Finite Group Databasebeta

← #30598367

Subgroups — #30598367

Order 15616.

818 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 114 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#21461723C212yesmaximal, complemented
2#21461680C211yesmaximal, characteristic
3#21461259C212yesmaximal, complemented
4#21461056C211yesmaximal, characteristic
5C2×#16840472C211yesmaximal, characteristic
6C2×#14519836—21no—
7D61⋊(C22⋊C8)C2212yes—
8#16840704C2212yes—
9#16840640—21no—
10#16840635—24no—
11#16840616—24no—
12#16840607—21no—
13#16840606—21no—
14#16840472—24no—
15#16840472—24no—
16C2×#14519777C411yescharacteristic, complemented
17C2×#14519769—21no—
18C2×#14519755—21no—
19C2×#14519755—21no—
20C2×#14519753C411yescharacteristic, complemented
21C2×#14519753C2211yescharacteristic
22C2×C4×D244—21no—
23C2×C4×D244C2211yescharacteristic
24C2×C4⋊C4×D61C411yescharacteristic, complemented
25C2×C4⋊C4×D61C2211yescharacteristic
26C2×C4×Q8×C61C411yescharacteristic, complemented
27C2×D61⋊C8—22no—
28C2×C4⋊#6923—22no—
29C2×C22⋊#6923—22no—
30#14519836—42no—
31#14519834—22no—
32C22⋊#92420—22no—
33(C2×#6923)⋊C4—22no—
34C22×Dic122—21no—
35C22×D244D411yescharacteristic
36C22×C4×D61—21no—
37C22×C4×D61—21no—
38C22×C4×D61C2×C411yescharacteristic
39C22×C4×D61C2311yescharacteristic
40#14519777—22no—
41#14519777—22no—
42#14519769—28no—
43#14519755—28no—
44#14519755—28no—
45#14519753—22no—
46#14519753—22no—
47#14519753—22no—
48#14519753D414yes—
49C4×D244—28no—
50C4×D244—22no—
51C4×D244—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2×C411yescharacteristic
54C2×C4⋊Dic61C2×C411yescharacteristic
55C2×C4⋊Dic61D411yescharacteristic
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×#11758196—21no—
59C2×#11758196—21no—
60C4⋊C4×D61—22no—
61C4⋊C4×D61—22no—
62C4⋊C4×D61—22no—
63C4⋊C4×D61D414yes—
64C2×#11758198—21no—
65C2×#11758198—21no—
66C2×#11758198D411yescharacteristic
67C2×#11758198D411yescharacteristic
68C22×Q8×C61C2×C411yescharacteristic
69C4×Q8×C61—22no—
70C4×Q8×C61—22no—
71C2×C4⋊C4×C61—21no—
72C2×C4⋊C4×C61C2×C411yescharacteristic
73C2×C42×C61—21no—
74C2×C42×C61C2×C411yescharacteristic
75C23×D61C22⋊C411yescharacteristic
76C23×D61C2×D411yescharacteristic
77C22×#6923—42no—
78C2×Q8×C61—21no—
79C2×Q8×C61—42no—
80C2×Q8×C61C22⋊C412yes—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C22⋊C411yescharacteristic
85C22×C4×C61C22×C411yescharacteristic
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61C22⋊C411yescharacteristic
89C22×Dic61C2×D411yescharacteristic
90C2×D244—22no—
91C2×D244—22no—
92C2×D244SD1612yes—
93C2×D244Q1612yes—
94C2×C4×D61—21no—
95C2×C4×D61—21no—
96C2×C4×D61—24no—
97C2×C4×D61—24no—
98C2×C4×D61—42no—
99C2×C4×D61—42no—
100C2×C4×D61—82no—
101C2×C4×D61—82no—
102C2×C4×D61C2×D412yes—
103C2×C4×D61C2×D412yes—
104C2×Dic122—24no—
105C2×Dic122—82no—
106C22⋊#6923—42no—
107C22⋊#6923—42no—
108C2×#92420—42no—
109C4⋊#6923—44no—
110D61⋊C8—44no—
111C4⋊C4×C61—24no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61C22⋊C412yes—
114C42×C61—24no—
115C42×C61—21no—
116C42×C61—21no—
117#11758198—24no—
118#11758198—24no—
119#11758198—22no—
120#11758198—22no—
121C4⋊Dic61—24no—
