—
245#398—1222no—
246#395—612no—
247#385—1221no—
248#381—1221no—
249#380—611no—
250#327—618no—
251#324C2×D6114yescomplemented
252C61#1829311yescharacteristic, complemented, Sylow
253C2×#41—1224no—
254C2×#41—1224no—
255C2×#41—2444no—
256C2×#41—2444no—
257C22×Q8—611no—
258C22×Q8—1221no—
259C22×D4—612no—
260C22×D4—1221no—
261C22×D4—1221no—
262C22×D4—1222no—
263C23×C4—1222no—
264C23×C4D24411yescharacteristic
265C23×C4D24411yescharacteristic
266C23×C4C22×D6111yescharacteristic
267C2×SD16—1224no—
268C22×C8—22no—
269C22⋊Q8—1228no—
270C22⋊Q8—1228no—
271C22⋊Q8—1222no—
272C22⋊Q8—1222no—
273#121—614no—
274#121—614no—
275#121—1228no—
276#121—1228no—
277#121—1224no—
278#121—1222no—
279#121—1224no—
280#121—1222no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—612no—
284C2×C4⋊C4—611no—
285C2×C4⋊C4—1221no—
286C2×C4⋊C4—1221no—
287C2×C4⋊C4—1222no—
288C2×C4⋊C4—1221no—
289C2×C4⋊C4D24412yes—
290C2×C4⋊C4C22×D6112yes—
291C2×C22⋊C4—1222no—
292C2×C22⋊C4—1222no—
293C2×C22⋊C4—1221no—
294C2×C22⋊C4—1221no—
295#106—1224no—
296#103—1224no—
297#102—1224no—
298C22⋊C8—22no—
299C22⋊C8C22×D6114yes—
300#95—612no—
301#95—1222no—
302#95—1222no—
303#95—1222no—
304C24—1222no—
305C24C2×D24411yescharacteristic
306#41—2448no—
307#41—24416no—
308#41—2448no—
309C2×Q8—1224no—
310C2×Q8—1222no—
311C2×Q8—4881no—
312C2×Q8—2444no—
313C2×D4—614no—
314C2×D4—1224no—
315C2×D4—1224no—
316C2×D4—1224no—
317C2×D4—1224no—
318C2×D4—1222no—
319C2×D4—4882no—
320C2×D4—4882no—
321C2×D4—2444no—
322C2×D4—2444no—
323C2×D4—2444no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4—1228no—
332C22×C4—1222no—
333C22×C4—1222no—
334C22×C4—1221no—
335C22×C4—1221no—
336C22×C4—2448no—
337C22×C4#1175819014yescomplemented
338C22×C4C2×D24412yes—
339C22×C4C2×D24412yes—
340C22×C4C2×D24411yescharacteristic
341C22×C4D4×D6112yes—
342C22×C4D4×D6112yes—
343C22×C4C23×D6111yescharacteristic
344SD16—4884no—
345C2×C8—24no—
346C2×C8—82no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—614no—
350C4⋊C4—1224no—
351C4⋊C4—1222no—
352C4⋊C4—1222no—
353C4⋊C4—4882no—
354C4⋊C4—4881no—
355C4⋊C4—2442no—
356C22⋊C4—1228no—
357C22⋊C4—1224no—
358C22⋊C4—1224no—
359C22⋊C4—2441no—
360C22⋊C4—2441no—
361C22⋊C4—2442no—
362C23—41no—
363C23—41no—
364C23—1224no—
365C23—2444no—
366C23—2444no—
367C23—2444no—
368C23#1451974711yescharacteristic
369C23#1451974711yescharacteristic
370C23C2×#1175819012yes—
371C23#1451980512yes—
372C23C22×D24411yescharacteristic
373Q8—2444no—
374Q8—2448no—
375D4—4882no—
376D4—4884no—
377D4—2444no—
378D4—2448no—
379D4—2444no—
380D4—2444no—
381D4—2448no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—81no—
394C2×C4—1224no—
395C2×C4—24416no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4—2442no—
399C2×C4—2442no—
400C2×C4—2441no—
401C2×C4—2441no—
402C2×C4—2442no—
403C2×C4—2444no—
404C2×C4#1451974712yes—
405C2×C4C2×#1175819014yes—
406C2×C4C2×D4×D6112yes—
407C8—44no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22—4888no—
417C22—2444no—
418C22—2444no—
419C22—2444no—
420C22#1683991512yes—
421C22#1683991512yes—
422C22C2×#1451974711yescharacteristic
423C22C22×#1175819011yescharacteristic
424C22C2×#1451980511yescharacteristic
425C4—22no—
426C4—22no—
427C4—24no—
428C4—42no—
429C4—4882no—
430C4—4881no—
431C4—4881no—
432C4—2444no—
433C2—21no—
434C2—22no—
435C2—21no—
436C2—22no—
437C2—4884no—
438C2C2×#1683991511yescharacteristic
439C2#2145518611yescharacteristic
440C2#2145521811yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#324C2
socleC22×C61C2×#120
prime radicalC2×#16840343C2
hypercenterC2×#324D61
hypocenterC61#18293
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #21456577C2semidirect
3. #21451632C2semidirect
4. #21444858C2semidirect
5. #21444790C2semidirect
6. #21444388C2semidirect
7. C2×C61×#324C2semidirect
18. #16840081C22semidirect
21. #16839903C22semidirect
22. #16839701C22semidirect
24. #16839682C22semidirect
28. #16839497C22semidirect
31. C61×#324C22semidirect
92. C22×C4×C61SD16semidirect
219. C2×#324D61semidirect
251. #324C2×D61semidirect
252. C61#18293semidirect
337. C22×C4#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×#168403432
#214565772
#214516322
#214448582
#214447902
#214443882
C2×C61×#3242
#1829361