Finite Group Databasebeta

← #29596820

Subgroups — #29596820

Order 15616.

516 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 89 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#21448989C211yesmaximal, characteristic
2C2×#16839556C211yesmaximal, characteristic
3#21446207C211yesmaximal, characteristic
4#21446100C211yesmaximal, characteristic
5#21443937C211yesmaximal, characteristic
6#21443936C211yesmaximal, characteristic
7C61×#1867C211yesmaximal, characteristic
8C2×#14519764C2211yescharacteristic
9#16839770—21no—
10C2×#14519694C2211yescharacteristic
11C2×#14519692C2211yescharacteristic
12#16839666—21no—
13#16839656—21no—
14#16839650—21no—
15#16839642C2211yescharacteristic
16#16839635—21no—
17#16839634—21no—
18#16839634—21no—
19#16839626—21no—
20#16839556C414yescomplemented
21#16839529—21no—
22#16839521—21no—
23C61×#518C2211yescharacteristic
24C2×#99×C61C2211yescharacteristic
25C61×#386C2211yescharacteristic
26C61×#385—21no—
27C61×#342—21no—
28C22×Dic122D411yescharacteristic
29C23×Dic61D411yescharacteristic
30C2×#11758194—21no—
31#14519764—21no—
32#14519764—21no—
33#14519764C2×C412yes—
34#14519764Q812yes—
35C2×#11758196—21no—
36C2×#11758196—21no—
37C2×#11758196—21no—
38C2×#11758196—41no—
39C2×C22⋊Dic61—21no—
40C2×C22⋊Dic61—21no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61D411yescharacteristic
44C2×C4×Dic61—21no—
45C2×C4×Dic61D411yescharacteristic
46#14519700—41no—
47#14519700—41no—
48#14519694—21no—
49#14519694—21no—
50#14519692—21no—
51#14519692C2×C412yes—
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—41no—
58C22×D4×C61C2311yescharacteristic
59C23×C4×C61D411yescharacteristic
60C4×D4×C61—41no—
61C4×D4×C61—41no—
62#117×C61—21no—
63C2×C4⋊C4×C61—41no—
64C2×C22⋊C4×C61—21no—
65C2×C22⋊C4×C61—21no—
66C2×C22⋊C4×C61—21no—
67C2×C22⋊C4×C61—21no—
68C2×C22⋊C4×C61D411yescharacteristic
69#99×C61—21no—
70#99×C61C2×C412yes—
71#95×C61—21no—
72#95×C61—41no—
73C24×C61C2×D411yescharacteristic
74C24×C61C2×D411yescharacteristic
75C2×D4×C61—41no—
76C2×D4×C61—41no—
77C2×D4×C61C22×C411yescharacteristic
78C2×D4×C61C2×Q811yescharacteristic
79C2×D4×C61#4111yescharacteristic
80C2×D4×C61#4111yescharacteristic
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—41no—
88C22×C4×C61—41no—
89C22×C4×C61—81no—
90C22×C4×C61C2×D411yescharacteristic
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—41no—
96C22×Dic61—41no—
97C22×Dic61—81no—
98C2×Dic122—21no—
99C2×Dic122—41no—
100C2×Dic122—41no—
101C2×Dic122C4⋊C412yescomplemented
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—82no—
105C22⋊C4×C61—22no—
106C22⋊C4×C61—41no—
107C22⋊C4×C61—41no—
108C22⋊C4×C61—41no—
109C22⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C22⋊C4×C61—81no—
112C42×C61—41no—
113C22⋊Dic61—22no—
114C22⋊Dic61—22no—
115C22⋊Dic61—22no—
116C22⋊Dic61—22no—
117C22⋊Dic61—22no—
118C22⋊Dic61—41no—
119C22⋊Dic61—41no—
120C22⋊Dic61—81no—
121#11758196—81no—
122#11758196—81no—
123#11758196—81no—
124#11758196—82no—
125C4×Dic61—21no—
126C4×Dic61—81no—
127C4×Dic61C4⋊C412yescomplemented
128#11758194—41no—
129#11758194—41no—
130C2×#92418—41no—
131C23×C61—21no—
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—21no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61C4×D411yescharacteristic
140C23×C61C4×D411yescharacteristic
141C23×C61#12011yescharacteristic
142C23×C61C22⋊Q811yescharacteristic
143C23×C61C22⋊Q811yescharacteristic
144D4×C61—41no—
145D4×C61—41no—
146D4×C61—41no—
147D4×C61—41no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—42no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—81no—
161C2×C4×C61—81no—
162C2×C4×C61—81no—
163C2×C4×C61—81no—
164C2×C4×C61—81no—
165C2×C4×C61—81no—
166C2×C4×C61—81no—
167C2×C4×C61C2×C4⋊C411yescharacteristic
168C2×C4×C61#12311yescharacteristic
169C2×Dic61—22no—
170C2×Dic61—22no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176C2×Dic61—41no—
177C2×Dic61—42no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—42no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—81no—
184C2×Dic61—81no—
185C2×Dic61—81no—
186Dic122—42no—
187Dic122—81no—
188#92418—81no—
189#12967—611nomaximal, Sylow
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—41no—
197C22×C61—41no—
198C22×C61—41no—
199C22×C61—41no—
200C22×C61—41no—
201C22×C61—41no—
202C22×C61—41no—
203C22×C61—41no—
204C22×C61—81no—
205C22×C61—81no—
206C22×C61#38511yescharacteristic
207C22×C61B(4,𝔽2)11yescharacteristic
208C22×C61#45811yescharacteristic
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—81no—
212C4×C61—82no—
213C4×C61—81no—
214C4×C61—81no—
