Finite Group Databasebeta

← #30601964

Subgroups — #30601964

Order 15616.

510 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 90 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#21462009C211yesmaximal, characteristic
2#21462008C211yesmaximal, characteristic
3#21461695C211yesmaximal, characteristic
4#21461555C211yesmaximal, characteristic
5#21461259C211yesmaximal, characteristic
6#21461258C211yesmaximal, characteristic
7C2×#16840002C211yesmaximal, characteristic
8C2×#14519845—21no—
9D61⋊(C22⋊C8)C2211yescharacteristic
10C2×#14519831—21no—
11#16840704C2211yescharacteristic
12#16840704C2211yescharacteristic
13C4×C22⋊#6923C2211yescharacteristic
14#16840640—22no—
15#16840640—22no—
16#16840636—21no—
17(C2×#6923)⋊C8—21no—
18#16840616—22no—
19#16840615—22no—
20C22×#11758202C411yescharacteristic, complemented
21C22×C8×D61C2211yescharacteristic
22C2×#14519757C2211yescharacteristic
23C2×#14519753C2211yescharacteristic
24#16840002—21no—
25#16840002—21no—
26#16840002—21no—
27#16840002—21no—
28C2×#14519710C411yescharacteristic, complemented
29C2×#14519702C411yescharacteristic, complemented
30C2×#103×C61C411yescharacteristic, complemented
31C2×D61⋊C8—21no—
32C2×C4⋊#6923—21no—
33C2×C4⋊#6923—21no—
34C2×C22⋊#6923—21no—
35C2×C22⋊#6923—21no—
36C2×C22⋊#6923D412yes—
37C2×C4×#6923—21no—
38#14519845—42no—
39#14519834D412yes—
40C22⋊#92420—21no—
41#14519831—42no—
42(C2×#6923)⋊C4—21no—
43(C2×#6923)⋊C4—21no—
44(C2×#6923)⋊C4—21no—
45C22×Dic122C2×C411yescharacteristic
46C22×C4×D61C2311yescharacteristic
47C2×#11758202—21no—
48C2×#11758202—21no—
49C2×#11758202—42no—
50C22×#92418C2×C411yescharacteristic
51C2×C8×D61—42no—
52C2×C8×D61D411yescharacteristic
53C2×C8×D61D411yescharacteristic
54C2×C8×D61D411yescharacteristic
55C2×C8×D61D411yescharacteristic
56#14519757—21no—
57#14519757—21no—
58#14519757—21no—
59#14519757—21no—
60#14519753—21no—
61#14519753—21no—
62#14519753—21no—
63#14519753—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61C2×C411yescharacteristic
66C2×#11758196—21no—
67C2×#11758198—21no—
68C2×#11758198—21no—
69#14519710—21no—
70#14519710—21no—
71#14519702—21no—
72#14519702—21no—
73C22×Q8×C61C2×C411yescharacteristic
74C22×C8×C61C2×C411yescharacteristic
75C2×C4⋊C4×C61C2×C411yescharacteristic
76#103×C61—21no—
77#103×C61—21no—
78C23×D61C2×D411yescharacteristic
79C22×#6923—22no—
80C22×#6923—41no—
81C22×#6923—41no—
82C2×Q8×C61—21no—
83C2×Q8×C61—21no—
84C2×Q8×C61—41no—
85C2×Q8×C61—41no—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61C22×C411yescharacteristic
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61#4111yescharacteristic
92C2×C4×D61—41no—
93C2×C4×D61—41no—
94C2×C4×D61C2×D411yescharacteristic
95C2×C4×D61C2×D411yescharacteristic
96C2×C4×D61C2×D411yescharacteristic
97C2×C4×D61#4111yescharacteristic
98C2×Dic122—21no—
99C2×Dic122—21no—
100C2×Dic122—41no—
101C2×Dic122—41no—
102C22⋊#6923—22no—
103C22⋊#6923—22no—
104C22⋊#6923—42no—
105C22⋊#6923—42no—
106C2×#92420—22no—
107C4⋊#6923—41no—
108C4⋊#6923—41no—
109C4⋊#6923—41no—
110C4⋊#6923—41no—
111C4×#6923—41no—
112C4×#6923—41no—
113D61⋊C8—41no—
114D61⋊C8—41no—
115C2×C8×C61—41no—
116C2×C8×C61C22⋊C411yescharacteristic
117C2×C8×C61C22⋊C411yescharacteristic
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120#11758202—42no—
