Finite Group Databasebeta

← #29580503

Subgroups — #29580503

Order 15616.

770 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 101 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#21456576C211yesmaximal, characteristic
2C2×#16839918C211yesmaximal, characteristic, complemented
3#21446804C211yesmaximal, characteristic, complemented
4#21445754C211yesmaximal, characteristic, complemented
5#21444387C211yesmaximal, characteristic
6#21443667C211yesmaximal, characteristic, complemented
7C61×#2436C211yesmaximal, characteristic
8C22×#11758202—21no—
9C2×#14519756C2211yescharacteristic
10#16840092—21no—
11#16840078—21no—
12#16839989—22no—
13#16839918—22no—
14#16839918—22no—
15#16839913—21no—
16#16839900—21no—
17C2×#14519701—21no—
18C2×#14519701—22no—
19C2×C22⋊#92418C2211yescharacteristic
20C2×C22⋊#92418C2211yescharacteristic
21#16839622C2211yescharacteristic
22#16839528—22no—
23#16839496—22no—
24C23×Q8×C61C2211yescharacteristic
25C2×#124×C61—21no—
26C2×#121×C61C2211yescharacteristic
27C61×#393—22no—
28C61×#391—21no—
29C61×#386—22no—
30C4×C22⋊C4×C61C2211yescharacteristic
31C22×Dic122D411yescharacteristic
32C2×#11758202—24no—
33C2×#11758202—44no—
34C2×#11758200—22no—
35C22×#92418—21no—
36C22×#92418D412yes—
37#14519756—21no—
38#14519756—21no—
39#14519756—21no—
40#14519756—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61D411yescharacteristic
43C2×#11758196—21no—
44C2×C22⋊Dic61—21no—
45C2×C22⋊Dic61—21no—
46#14519708—22no—
47#14519701—24no—
48#14519701—44no—
49#14519700—22no—
50#14519698—22no—
51C22⋊#92418—21no—
52C22⋊#92418—21no—
53C22⋊#92418—21no—
54C22⋊#92418D412yes—
55#14519684—21no—
56C22×Q8×C61—21no—
57C22×Q8×C61—21no—
58C22×Q8×C61—21no—
59C22×Q8×C61—21no—
60C22×Q8×C61—21no—
61C22×Q8×C61—44no—
62C22×Q8×C61D412yes—
63C22×D4×C61—21no—
64C22×D4×C61D411yescharacteristic
65C23×C4×C61—21no—
66C23×C4×C61C2311yescharacteristic
67#124×C61—44no—
68#121×C61—21no—
69#121×C61—21no—
70#121×C61—21no—
71#121×C61—21no—
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C22⋊C4×C61—24no—
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61D412yes—
76C2×C42×C61—21no—
77#95×C61—24no—
78#95×C61—21no—
79C24×C61—21no—
80C24×C61C2×D411yescharacteristic
81C2×Q8×C61—24no—
82C2×Q8×C61—24no—
83C2×Q8×C61—24no—
84C2×Q8×C61—22no—
85C2×Q8×C61—22no—
86C2×Q8×C61—44no—
87C2×Q8×C61—42no—
88C2×Q8×C61—42no—
89C2×Q8×C61—42no—
90C2×Q8×C61—44no—
91C2×Q8×C61—44no—
92C2×Q8×C61—41no—
93C2×Q8×C61—41no—
94C2×D4×C61—21no—
95C2×D4×C61—21no—
96C2×D4×C61—42no—
97C2×D4×C61—41no—
98C2×D4×C61—41no—
99C2×D4×C61—42no—
100C2×D4×C61—42no—
101C22×C4×C61—22no—
102C22×C4×C61—21no—
103C22×C4×C61—22no—
104C22×C4×C61—42no—
105C22×C4×C61—44no—
106C22×C4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×Dic61—21no—
115C22×Dic61—21no—
116C22×Dic61—41no—
117C2×Dic122—21no—
118C2×Dic122—21no—
119C2×Dic122—42no—
120C4⋊C4×C61—21no—
121C4⋊C4×C61—21no—
122C22⋊C4×C61—22no—
123C22⋊C4×C61—22no—
124C22⋊C4×C61—48no—
125C22⋊C4×C61—41no—
126C22⋊C4×C61—41no—
127C42×C61—42no—
128C22⋊Dic61—41no—
129C22⋊Dic61—41no—
130C22⋊Dic61—42no—
131#11758202—48no—
132#11758200—82no—
133C4⋊Dic61—21no—
134C4⋊Dic61—21no—
135C4⋊Dic61—42no—
136#11758196—41no—