122C4⋊Dic61—21no—
123C4⋊Dic61—21no—
124C4⋊Dic61—22no—
125C4⋊Dic61C22⋊C412yes—
126C4⋊Dic61C22⋊C412yes—
127#11758196—24no—
128#11758196—24no—
129#11758196—24no—
130#11758196—24no—
131C23×C61C2×C22⋊C411yescharacteristic
132C22×D61—21no—
133C22×D61—21no—
134C22×D61—41no—
135C22×D61—41no—
136C22×D61—41no—
137C22×D61—41no—
138C22×D61#10312yes—
139C22×D61#12012yes—
140C22×D61C2×SD1611yescharacteristic
141C22×D61C2×Q1611yescharacteristic
142C22×D61#13712yes—
143C2×#6923—82no—
144C2×#6923—82no—
145C2×#6923—82no—
146Q8×C61—42no—
147Q8×C61—42no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—21no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—41no—
157C2×C4×C61#10312yes—
158C2×C4×C61C2×C22⋊C411yescharacteristic
159C2×C4×C61C2×C22⋊C411yescharacteristic
160C2×Dic61—21no—
161C2×Dic61—22no—
162C2×Dic61—22no—
163C2×Dic61—41no—
164C2×Dic61—42no—
165C2×Dic61—42no—
166C2×Dic61—41no—
167C2×Dic61#12012yes—
168D244—24no—
169D244—24no—
170C4×D61—24no—
171C4×D61—42no—
172C4×D61—42no—
173C4×D61—42no—
174C4×D61—84no—
175C4×D61—84no—
176Dic122—84no—
177#92420—82no—
178#8380—611nomaximal, Sylow
179C22×C61—21no—
180C22×C61—21no—
181C22×C61#37911yescharacteristic
182C22×C61C2×#10311yescharacteristic
183C22×C61#41911yescharacteristic
184C2×D61—22no—
185C2×D61—22no—
186C2×D61—22no—
187C2×D61—41no—
188C2×D61—41no—
189C2×D61—42no—
190C2×D61—42no—
191C2×D61—41no—
192C2×D61—41no—
193C2×D61—42no—
194C2×D61#44812yes—
195C2×D61#45112yes—
196#6923—162no—
197C4×C61—21no—
198C4×C61—22no—
199C4×C61—41no—
200C4×C61—42no—
201C4×C61—42no—
202C4×C61#41012yes—
203Dic61—22no—
204Dic61—42no—
205Dic61—42no—
206Dic61—41no—
207C2×#554—611no—
208#1824—612no—
209#1799—611no—
210#1520—612no—
211#1459—611no—
212C2×C61—21no—
213C2×C61—21no—
214C2×C61#178211yescharacteristic
215C2×C61#178711yescharacteristic
216C2×C61#179511yescharacteristic
217D61—22no—
218D61—22no—
219D61—42no—
220#554—1224no—
221#554—1224no—
222C2×C4⋊Q8—611no—
223C2×#125—1221no—
224C2×C22⋊Q8—611no—
225C2×C22⋊Q8—611no—
226C2×C22⋊Q8—1221no—
227C2×C22⋊Q8—1221no—
228C2×C4×Q8#692311yescharacteristic, complemented
229C2×C4×D4—611no—
230C2×C4×D4—1221no—
231C22×C4⋊C4—611no—
232C22×C4⋊C4—611no—
233C2×#99—1221no—
234C2×C22⋊C8—612no—
235#380—612no—
236#342—1221no—
237#340—1224no—
238#328—1224no—
239#324—1221no—
240#323—1221no—
241C61#838011yescharacteristic, complemented, Sylow
242C22×Q8—1221no—
243C22×Q8C2×#692311yescharacteristic
244C22×D4—611no—
245C23×C4—611no—
246C23×C4—611no—
247C23×C4—1221no—
248C23×C4—1221no—
249C22×C8—1222no—
250C4⋊Q8—1222no—
251C4⋊Q8—1222no—
252#125—1228no—
253C22⋊Q8—614no—
254C22⋊Q8—1228no—
255C22⋊Q8—1228no—
256C22⋊Q8—1222no—
257C22⋊Q8—1222no—
258C22⋊Q8—1222no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×D4—1222no—
262C4×D4—1222no—
263C4×D4—1228no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—611no—
266C2×C4⋊C4—611no—
267C2×C4⋊C4—614no—
268C2×C4⋊C4—611no—
269C2×C4⋊C4—1222no—
270C2×C4⋊C4—1222no—
271C2×C4⋊C4—1221no—
272C2×C4⋊C4—1221no—
273C2×C4⋊C4—1221no—
274C2×C4⋊C4—1221no—
275C2×C4⋊C4—1221no—
276C2×C4⋊C4—1222no—
277C2×C4⋊C4—1222no—
278C2×C4⋊C4C2×#692311yescharacteristic
279C2×C22⋊C4—611no—
280C2×C22⋊C4—611no—
