Finite Group Databasebeta

← #29572545

Subgroups — #29572545

Order 15616.

694 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#16840389C211yesmaximal, characteristic
2#21444536C211yesmaximal, characteristic, complemented
3#21444445C211yesmaximal, characteristic, complemented
4#21444445C211yesmaximal, characteristic, complemented
5#21444151C211yesmaximal, characteristic
6#21444151C211yesmaximal, characteristic
7C61×#1556C211yesmaximal, characteristic, complemented
8#16840389—22no—
9#16840389—22no—
10#16840389—22no—
11#16840389—22no—
12C2×#14519776C411yescharacteristic
13C2×#14519775—21no—
14C2×#14519769C2211yescharacteristic
15C2×#14519768C411yescharacteristic
16C2×#14519767—21no—
17C2×#14519755—21no—
18C2×#14519753—21no—
19C2×#14519735C411yescharacteristic
20C2×#14519734C2211yescharacteristic
21C2×C4⋊C4×D61C411yescharacteristic
22#16839819C2211yescharacteristic
23#16839819C2211yescharacteristic
24#16839508—22no—
25#16839503—22no—
26#16839503—22no—
27#16839487—21no—
28#16839487—21no—
29#16839485—21no—
30#16839485—21no—
31C2×C4⋊Q8×C61C2211yescharacteristic
32C61×#432C2211yescharacteristic
33C61×#432C2211yescharacteristic
34C61×#332—22no—
35C22×Dic122—21no—
36C22×C4×D61—21no—
37C22×C4×D61C2×C411yescharacteristic
38#14519776—22no—
39#14519776—22no—
40#14519775—24no—
41#14519775—24no—
42#14519769—21no—
43#14519769—21no—
44#14519769D412yes—
45#14519769D412yes—
46#14519768—22no—
47#14519768—22no—
48#14519767—24no—
49#14519767—24no—
50#14519755—24no—
51#14519755—24no—
52#14519753—24no—
53#14519753—24no—
54#14519735—22no—
55#14519735—22no—
56#14519734—21no—
57#14519734—21no—
58#14519734—21no—
59#14519734—21no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61C2×C411yescharacteristic
62C2×C4⋊Dic61C2×C411yescharacteristic
63C2×#11758196—21no—
64C2×#11758196—21no—
65C2×#11758196—21no—
66C2×#11758196—21no—
67C2×#11758196—21no—
68C2×#11758196D411yescharacteristic
69C2×#11758196D411yescharacteristic
70C4⋊C4×D61—22no—
71C4⋊C4×D61—22no—
72C2×#11758198—21no—
73C2×#11758198D411yescharacteristic
74C2×#11758198D411yescharacteristic
75C2×C4×Dic61—21no—
76C2×C4×Dic61C2×C411yescharacteristic
77C4⋊#92418—21no—
78C4⋊#92418—21no—
79C22⋊#92418—21no—
80C22⋊#92418—21no—
81#14519688—21no—
82#14519688—21no—
83C22×Q8×C61—21no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61D412yes—
87C4⋊Q8×C61D412yes—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61C2×C411yescharacteristic
90C2×C4⋊C4×C61C2×C411yescharacteristic
91C2×C42×C61C2311yescharacteristic
92C4⋊C8×C61—21no—
93C4⋊C8×C61—21no—
94C22⋊C8×C61—21no—
95C22⋊C8×C61—21no—
96#97×C61—21no—
97#97×C61—21no—
98C23×D61C22⋊C411yescharacteristic
99C2×Q8×C61—22no—
100C2×Q8×C61—22no—
101C2×Q8×C61—82no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61C22×C411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×Dic61—21no—
108C22×Dic61—21no—
109C22×Dic61C22⋊C411yescharacteristic
110C22×Dic61C22⋊C411yescharacteristic
111C22×Dic61C22⋊C411yescharacteristic
112C2×C4×D61—22no—
113C2×C4×D61—21no—
114C2×C4×D61—21no—
115C2×C4×D61—22no—
116C2×C4×D61—24no—
117C2×C4×D61—82no—
118C2×Dic122—24no—
119C2×Dic122—82no—
120C2×C8×C61—41no—
121C2×C8×C61—41no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61C22⋊C412yes—
127C4⋊C4×C61C22⋊C412yes—
128C42×C61C2×D411yescharacteristic
129C42×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C42×C61C2×D411yescharacteristic