137#11758196—41no—
138C2×#92418—22no—
139C2×#92418—22no—
140C2×#92418—42no—
141C2×#92418—42no—
142C23×C61—41no—
143C23×C61—41no—
144C23×C61—42no—
145C23×C61—42no—
146C23×C61—41no—
147C23×C61—42no—
148C23×C61#12011yescharacteristic
149C23×C61#12111yescharacteristic
150C23×C61#12711yescharacteristic
151C23×C61#13512yes—
152C23×C61#13712yes—
153Q8×C61—24no—
154Q8×C61—24no—
155Q8×C61—42no—
156Q8×C61—44no—
157Q8×C61—42no—
158Q8×C61—44no—
159D4×C61—42no—
160D4×C61—81no—
161D4×C61—82no—
162D4×C61—81no—
163D4×C61—81no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61—24no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—42no—
172C2×C4×C61—44no—
173C2×C4×C61—44no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61—41no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61#12011yescharacteristic
181C2×C4×C61#12011yescharacteristic
182C2×C4×C61#12111yescharacteristic
183C2×C4×C61#12111yescharacteristic
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61—81no—
187C2×Dic61—81no—
188C2×Dic61—81no—
189C2×Dic61—81no—
190C2×Dic61—81no—
191Dic122—42no—
192Dic122—42no—
193#92418—42no—
194#92418—42no—
195#9956—611nomaximal, Sylow
196C22×C61—21no—
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—42no—
203C22×C61—42no—
204C22×C61—84no—
205C22×C61—81no—
206C22×C61—81no—
207C22×C61—81no—
208C22×C61#39211yescharacteristic
209C22×C61#45212yes—
210C22×C61#46412yes—
211C22×C61#49511yescharacteristic
212C22×C61#49711yescharacteristic
213C4×C61—21no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—44no—
217C4×C61—82no—
218C4×C61—81no—
219C4×C61B(4,𝔽2)12yes—
220Dic61—81no—
221Dic61—81no—
222Dic61—161no—
223#3187—611no—
224C2×#451—611no—
225#2436D6111yescharacteristic
226#1996—611no—
227#1787—611no—
228#1519—611no—
229#1282—611no—
230C2×C61—21no—
231C2×C61—22no—
232C2×C61—22no—
233C2×C61—21no—
234C2×C61—161no—
235C2×C61#200911yescharacteristic
236C2×C61#201511yescharacteristic
237C2×C61#202111yescharacteristic
238C23×Q8C2×D6111yescharacteristic
239C22×Q16—1221no—
240C2×#124—21no—
241C2×C22⋊Q8—611no—
242C2×#121C2×D6111yescharacteristic
243#483—1221no—
244#481—1221no—
245#465—1222no—
246#451—1222no—
247#451—1222no—
248#450—1221no—
249#448—1221no—
250C2×#103—1221no—
251C2×#103—1222no—
252C2×C22⋊C8—611no—
253C2×C22⋊C8—611no—
254#393—22no—
255#391—21no—
256#386—22no—
257#380—611no—
258C4×C22⋊C4C2×D6111yescharacteristic
259#342—1222no—
260#327—1222no—
261C61#995611yescharacteristic, complemented, Sylow
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8—21no—
265C22×Q8—21no—
266C22×Q8—21no—
267C22×Q8—44no—
268C22×Q8—611no—
269C22×Q8C22⋊D6112yes—
270C22×D4—21no—
271C22×D4C22⋊D6111yescharacteristic
272C23×C4—21no—
273C23×C4C22×D6111yescharacteristic
274C2×Q16—1224no—
275C2×Q16—2444no—
276C2×SD16—1222no—
277C22×C8—612no—
278C22×C8—1221no—
279#124—44no—
280C22⋊Q8—1221no—
281C22⋊Q8—1221no—
282C22⋊Q8—1221no—
283C22⋊Q8—1221no—
284#121—21no—
285#121—21no—
286#121—21no—
287#121—21no—
288C2×C4⋊C4—611no—
289C2×C4⋊C4—1221no—
290C2×C4⋊C4—1221no—
291C2×C4⋊C4C22⋊D6111yescharacteristic
292C2×C22⋊C4—24no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—1221no—
295C2×C22⋊C4—1221no—
296C2×C22⋊C4C22⋊D6112yes—