281C2×C22⋊C4—1221no—
282C2×C22⋊C4—1221no—
283C2×C22⋊C4—1222no—
284C2×C42—21no—
285C2×C42C2×#692311yescharacteristic
286#99—2442no—
287C22⋊C8—1222no—
288C22⋊C8—1222no—
289#95—1222no—
290C24—611no—
291C24—611no—
292C2×Q8—21no—
293C2×Q8—42no—
294C2×Q8—1224no—
295C2×Q8—4882no—
296C2×Q8C22⋊#692312yes—
297C2×D4—612no—
298C2×D4—612no—
299C2×D4—1222no—
300C2×D4—1222no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—611no—
304C22×C4—612no—
305C22×C4—612no—
306C22×C4—611no—
307C22×C4—1224no—
308C22×C4—1224no—
309C22×C4—1221no—
310C22×C4—1221no—
311C22×C4—1221no—
312C22×C4—1221no—
313C22×C4—4882no—
314C22×C4—4882no—
315C22×C4—2442no—
316C22×C4—2442no—
317C22×C4—2442no—
318C22×C4C22⋊#692311yescharacteristic
319C22×C4C22⋊#692311yescharacteristic
320C22×C4C22×#692311yescharacteristic
321C2×C8—2442no—
322C2×C8—2444no—
323C4⋊C4—24no—
324C4⋊C4—21no—
325C4⋊C4—612no—
326C4⋊C4—612no—
327C4⋊C4—1224no—
328C4⋊C4—1222no—
329C4⋊C4—1224no—
330C4⋊C4—1224no—
331C4⋊C4—1224no—
332C4⋊C4—1224no—
333C4⋊C4—1221no—
334C4⋊C4—1221no—
335C4⋊C4—2444no—
336C4⋊C4C22⋊#692312yes—
337C22⋊C4—1222no—
338C22⋊C4—1222no—
339C22⋊C4—1224no—
340C22⋊C4—1224no—
341C22⋊C4—2442no—
342C22⋊C4—2442no—
343C42—24no—
344C42—21no—
345C42—21no—
346C23—612no—
347C23—611no—
348C23—611no—
349C23—612no—
350C23—612no—
351C23—1221no—
352C23—1221no—
353C23—2441no—
354C23—2441no—
355C23—2441no—
356C23—2441no—
357C23C2×C22⋊#692311yescharacteristic
358Q8—42no—
359Q8—42no—
360Q8—4884no—
361D4—1224no—
362D4—1224no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—612no—
373C2×C4—1224no—
374C2×C4—1222no—
375C2×C4—1222no—
376C2×C4—1221no—
377C2×C4—4884no—
378C2×C4—4884no—
379C2×C4—4882no—
380C2×C4—4882no—
381C2×C4—4882no—
382C2×C4—2442no—
383C2×C4—2442no—
384C2×C4—2441no—
385C2×C4—2442no—
386C2×C4—2442no—
387C2×C4—2442no—
388C2×C4—2441no—
389C2×C4#1451984012yes—
390C2×C4C2×C22⋊#692311yescharacteristic
391C2×C4C2×C22⋊#692311yescharacteristic
392C8—4882no—
393C22—21no—
394C22—21no—
395C22—612no—
396C22—612no—
397C22—1222no—
398C22—1222no—
399C22—1222no—
400C22—2441no—
401C22—2441no—
402C22—2442no—
403C22—2442no—
404C22—2442no—
405C22—2441no—
406C22—2441no—
407C22#1684070311yescharacteristic
408C22C2×#1451984011yescharacteristic
409C22#1684075511yescharacteristic
410C4—21no—
411C4—22no—
412C4—42no—
413C4—42no—
414C4—41no—
415C4—1222no—
416C4—9762no—
417C4—2441no—
418C4—2442no—
419C4—2442no—
420C4#1684074112yes—
421C2—21no—
422C2—21no—
423C2—1222no—
424C2—1222no—
425C2—2442no—
426C2#2146164811yescharacteristic
427C2#2146165811yescharacteristic
428C2#2146167011yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61#379
prime radicalC22×D244D4
cosocleC22×C4×D61C23
hypercenterC2×C4×Q8#6923
hypocenterC61#8380
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #21461723C2semidirect
3. #21461259C2semidirect
16. C2×#14519777C4semidirect
20. C2×#14519753C4semidirect
24. C2×C4⋊C4×D61C4semidirect
26. C2×C4×Q8×C61C4semidirect
228. C2×C4×Q8#6923semidirect
241. C61#8380semidirect

Maximal subgroups

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

subgroupindex
#214617232
#214616802
#214612592
#214610562
C2×#168404722
#838061