215C4×C61—81no—
216Dic61—42no—
217Dic61—42no—
218Dic61—81no—
219Dic61—82no—
220Dic61—81no—
221#2379—611no—
222C2×#354—611no—
223#1885—611no—
224#1867—611no—
225#1867D6111yescharacteristic
226#1390—611no—
227#1390—611no—
228C2×C61—21no—
229C2×C61—21no—
230C2×C61—41no—
231C2×C61—41no—
232C2×C61—41no—
233C2×C61—41no—
234C2×C61#186711yescharacteristic
235C2×C61#219911yescharacteristic
236C2×C61#220111yescharacteristic
237C2×#124—611no—
238#518C2×D6111yescharacteristic
239#419—1221no—
240C2×#100—611no—
241C2×#99—611no—
242C2×#99C2×D6111yescharacteristic
243#391—1221no—
244#388—1221no—
245#388—1221no—
246#386—611no—
247#386C2×D6111yescharacteristic
248#385—21no—
249#385—1221no—
250#385—1221no—
251#385—1221no—
252#382—1221no—
253#354—614no—
254#342—21no—
255#342—1221no—
256#339—1221no—
257C61#1296711yescharacteristic, complemented, Sylow
258C22×Q8—611no—
259C22×D4C22×D6111yescharacteristic
260C23×C4—611no—
261C23×C4C22⋊D6111yescharacteristic
262C2×M16—1221no—
263#124—612no—
264#124—612no—
265#124—1221no—
266#124—1221no—
267C4×D4—41no—
268C4×D4—41no—
269#117—21no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—1221no—
272C2×C4⋊C4—1221no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4—2441no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—611no—
280C2×C22⋊C4—1221no—
281C2×C22⋊C4—1221no—
282C2×C22⋊C4—1221no—
283C2×C22⋊C4—1221no—
284C2×C22⋊C4C22⋊D6111yescharacteristic
285C2×C42—611no—
286C2×C42—1221no—
287#103—2441no—
288#103—2441no—
289#100—1221no—
290#100—1221no—
291#99—21no—
292#99—612no—
293#99—1221no—
294#99C4×D6112yes—
295#95—21no—
296#95—41no—
297#95—1221no—
298#95—1221no—
299#95—1221no—
300#95—1221no—
301#95—1221no—
302#95—2441no—
303C24D4×D6111yescharacteristic
304C24C2×C22⋊D6111yescharacteristic
305C2×Q8—612no—
306C2×Q8—1221no—
307C2×Q8—2441no—
308C2×Q8—2441no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4C2×C4×D6111yescharacteristic
312C2×D4#1175822211yescharacteristic
313C2×D4Q8×D6111yescharacteristic
314C2×D4#1175822611yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—81no—
324C22×C4—1221no—
325C22×C4—1221no—
326C22×C4—1221no—
327C22×C4—1221no—
328C22×C4—4881no—
329C22×C4—2441no—
330C22×C4—2441no—
331C22×C4D4×D6111yescharacteristic
332M16—2441no—
333M16—2441no—
334C2×C8—2441no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—82no—
338C4⋊C4—4881no—
339C4⋊C4—4881no—
340C4⋊C4—4881no—
341C4⋊C4—4882no—
342C22⋊C4—22no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—81no—
349C22⋊C4—1222no—
350C22⋊C4—1222no—
351C22⋊C4—1222no—
352C22⋊C4—1222no—
353C22⋊C4—1222no—
354C22⋊C4—4881no—
355C22⋊C4—2441no—
356C22⋊C4—2441no—
357C42—41no—
358C42—612no—
359C42—1221no—
360C42—4881no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23C4×C22⋊D6111yescharacteristic
370C23#1451973911yescharacteristic
371C23#1451974511yescharacteristic
372C23#1451975311yescharacteristic
373C23#1451975511yescharacteristic
374Q8—4881no—
375Q8—2442no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—41no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—1222no—
400C2×C4—1222no—
401C2×C4—4881no—
402C2×C4—4881no—
403C2×C4—4881no—
404C2×C4—2442no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4—2442no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4—2441no—
413C2×C4—2441no—
414C2×C4—2441no—
415C2×C4—2441no—
416C2×C4C4⋊C4×D6111yescharacteristic
417C2×C4#1451976211yescharacteristic
418C8—4881no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—81no—
434C22—81no—
435C22#1683963311yescharacteristic
436C22#1683994811yescharacteristic
437C22#1683995111yescharacteristic
438C4—21no—
439C4—21no—
440C4—81no—
441C4—82no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—4881no—
446C4—4882no—
447C4—4881no—
448C4—2442no—
449C4—2442no—
450C2—21no—
451C2—21no—
452C2—41no—
453C2—41no—
454C2—41no—
455C2—41no—
456C2#2144609811yescharacteristic
457C2#2144781811yescharacteristic
458C2#2144783011yescharacteristic
subgroupG/H
centerC22#16839948
commutatorC2×D4×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#1867C2
socleC22×C61B(4,𝔽2)
prime radical, cosocleC22×D4×C61C23
hypercenter#1867D61
hypocenterC61#12967
LCSG ▷ C2×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1867

Complements

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

NH ≅ G/Ntype
20. #16839556C4semidirect
101. C2×Dic122C4⋊C4semidirect
127. C4×Dic61C4⋊C4semidirect
257. C61#12967semidirect

Maximal subgroups

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

subgroupindex
#214489892
C2×#168395562
#214462072
#214461002
#214439372
#214439362
C61×#18672
#1296761