297C2×C42—21no—
298#106—1222no—
299#103—1222no—
300#103—1224no—
301#103—2444no—
302#102—1222no—
303C22⋊C8—612no—
304C22⋊C8—1221no—
305C22⋊C8—1221no—
306C22⋊C8—1221no—
307#95—24no—
308#95—21no—
309#95—1221no—
310C24—21no—
311C24C2×C22⋊D6111yescharacteristic
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—44no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—1221no—
326C2×Q8—1221no—
327C2×Q8—2442no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—42no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—42no—
334C2×D4—42no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—1221no—
343C22×C4—1221no—
344C22×C4—2441no—
345C22×C4D4×D6111yescharacteristic
346C22×C4D4×D6111yescharacteristic
347C22×C4D4×D6111yescharacteristic
348C22×C4#1175822411yescharacteristic
349C22×C4C2×C22⋊D6111yescharacteristic
350C22×C4C2×C22⋊D6111yescharacteristic
351Q16—2448no—
352SD16—4882no—
353C2×C8—1222no—
354C2×C8—1222no—
355C2×C8—2442no—
356C2×C8—2442no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—1221no—
360C4⋊C4—1221no—
361C4⋊C4—2442no—
362C4⋊C4—2441no—
363C4⋊C4—2441no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—48no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—2442no—
370C22⋊C4—2441no—
371C22⋊C4—2441no—
372C42—42no—
373C23—41no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—41no—
379C23#1451974611yescharacteristic
380C23#1451974811yescharacteristic
381C23#1451977311yescharacteristic
382C23#1451979712yes—
383C23#1451980712yes—
384Q8—24no—
385Q8—24no—
386Q8—42no—
387Q8—44no—
388Q8—42no—
389Q8—44no—
390Q8—2442no—
391Q8—2442no—
392D4—42no—
393D4—81no—
394D4—82no—
395D4—81no—
396D4—81no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—4881no—
414C2×C4—4881no—
415C2×C4—4881no—
416C2×C4—4881no—
417C2×C4—4881no—
418C2×C4—2441no—
419C2×C4—2441no—
420C2×C4#1451974511yescharacteristic
421C2×C4#1451974511yescharacteristic
422C2×C4#1451974911yescharacteristic
423C2×C4#1451974911yescharacteristic
424C8—2442no—
425C8—2442no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—42no—
433C22—42no—
434C22—84no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22#1683967011yescharacteristic
439C22#1683992412yes—
440C22#1683998112yes—
441C22#1684014411yescharacteristic
442C22#1684015711yescharacteristic
443C4—21no—
444C4—21no—
445C4—21no—
446C4—44no—
447C4—82no—
448C4—81no—
449C4—9761no—
450C4—4881no—
451C4—4881no—
452C4#1683994812yes—
453C2—22no—
454C2—21no—
455C2—22no—
456C2—21no—
457C2—161no—
458C2#2144687911yescharacteristic
459C2#2144691311yescharacteristic
460C2#2144694211yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#2436C2
socleC22×C61#392
prime radicalC2×#121×C61C22
hypercenter#2436D61
hypocenterC61#9956
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2436

Complements

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

NH ≅ G/Ntype
2. C2×#16839918C2semidirect
3. #21446804C2semidirect
4. #21445754C2semidirect
6. #21443667C2semidirect
261. C61#9956semidirect

Maximal subgroups

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

subgroupindex
#214565762
C2×#168399182
#214468042
#214457542
#214443872
#214436672
C61×#24362
#995661