Finite Group Databasebeta

← #30601017

Subgroups — #30601017

Order 15616.

594 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 96 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#21462668C211yesmaximal, characteristic
2#21461684C211yesmaximal, characteristic
3#21461334C212yesmaximal, complemented
4#21460717C212yesmaximal, complemented
5#21450767C211yesmaximal, characteristic
6#16840789—24no—
7#16840779C2212yes—
8#16840765C2212yes—
9C2×#14519837—21no—
10D61⋊(C22⋊C8)—21no—
11C2×C22⋊#92420—21no—
12#16840642—21no—
13#16840621—24no—
14C2×#14519776C2211yescharacteristic
15C2×#14519755—21no—
16C2×#14519754—21no—
17C2×#14519737C2211yescharacteristic
18#16839699C411yescharacteristic
19#16839698C411yescharacteristic
20#16839697C411yescharacteristic
21#16839662—21no—
22#16839631—21no—
23#16839613C2211yescharacteristic
24C61×#399C411yescharacteristic
25C2×#11758217—22no—
26C2×D61⋊C8—22no—
27C22×#92420—22no—
28#14519844—22no—
29#14519844D414yes—
30C4⋊#92420—22no—
31#14519837—42no—
32#14519834—22no—
33C22⋊#92420—22no—
34C22⋊#92420—22no—
35C4×#92420—22no—
36C22×Dic122—21no—
37C22×C22⋊D61D411yescharacteristic
38C23×Dic61D411yescharacteristic
39C22×C4×D61D411yescharacteristic
40#14519776—22no—
41#14519776—22no—
42#14519755—44no—
43#14519754—44no—
44#14519737—22no—
45#14519737—22no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61C2×C411yescharacteristic
48C2×#11758196—21no—
49C2×#11758196—21no—
50C2×#11758196—21no—
51C2×#11758196—21no—
52C2×#11758196D411yescharacteristic
53C2×C22⋊Dic61—21no—
54C2×#11758198—21no—
55C2×#11758198—21no—
56C2×#11758198C2×C411yescharacteristic
57C2×C4×Dic61—21no—
58C2×C4×Dic61C2311yescharacteristic
59#14519684C2×C411yescharacteristic
60C2×C4⋊C4×C61—21no—
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C22⋊C4×C61—21no—
63C2×C22⋊C4×C61C2×C411yescharacteristic
64#95×C61C2×C411yescharacteristic
65C24×C61C22⋊C411yescharacteristic
66C23×D61C22⋊C411yescharacteristic
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C2×C22⋊D61—22no—
72C2×C22⋊D61—22no—
73C2×C22⋊D61SD1612yes—
74C2×C22⋊D61Q1612yes—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—42no—
80C22×Dic61C22⋊C411yescharacteristic
81C22×Dic61C2×D411yescharacteristic
82C22×Dic61C2×D411yescharacteristic
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—42no—
86C2×Dic122—42no—
87C2×Dic122—42no—
88C2×Dic122—41no—
89C2×Dic122—41no—
90#11758217—42no—
91#11758217—42no—
92C2×#92420—24no—
93C2×#92420—42no—
94C2×#92420—82no—
95D61⋊C8—82no—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—42no—
98C22⋊C4×C61—22no—
99C22⋊C4×C61—42no—
100C22⋊Dic61—42no—
101#11758198—22no—
102#11758198—42no—
103#11758198—42no—
104C4⋊Dic61—22no—
105C4⋊Dic61—42no—
106#11758196—24no—
107#11758196—22no—
108#11758196—42no—
109#11758196—42no—
110#11758196—42no—
111C4×Dic61—42no—
112C4×Dic61C2×D412yes—
113C4×Dic61#4112yes—
114C23×C61—21no—
115C23×C61—21no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61#10312yes—
119C23×C61C2×C22⋊C411yescharacteristic
120C22×D61—21no—
121C22×D61—21no—
122C22×D61—41no—
123C22×D61—41no—
124C22×D61#10312yes—
125C2×C4×C61—22no—
126C2×C4×C61—21no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61C4×D412yes—
135C22⋊D61—24no—
136C22⋊D61—24no—
137C2×Dic61—22no—
138C2×Dic61—22no—
139C2×Dic61—21no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—42no—
143C2×Dic61—41no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—81no—
148C2×Dic61#12011yescharacteristic
149C2×Dic61#12112yes—
150C2×Dic61#12311yescharacteristic
151C2×Dic61C2×SD1611yescharacteristic
152C2×Dic61C2×Q1611yescharacteristic
153C2×Dic61#13512yes—
154C4×D61—42no—
155C4×D61—81no—
156Dic122—82no—
157Dic122—82no—
158#92420—44no—
159#92420—82no—
160#9395—611nomaximal, Sylow
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—22no—
164C22×C61—42no—
165C22×C61—41no—
166C22×C61—41no—
167C22×C61—81no—
168C22×C61#38611yescharacteristic
169C22×C61C2×#10311yescharacteristic
170C22×C61#41611yescharacteristic
171C2×D61—22no—
172C2×D61—41no—
173C2×D61—41no—
174C2×D61—42no—
175C2×D61—81no—
176C4×C61—22no—
177C4×C61—42no—
178C4×C61—41no—
179C4×C61—81no—
180Dic61—21no—