121#11758202—42no—
122#11758198—41no—
123#11758198—41no—
124#11758198—41no—
125#11758198—41no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4⋊Dic61—41no—
129C4⋊Dic61—41no—
130#11758196—41no—
131#11758196—41no—
132C2×#92418—22no—
133C2×#92418C22⋊C411yescharacteristic
134C2×#92418C22⋊C411yescharacteristic
135C8×D61—22no—
136C8×D61—22no—
137C8×D61—41no—
138C8×D61—41no—
139C23×C61C4×D411yescharacteristic
140C22×D61—41no—
141C22×D61—41no—
142C22×D61#12111yescharacteristic
143C22×D61#12711yescharacteristic
144C22×D61#13712yes—
145C22×D61#13712yes—
146C2×#6923—42no—
147C2×#6923—42no—
148C2×#6923—42no—
149C2×#6923—81no—
150C2×#6923—81no—
151C2×#6923—81no—
152C2×#6923—81no—
153C2×#6923—81no—
154C2×#6923—81no—
155Q8×C61—41no—
156Q8×C61—41no—
157Q8×C61—81no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—81no—
163C2×C4×C61—81no—
164C2×C4×C61C2×C22⋊C411yescharacteristic
165C2×C4×C61C4×D411yescharacteristic
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—81no—
170C2×Dic61—81no—
171C2×Dic61#12111yescharacteristic
172C2×Dic61#12411yescharacteristic
173C4×D61—21no—
174C4×D61—21no—
175C4×D61—21no—
176C4×D61—21no—
177C4×D61—41no—
178C4×D61—41no—
179Dic122—41no—
180Dic122—41no—
181Dic122—81no—
182#92420—42no—
183C8×C61—21no—
184C8×C61—21no—
185#92418—21no—
186#92418#9912yes—
187#9784—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—21no—
190C22×C61#39011yescharacteristic
191C22×C61#44112yes—
192C2×D61—21no—
193C2×D61—21no—
194C2×D61—21no—
195C2×D61—21no—
196C2×D61—42no—
197C2×D61—42no—
198C2×D61#47012yes—
199C2×D61#49711yescharacteristic
200C2×D61#49711yescharacteristic
201#6923—82no—
202#6923—161no—
203#6923—161no—
204C4×C61—21no—
205C4×C61—81no—
206C4×C61—81no—
207C4×C61C2×#9911yescharacteristic
208C4×C61#41011yescharacteristic
209Dic61—21no—
210Dic61—21no—
211Dic61—81no—
212Dic61—81no—
213C2×#467—611no—
214#1996—611no—
215#1996—611no—
216#1807—611no—
217#1752—611no—
218#1520—611no—
219#1520—611no—
220C2×C61—22no—
221C2×C61#196411yescharacteristic
222C2×C61#197111yescharacteristic
223C2×C61#197611yescharacteristic
224D61—22no—
225D61—21no—
226D61—21no—
227C22×Q16—611no—
228C23×C8—611no—
229C2×C22⋊Q8—611no—
230C2×C22⋊Q8—611no—
231#467—1221no—
232#467—1221no—
233#467—1221no—
234#467—1221no—
235C2×#107—611no—
236C2×#106—1221no—
237C2×#103—611no—
238C2×#103#692311yescharacteristic, complemented
239C2×C22⋊C8—611no—
240C2×#97—1221no—
241#380—611no—
242#380—611no—
243C4×C22⋊C4—611no—
244#342—1222no—
245#342—1222no—
246#340—1221no—
247#336—1221no—
248#328—1222no—
249#328—1222no—
250C61#978411yescharacteristic, complemented, Sylow
251C22×Q8—611no—
252C22×Q8C2×#692311yescharacteristic
253C23×C4—611no—
254C2×Q16—1221no—
255C2×Q16—1221no—
256C2×Q16—2442no—
257C22×C8—611no—
258C22×C8—611no—
259C22×C8—611no—
260C22×C8—611no—
261C22×C8—611no—
262C22×C8—1221no—
263C22×C8—2442no—
264C22×C8C2×#692311yescharacteristic
265C22⋊Q8—1221no—
266C22⋊Q8—1221no—
267C22⋊Q8—1221no—
268C22⋊Q8—1221no—
269C22⋊Q8—1221no—
270C22⋊Q8—1221no—
271C22⋊Q8—1221no—
272C22⋊Q8—1221no—
273C2×C4⋊C4—611no—
274C2×C4⋊C4—1221no—
275C2×C4⋊C4—1221no—
276C2×C4⋊C4—1221no—
277C2×C4⋊C4—1221no—
278C2×C4⋊C4C2×#692311yescharacteristic
279C2×C22⋊C4—612no—
280C2×C22⋊C4—1221no—
281C2×C22⋊C4—1221no—
