Finite Group Databasebeta

← #29652468

Subgroups — #29652468

Order 15616.

1170 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1170 conjugacy classes of subgroups — too many to draw. Only its 183 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#21455077C212yesmaximal
2#21449154C211yesmaximal, characteristic
3#21444793C214yesmaximal, complemented
4#21444759C214yesmaximal, complemented
5#21444352C212yesmaximal, complemented
6#21444149C211yesmaximal, characteristic, complemented
7C2×C61×#544C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C2×#14519808—22no—
10C2×#14519760—21no—
11#16839922—24no—
12#16839917C2214yes—
13#16839912C2214yes—
14#16839761—22no—
15#16839726C2212yes—
16#16839682C2212yes—
17#16839679C2212yes—
18#16839641—21no—
19#16839633—22no—
20#16839625C2211yescharacteristic
21#16839499C2218yes—
22#16839488C2214yes—
23C61×#544—24no—
24C61×#544—22no—
25C61×#544C2214yes—
26C2×C4⋊Q8×C61—22no—
27C2×#125×C61—22no—
28C2×C22⋊Q8×C61—21no—
29C2×C22⋊Q8×C61C2212yes—
30C2×#117×C61—21no—
31C22×C4⋊C4×C61C2211yescharacteristic
32C2×#11758222—22no—
33C2×#11758222—22no—
34C2×#11758222—44no—
35C22×Dic122C2311yescharacteristic
36C22×D244C2311yescharacteristic
37C22×C22⋊D61—21no—
38C22×C4×D61—21no—
39#14519808—48no—
40C2×#11758202—24no—
41C2×#11758201—24no—
42C2×#11758194—22no—
43#14519760—44no—
44C2×#11758196—22no—
45C2×C22⋊Dic61—21no—
46C2×#11758198—22no—
47C2×#11758198—22no—
48#14519700—24no—
49#14519697—24no—
50C22⋊#92418—22no—
51C22⋊#92418C2314yes—
52#14519684—22no—
53#14519684—21no—
54#14519684C2312yes—
55C22×Q8×C61—21no—
56C22×Q8×C61D412yes—
57C23×C4×C61D411yescharacteristic
58C23×C4×C61D411yescharacteristic
59C23×C4×C61C2311yescharacteristic
60C4⋊Q8×C61—28no—
61C4⋊Q8×C61—28no—
62#125×C61—28no—
63#125×C61—28no—
64C22⋊Q8×C61—24no—
65C22⋊Q8×C61—22no—
66C22⋊Q8×C61—22no—
67C22⋊Q8×C61—24no—
68C22⋊Q8×C61D414yes—
69C22⋊Q8×C61C2314yes—
70#117×C61—24no—
71#117×C61—24no—
72C2×C4⋊C4×C61—21no—
73C2×C4⋊C4×C61—22no—
74C2×C4⋊C4×C61—21no—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61—22no—
77C2×C4⋊C4×C61—22no—
78C2×C4⋊C4×C61—22no—
79C2×C4⋊C4×C61D412yes—
80C2×C4⋊C4×C61D412yes—
81C2×C4⋊C4×C61C2312yes—
82C2×C22⋊C4×C61—22no—
83C2×C42×C61—22no—
84C24×C61C2×D411yescharacteristic
85C23×D61—21no—
86C2×Q8×C61—24no—
87C2×Q8×C61—22no—
88C2×Q8×C61—44no—
89C2×Q8×C61—82no—
90C2×Q8×C61C2×D414yes—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—24no—
94C22×C4×C61—22no—
95C22×C4×C61—22no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—42no—
99C22×C4×C61—42no—
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2×D412yes—
102C22×C4×C61C2×D412yes—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D412yes—
105C22×C4×C61C2411yescharacteristic
106C2×C22⋊D61—24no—
107C2×C22⋊D61—42no—
108C2×C22⋊D61—41no—
109C2×C22⋊D61—41no—
110C22×Dic61—21no—
111C22×Dic61—22no—
112#11758222—28no—
113#11758222—44no—
114#11758222—44no—
115#11758222—44no—
116C2×D244—22no—
117C2×D244—42no—
118C2×C4×D61—42no—
119C2×C4×D61—42no—
120C2×C4×D61—42no—
121C2×Dic122—22no—
122C2×Dic122—42no—