132#11758198—24no—
133#11758198—21no—
134#11758198—21no—
135#11758198—21no—
136#11758198—21no—
137C4⋊Dic61—24no—
138C4⋊Dic61—21no—
139C4⋊Dic61—21no—
140C4⋊Dic61C22⋊C412yes—
141C4⋊Dic61C22⋊C412yes—
142#11758196—24no—
143#11758196—24no—
144#11758196—24no—
145#11758196—22no—
146#11758196—22no—
147#11758196—22no—
148#11758196—22no—
149#11758196—21no—
150#11758196—21no—
151#11758196—21no—
152#11758196—21no—
153C4×Dic61—24no—
154C4×Dic61—21no—
155C4×Dic61—21no—
156C2×#92418—41no—
157C2×#92418—41no—
158C23×C61C2×C22⋊C411yescharacteristic
159C22×D61—21no—
160C22×D61—21no—
161C22×D61—41no—
162C22×D61—41no—
163C22×D61#10112yes—
164Q8×C61—82no—
165Q8×C61—82no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61C2×C22⋊C411yescharacteristic
175C2×C4×C61C2×C22⋊C411yescharacteristic
176C2×C4×C61#12011yescharacteristic
177C2×C4×C61#12011yescharacteristic
178C2×C4×C61#12011yescharacteristic
179C2×C4×C61#12011yescharacteristic
180C2×Dic61—21no—
181C2×Dic61—21no—
182C2×Dic61—21no—
183C2×Dic61—22no—
184C2×Dic61—21no—
185C2×Dic61—22no—
186C2×Dic61—42no—
187C2×Dic61—42no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61#10112yes—
191C4×D61—24no—
192C4×D61—24no—
193C4×D61—84no—
194Dic122—84no—
195C8×C61—81no—
196C8×C61—81no—
197#92418—81no—
198#92418—81no—
199#8525—611nomaximal, Sylow
200C22×C61—21no—
201C22×C61—21no—
202C22×C61#37911yescharacteristic
203C22×C61C2×#10111yescharacteristic
204C22×C61C2×#10111yescharacteristic
205C2×D61—22no—
206C2×D61—42no—
207C2×D61—41no—
208C2×D61—41no—
209C2×D61—42no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—42no—
217C4×C61—42no—
218Dic61—22no—
219Dic61—41no—
220Dic61—42no—
221Dic61—41no—
222Dic61—42no—
223C2×#544—611no—
224#1556—611no—
225#1556D6111yescharacteristic, complemented
226#1534—611no—
227#1534—611no—
228#1464—611no—
229#1464—611no—
230C2×C61—21no—
231C2×C61—21no—
232C2×C61#178411yescharacteristic
233C2×C61#179511yescharacteristic
234C2×C61#179611yescharacteristic
235D61—42no—
236#544—1222no—
237#544—1222no—
238#544—1222no—
239#544—1222no—
240C2×C4⋊Q8—611no—
241C2×C4⋊Q8—1221no—
242C2×C4⋊Q8C2×D6111yescharacteristic
243C2×#125—611no—
244C2×#125—611no—
245C2×#125—1221no—
246C2×C22⋊Q8—1221no—
247C2×C22⋊Q8—1221no—
248C2×#117—611no—
249C2×#117—611no—
250C22×C4⋊C4—611no—
251#432—611no—
252#432—611no—
253#432C2×D6111yescharacteristic
254#432C2×D6111yescharacteristic
255#332—22no—
256#332—1222no—
257#330—1222no—
258#330—1222no—
259#324—1221no—
260#324—1221no—
261#323—1221no—
262#323—1221no—
263C61#852511yescharacteristic, complemented, Sylow
264C22×Q8—21no—
265C22×Q8—1221no—
266C23×C4—611no—
267C23×C4—1221no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—1222no—
271C4⋊Q8—1224no—
272C4⋊Q8—1224no—
273C4⋊Q8—1222no—
274C4⋊Q8D24412yes—
275C4⋊Q8C22⋊D6112yes—
276#125—612no—
277#125—612no—
278#125—1222no—
279#125—1224no—
280#125—1224no—
281#125—1222no—
282#125—1221no—
283#125—1221no—
284C22⋊Q8—1224no—
285C22⋊Q8—1224no—
286C22⋊Q8—1224no—
287C22⋊Q8—1224no—
288#117—1222no—
289#117—1222no—
290#117—1221no—
291#117—1221no—
292#117—1221no—
293#117—1221no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—611no—
296C2×C4⋊C4—611no—
297C2×C4⋊C4—611no—
298C2×C4⋊C4—611no—
299C2×C4⋊C4—1222no—
300C2×C4⋊C4—1221no—
301C2×C4⋊C4—1221no—
302C2×C4⋊C4—1221no—
303C2×C4⋊C4—1221no—
304C2×C4⋊C4—1221no—