181Dic61—42no—
182Dic61—41no—
183Dic61—81no—
184Dic61#45312yes—
185Dic61#46512yes—
186Dic61#46712yes—
187#2972—611no—
188#2542—611no—
189#1800—611no—
190#1549—612no—
191#1303—612no—
192C2×C61—21no—
193C2×C61—21no—
194C2×C61—42no—
195C2×C61#188711yescharacteristic
196C2×C61#189211yescharacteristic
197C2×C61#191511yescharacteristic
198D61—42no—
199C2×C4⋊Q8—611no—
200C2×C22⋊Q8—1221no—
201C2×C22⋊Q8—1221no—
202C2×C4×D4—611no—
203C8×D4—1224no—
204#431—612no—
205#423—612no—
206C2×#100—1221no—
207C2×C22⋊C8—1221no—
208C2×C22⋊C8—1221no—
209#399—611no—
210#399—611no—
211#399—611no—
212#399#692311yescharacteristic
213#390—1221no—
214#384—1221no—
215C4×C4⋊C4—611no—
216#343—1221no—
217#332—1224no—
218C61#939511yescharacteristic, complemented, Sylow
219C22×Q8—1221no—
220C22×D4—611no—
221C23×C4—611no—
222C23×C4—611no—
223C2×M16—1222no—
224C22×C8—1222no—
225C22×C8—1222no—
226C4⋊Q8—1222no—
227C4⋊Q8—1222no—
228C22⋊Q8—2444no—
229C22⋊Q8—2444no—
230C4×D4—1222no—
231C4×D4—1222no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—611no—
234C2×C4⋊C4—611no—
235C2×C4⋊C4—1221no—
236C2×C4⋊C4—1221no—
237C2×C4⋊C4—1221no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4—1221no—
240C2×C4⋊C4C2×#692311yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—611no—
243C2×C22⋊C4—1221no—
244C2×C22⋊C4—1221no—
245C2×C22⋊C4—1221no—
246C2×C22⋊C4C2×#692311yescharacteristic
247C2×C42—611no—
248C2×C42—1221no—
249C4⋊C8—614no—
250C4⋊C8—1222no—
251C4⋊C8—1222no—
252#100—2442no—
253C22⋊C8—1222no—
254C22⋊C8—1222no—
255C22⋊C8—1222no—
256C4×C8—1222no—
257#95—611no—
258#95C2×#692311yescharacteristic
259C24—611no—
260C24C22⋊#692311yescharacteristic
261C2×Q8—2441no—
262C2×Q8—2441no—
263C2×Q8—2442no—
264C2×Q8—2442no—
265C2×D4—612no—
266C2×D4—612no—
267C2×D4—1222no—
268C2×D4—1222no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—611no—
272C22×C4—611no—
273C22×C4—611no—
274C22×C4—1221no—
275C22×C4—1221no—
276C22×C4—1221no—
277C22×C4—1221no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—2442no—
281C22×C4—2442no—
282C22×C4C22⋊#692311yescharacteristic
283C22×C4C22×#692311yescharacteristic
284M16—2442no—
285M16—2442no—
286C2×C8—1224no—
287C2×C8—4882no—
288C2×C8—4882no—
289C2×C8—2442no—
290C4⋊C4—22no—
291C4⋊C4—42no—
292C4⋊C4—1222no—
293C4⋊C4—1224no—
294C4⋊C4—1222no—
295C4⋊C4—2442no—
296C4⋊C4—2442no—
297C4⋊C4—2442no—
298C4⋊C4—2442no—
299C22⋊C4—22no—
300C22⋊C4—42no—
301C22⋊C4—1222no—
302C22⋊C4—2442no—
303C22⋊C4—2442no—
304C22⋊C4—2442no—
305C42—612no—
306C42—612no—
307C42—2442no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—612no—
313C23—1221no—
314C23—1221no—
315C23—2441no—
316C23—2441no—
317C23#1451984012yes—
318C23C2×C22⋊#692311yescharacteristic
319Q8—4882no—
320Q8—4882no—
321D4—1224no—
322D4—1224no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—611no—
333C2×C4—612no—
334C2×C4—611no—
335C2×C4—611no—
336C2×C4—612no—
337C2×C4—611no—
338C2×C4—1222no—
339C2×C4—1222no—
340C2×C4—1221no—
341C2×C4—4881no—
342C2×C4—4881no—
343C2×C4—2442no—
344C2×C4—2442no—
345C2×C4—2441no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4D4×#692312yes—
352C8—4882no—
353C8—2444no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—42no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—1222no—
362C22—4881no—
363C22—2442no—
364C22—2441no—
365C22—2441no—
366C22#1684071011yescharacteristic
367C22C2×#1451984011yescharacteristic
368C22#1684075211yescharacteristic
369C4—22no—
370C4—42no—
371C4—41no—
372C4—81no—
373C4—612no—
374C4—612no—
375C4—612no—
376C4—1221no—
377C4—4881no—
378C4—2442no—
379C4—2441no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—2442no—
384C2#2146182611yescharacteristic
385C2#2146183211yescharacteristic
386C2#2146187111yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#399C4
socleC22×C61#386
prime radicalC22×C22⋊D61D4
cosocleC2×C4×Dic61C23
hypercenter#399#6923
hypocenterC61#9395
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
3. #21461334C2semidirect
4. #21460717C2semidirect
218. C61#9395semidirect

Maximal subgroups

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

subgroupindex
#214626682
#214616842
#214613342
#214607172
#214507672
#939561