123C4⋊C4×C61—24no—
124C4⋊C4×C61—24no—
125C4⋊C4×C61—24no—
126C4⋊C4×C61—28no—
127C4⋊C4×C61—24no—
128C4⋊C4×C61—24no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—24no—
131C4⋊C4×C61—24no—
132C4⋊C4×C61C2×D414yes—
133C22⋊C4×C61—24no—
134C22⋊C4×C61—24no—
135C42×C61—28no—
136C22⋊Dic61—42no—
137#11758202—48no—
138#11758201—48no—
139#11758198—44no—
140#11758198—44no—
141#11758196—44no—
142#11758194—44no—
143C2×#92418—24no—
144C23×C61—22no—
145C23×C61—22no—
146C23×C61—41no—
147C23×C61#12011yescharacteristic
148C23×C61#12011yescharacteristic
149C23×C61C22×D411yescharacteristic
150C22×D61—22no—
151C22×D61—41no—
152C22×D61—42no—
153C22×D61—41no—
154C22×D61—42no—
155Q8×C61—44no—
156Q8×C61—82no—
157Q8×C61—84no—
158C2×C4×C61—24no—
159C2×C4×C61—24no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—24no—
163C2×C4×C61—24no—
164C2×C4×C61—24no—
165C2×C4×C61—22no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—42no—
169C2×C4×C61—44no—
170C2×C4×C61—42no—
171C2×C4×C61—44no—
172C2×C4×C61—42no—
173C2×C4×C61—81no—
174C2×C4×C61#12012yes—
175C2×C4×C61#13714yescomplemented
176C2×C4×C61C22×D412yes—
177C22⋊D61—44no—
178C22⋊D61—44no—
179C2×Dic61—22no—
180C2×Dic61—41no—
181C2×Dic61—41no—
182C2×Dic61—42no—
183C2×Dic61—42no—
184C2×Dic61—42no—
185D244—24no—
186D244—42no—
187C4×D61—48no—
188Dic122—24no—
189Dic122—42no—
190#92418—44no—
191#18143—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×C61B(4,𝔽2)12yes—
201C22×C61C2×#12011yescharacteristic
202C22×C61C2×#13712yes—
203C2×D61—44no—
204C2×D61—42no—
205C2×D61—82no—
206C2×D61—81no—
207C2×D61—81no—
208C4×C61—24no—
209C4×C61—22no—
210C4×C61—22no—
211C4×C61—44no—
212C4×C61—42no—
213C4×C61—44no—
214Dic61—42no—
215Dic61—82no—
216C2×#544D6111yescharacteristic, complemented
217#3007—612no—
218#2406—611no—
219#1605—614no—
220#1599—614no—
221#1511—612no—
222#1463—611no—
223C2×C61—22no—
224C2×C61—22no—
225C2×C61—41no—
226C2×C61#300411yescharacteristic
227C2×C61#302011yescharacteristic
228C2×C61C2×B(4,𝔽2)11yescharacteristic
229D61—82no—
230C22×#41—611no—
231C2×#137—1222no—
232#544—24no—
233#544—22no—
234#544C2×D6114yes—
235C2×C4⋊Q8—22no—
236C2×#125—22no—
237C2×#123—1221no—
238C2×C22⋊Q8—21no—
239C2×C22⋊Q8C2×D6112yes—
240C2×#117—21no—
241C22×C4⋊C4C2×D6111yescharacteristic
242#452—1224no—
243#451—614no—
244#450—614no—
245#417—1222no—
246#407—612no—
247#395—612no—
248#394—612no—
249#386—1221no—
250#385—1222no—
251#381—611no—
252#328—618no—
253#324—614no—
254C61#1814311yescharacteristic, complemented, Sylow
255C2×#41—1222no—
256C2×#41—1222no—
257C2×#41—2444no—
258C22×Q8—21no—
259C22×Q8—611no—
260C22×Q8C22⋊D6112yes—
261C22×D4—611no—
262C22×D4—1221no—
263C23×C4—1221no—
264C23×C4C22⋊D6111yescharacteristic
265C23×C4C22⋊D6111yescharacteristic
266C23×C4C22×D6111yescharacteristic
267#137—2448no—
268C2×Q16—1224no—
269C2×SD16—1224no—
270C2×M16—1222no—
271C4⋊Q8—28no—
272C4⋊Q8—28no—
273#125—28no—
274#125—28no—
275#123—2444no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—22no—
279C22⋊Q8—22no—
280C22⋊Q8C22⋊D6114yes—
281C22⋊Q8C22×D6114yes—
282#117—24no—