305C2×C4⋊C4—1221no—
306C2×C4⋊C4—1222no—
307C2×C4⋊C4C4×D6111yescharacteristic
308C2×C4⋊C4C4×D6111yescharacteristic
309C2×C22⋊C4—611no—
310C2×C22⋊C4—611no—
311C2×C22⋊C4—1221no—
312C2×C42—611no—
313C2×C42—1221no—
314C2×C42C22×D6111yescharacteristic
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—1221no—
318C4⋊C8—1221no—
319C22⋊C8—21no—
320C22⋊C8—21no—
321C22⋊C8—1221no—
322C22⋊C8—1221no—
323#97—21no—
324#97—21no—
325#97—1221no—
326#97—1221no—
327C24—611no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—82no—
331C2×Q8—1224no—
332C2×Q8—4882no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—611no—
336C22×C4—611no—
337C22×C4—611no—
338C22×C4—1222no—
339C22×C4—1222no—
340C22×C4—1224no—
341C22×C4—1221no—
342C22×C4—1221no—
343C22×C4—1221no—
344C22×C4—1221no—
345C22×C4—4882no—
346C22×C4C2×C4×D6111yescharacteristic
347C22×C4D4×D6111yescharacteristic
348C22×C4D4×D6111yescharacteristic
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—2441no—
352C2×C8—2441no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—612no—
358C4⋊C4—612no—
359C4⋊C4—1224no—
360C4⋊C4—1224no—
361C4⋊C4—1224no—
362C4⋊C4—1224no—
363C4⋊C4—1221no—
364C4⋊C4—1221no—
365C4⋊C4—1221no—
366C4⋊C4—1221no—
367C4⋊C4—1222no—
368C4⋊C4—1222no—
369C4⋊C4—1221no—
370C4⋊C4—1221no—
371C4⋊C4—1222no—
372C4⋊C4—1222no—
373C4⋊C4#1175819812yes—
374C4⋊C4#1175819812yes—
375C22⋊C4—1224no—
376C22⋊C4—1221no—
377C22⋊C4—1221no—
378C22⋊C4—1221no—
379C22⋊C4—1221no—
380C42—1224no—
381C42—1221no—
382C42—1221no—
383C42C2×D24411yescharacteristic
384C42D4×D6111yescharacteristic
385C42D4×D6111yescharacteristic
386C42C2×C22⋊D6111yescharacteristic
387C23—612no—
388C23—1221no—
389C23—1221no—
390C23—2441no—
391C23—2441no—
392C23C22⋊C4×D6111yescharacteristic
393Q8—82no—
394Q8—82no—
395Q8—4884no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—612no—
405C2×C4—1224no—
406C2×C4—1224no—
407C2×C4—1221no—
408C2×C4—1221no—
409C2×C4—1222no—
410C2×C4—1221no—
411C2×C4—1221no—
412C2×C4—1222no—
413C2×C4—4884no—
414C2×C4—2442no—
415C2×C4—2442no—
416C2×C4—2441no—
417C2×C4—2441no—
418C2×C4C22⋊C4×D6111yescharacteristic
419C2×C4C2×#1175819811yescharacteristic
420C2×C4#1451974511yescharacteristic
421C2×C4#1451974511yescharacteristic
422C2×C4#1451974711yescharacteristic
423C2×C4#1451974711yescharacteristic
424C8—81no—
425C8—81no—
426C8—4881no—
427C8—4881no—
428C22—21no—
429C22—21no—
430C22—1222no—
431C22—2442no—
432C22—2442no—
433C22—2441no—
434C22—2441no—
435C22#1683961711yescharacteristic
436C22#101×D6111yescharacteristic
437C22#101×D6111yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—42no—
445C4—42no—
446C4—1222no—
447C4—2442no—
448C4—2441no—
449C4—2441no—
450C4—2442no—
451C2—21no—
452C2—21no—
453C2—2442no—
454C2#2144574111yescharacteristic
455C2#2144578611yescharacteristic
456C2#2144579111yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1556C2
socleC22×C61#379
prime radicalC23×D61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1556D61
hypocenterC61#8525
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
2. #21444536C2semidirect
3. #21444445C2semidirect
4. #21444445C2semidirect
7. C61×#1556C2semidirect
225. #1556D61semidirect
263. C61#8525semidirect

Maximal subgroups

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

subgroupindex
C2×#168403892
#214445362
#214444452
#214441512
C61×#15562
#852561