282C2×C22⋊C4—1221no—
283C2×C22⋊C4—1221no—
284C2×C42—1221no—
285#107—1221no—
286#107—1221no—
287#106—2442no—
288#103—21no—
289#103—21no—
290#103—1221no—
291#103—1221no—
292C22⋊C8—612no—
293C22⋊C8—1221no—
294#97—2442no—
295#95—1221no—
296#95—1221no—
297#95—1221no—
298C24—611no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—1221no—
304C2×Q8—1221no—
305C2×Q8—2441no—
306C2×Q8—2441no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—611no—
310C22×C4—611no—
311C22×C4—611no—
312C22×C4—611no—
313C22×C4—611no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—1222no—
317C22×C4—2441no—
318C22×C4—2441no—
319C22×C4—2441no—
320C22×C4—2441no—
321C22×C4C22×#692311yescharacteristic
322Q16—2442no—
323Q16—2442no—
324C2×C8—41no—
325C2×C8—611no—
326C2×C8—611no—
327C2×C8—1222no—
328C2×C8—1222no—
329C2×C8—1222no—
330C2×C8—1222no—
331C2×C8—2441no—
332C2×C8—2441no—
333C2×C8—2441no—
334C2×C8—2441no—
335C2×C8C22⋊#692311yescharacteristic
336C2×C8C22⋊#692311yescharacteristic
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—1221no—
340C4⋊C4—1221no—
341C4⋊C4—2441no—
342C4⋊C4—2441no—
343C4⋊C4—2441no—
344C4⋊C4—2441no—
345C4⋊C4—2441no—
346C4⋊C4—2441no—
347C4⋊C4—2441no—
348C4⋊C4—2441no—
349C22⋊C4—1222no—
350C22⋊C4—1222no—
351C22⋊C4—2441no—
352C22⋊C4—2441no—
353C22⋊C4—2441no—
354C22⋊C4—2441no—
355C22⋊C4—2442no—
356C22⋊C4—2442no—
357C42—2441no—
358C42—2441no—
359C23—611no—
360C23—611no—
361C23—612no—
362C23—612no—
363C23—2441no—
364C23—2441no—
365C23D4×#692311yescharacteristic
366Q8—41no—
367Q8—41no—
368Q8—81no—
369Q8—4881no—
370Q8—2441no—
371Q8—2441no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—611no—
379C2×C4—611no—
380C2×C4—1221no—
381C2×C4—1221no—
382C2×C4—1221no—
383C2×C4—1221no—
384C2×C4—4881no—
385C2×C4—4881no—
386C2×C4—4881no—
387C2×C4—4881no—
388C2×C4—4881no—
389C2×C4—4881no—
390C2×C4—4881no—
391C2×C4—4881no—
392C2×C4—2441no—
393C2×C4—2441no—
394C2×C4—2441no—
395C2×C4—2442no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4C2×C22⋊#692311yescharacteristic
401C2×C4D4×#692311yescharacteristic
402C8—21no—
403C8—21no—
404C8—612no—
405C8—1221no—
406C8—2442no—
407C22—21no—
408C22—21no—
409C22—611no—
410C22—611no—
411C22—612no—
412C22—1221no—
413C22—1221no—
414C22—1221no—
415C22—1221no—
416C22—2442no—
417C22—2442no—
418C22#1684071411yescharacteristic
419C22#1684080212yes—
420C4—21no—
421C4—81no—
422C4—81no—
423C4—1221no—
424C4—1221no—
425C4—9761no—
426C4—9761no—
427C4—4882no—
428C4—4881no—
429C4—4881no—
430C4D61⋊#9911yescharacteristic
431C4#1684074111yescharacteristic
432C2—22no—
433C2—1221no—
434C2—1221no—
435C2—1222no—
436C2#2146196011yescharacteristic
437C2#2146196711yescharacteristic
438C2#2146197711yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#103×C61C4
socleC22×C61#390
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenterC2×#103#6923
hypocenterC61#9784
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
20. C22×#11758202C4semidirect
28. C2×#14519710C4semidirect
29. C2×#14519702C4semidirect
30. C2×#103×C61C4semidirect
238. C2×#103#6923semidirect
250. C61#9784semidirect

Maximal subgroups

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

subgroupindex
#214620092
#214620082
#214616952
#214615552
#214612592
#214612582
C2×#168400022
#978461