283#117—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—1222no—
292C2×C4⋊C4C22⋊D6112yes—
293C2×C4⋊C4C22⋊D6112yes—
294C2×C4⋊C4C22×D6112yes—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—1221no—
297C2×C22⋊C4—1222no—
298C2×C22⋊C4—1222no—
299C2×C42—22no—
300#103—1224no—
301#102—1224no—
302C22⋊C8—614no—
303C22⋊C8—1222no—
304#95—612no—
305#95—1221no—
306#95—1222no—
307C24—1221no—
308C24C2×C22⋊D6111yescharacteristic
309#41—1228no—
310#41—2444no—
311#41—2444no—
312#41—2444no—
313C2×Q8—24no—
314C2×Q8—22no—
315C2×Q8—44no—
316C2×Q8—82no—
317C2×Q8—1222no—
318C2×Q8—2442no—
319C2×Q8C2×C22⋊D6114yes—
320C2×D4—1224no—
321C2×D4—1222no—
322C2×D4—2442no—
323C2×D4—2442no—
324C2×D4—2441no—
325C2×D4—2441no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—1221no—
336C22×C4—1222no—
337C22×C4—2442no—
338C22×C4—2442no—
339C22×C4—2442no—
340C22×C4C2×C22⋊D6112yes—
341C22×C4C2×C22⋊D6112yes—
342C22×C4C2×C22⋊D6112yes—
343C22×C4C2×C22⋊D6112yes—
344C22×C4C2×C22⋊D6111yescharacteristic
345C22×C4C23×D6111yescharacteristic
346Q16—2448no—
347SD16—2448no—
348M16—2444no—
349C2×C8—1224no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—28no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—2444no—
360C4⋊C4C2×C22⋊D6114yes—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—2444no—
364C22⋊C4—2442no—
365C22⋊C4—2444no—
366C42—28no—
367C23—22no—
368C23—22no—
369C23—41no—
370C23—1222no—
371C23—2441no—
372C23—2442no—
373C23—2441no—
374C23—2442no—
375C23#1451974611yescharacteristic
376C23#1451974611yescharacteristic
377C23C22×C22⋊D6111yescharacteristic
378Q8—44no—
379Q8—82no—
380Q8—84no—
381Q8—1224no—
382Q8—2442no—
383D4—1224no—
384D4—2442no—
385D4—2444no—
386D4—2444no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—81no—
403C2×C4—1222no—
404C2×C4—2448no—
405C2×C4—2442no—
406C2×C4—2442no—
407C2×C4—2442no—
408C2×C4—2441no—
409C2×C4—2441no—
410C2×C4#1451974612yes—
411C2×C4#1451980814yescomplemented
412C2×C4C22×C22⋊D6112yes—
413C8—2444no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—44no—
422C22—4882no—
423C22—4881no—
424C22—4881no—
425C22—2444no—
426C22—2442no—
427C22#1683994912yes—
428C22C2×#1451974611yescharacteristic
429C22C2×#1451980812yes—
430C4—22no—
431C4—22no—
432C4—24no—
433C4—44no—
434C4—42no—
435C4—44no—
436C4—4882no—
437C4—2442no—
438C2—22no—
439C2—22no—
440C2—41no—
441C2—4882no—
442C2#2145505311yescharacteristic
443C2#2145518311yescharacteristic
444C2C2×#1683994911yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#544C2
socleC22×C61C2×#120
prime radicalC22×#11758222C22
hypercenterC2×#544D61
hypocenterC61#18143
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #21444793C2semidirect
4. #21444759C2semidirect
5. #21444352C2semidirect
6. #21444149C2semidirect
7. C2×C61×#544C2semidirect
175. C2×C4×C61#137semidirect
216. C2×#544D61semidirect
254. C61#18143semidirect
411. C2×C4#14519808semidirect

Maximal subgroups

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

subgroupindex
#214550772
#214491542
#214447932
#214447592
#214443522
#214441492
C2×C61×#5442
#1814361