Finite Group Databasebeta

← #29598332

Subgroups — #29598332

Order 15616.

602 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16839557C212yesmaximal, complemented
2C2×#16839555C212yesmaximal
3#21445807C211yesmaximal, characteristic, complemented
4C4×#14519692C211yesmaximal, characteristic, complemented
5C61×#2544C211yesmaximal, characteristic
6C2×#14519692C2212yes—
7C2×#14519692C2212yes—
8#16839622—21no—
9C4×C22⋊Dic61—21no—
10#16839557—24no—
11#16839555—24no—
12#16839529—21no—
13#16839529—21no—
14C2×#127×C61C2211yescharacteristic
15C2×#124×C61C411yescharacteristic, complemented
16C2×#124×C61C2211yescharacteristic
17C2×C4×D4×C61C2211yescharacteristic
18C61×#394—21no—
19C61×#394—21no—
20C61×#391C411yescharacteristic, complemented
21C61×#390C411yescharacteristic, complemented
22C61×#386—21no—
23C61×#386—21no—
24C4×C4⋊C4×C61C411yescharacteristic, complemented
25C2×C4⋊Dic61—41no—
26C2×C22⋊Dic61—22no—
27C2×C22⋊Dic61—22no—
28C2×C4×Dic61—41no—
29#14519692—22no—
30#14519692—22no—
31#14519692—22no—
32#14519692D414yes—
33#14519684—41no—
34#14519684—41no—
35C22×Q8×C61C2×C411yescharacteristic
36C22×D4×C61—21no—
37C22×D4×C61—21no—
38C22×D4×C61C2×C411yescharacteristic
39C22×D4×C61C2311yescharacteristic
40C23×C4×C61D411yescharacteristic
41C23×C4×C61D411yescharacteristic
42#127×C61—22no—
43#127×C61—22no—
44#124×C61—22no—
45#124×C61—22no—
46#124×C61—41no—
47#124×C61—41no—
48C4×D4×C61—41no—
49C4×D4×C61—41no—
50C2×C4⋊C4×C61—21no—
51C2×C4⋊C4×C61C2×C411yescharacteristic
52C2×C22⋊C4×C61—21no—
53C2×C22⋊C4×C61—21no—
54C2×C22⋊C4×C61—21no—
55C2×C22⋊C4×C61—21no—
56C2×C22⋊C4×C61—21no—
57C2×C22⋊C4×C61—21no—
58C2×C22⋊C4×C61D411yescharacteristic
59C2×C22⋊C4×C61D411yescharacteristic
60C2×C42×C61C2×C411yescharacteristic
61C2×C42×C61C2×C411yescharacteristic
62C2×C42×C61C2×C411yescharacteristic
63#95×C61—21no—
64#95×C61—21no—
65#95×C61—21no—
66C24×C61—21no—
67C24×C61C2×D411yescharacteristic
68C24×C61C2×D411yescharacteristic
69C2×Q8×C61—22no—
70C2×Q8×C61—41no—
71C2×Q8×C61—41no—
72C2×D4×C61—22no—
73C2×D4×C61—41no—
74C2×D4×C61—41no—
75C2×D4×C61—42no—
76C2×D4×C61—42no—
77C2×D4×C61—41no—
78C2×D4×C61—41no—
79C2×D4×C61—42no—
80C2×D4×C61—41no—
81C2×D4×C61—41no—
82C2×D4×C61—81no—
83C2×D4×C61—81no—
84C2×D4×C61C2×D412yes—
85C2×D4×C61#4112yes—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—21no—
93C22×C4×C61—81no—
94C22×C4×C61—81no—
95C22×C4×C61C22⋊C411yescharacteristic
96C22×C4×C61C22⋊C411yescharacteristic
97C22×C4×C61C22⋊C411yescharacteristic
98C22×C4×C61C22⋊C411yescharacteristic
99C22×C4×C61C22×C411yescharacteristic
100C22×Dic61—42no—
101C22×Dic61—42no—
102C4⋊C4×C61—41no—
103C4⋊C4×C61—42no—
104C22⋊C4×C61—41no—
105C22⋊C4×C61—41no—
106C22⋊C4×C61—42no—
107C22⋊C4×C61—42no—
108C22⋊C4×C61—41no—
109C22⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—81no—
113C22⋊C4×C61—81no—
114C42×C61—21no—
115C42×C61—21no—
116C42×C61—22no—
117C42×C61—22no—
118C22⋊Dic61—24no—
119C22⋊Dic61—42no—
120C22⋊Dic61—42no—
121C22⋊Dic61—42no—
122C4⋊Dic61—82no—
123C4×Dic61—82no—
124C23×C61—21no—
125C23×C61—21no—
126C23×C61—21no—
127C23×C61—21no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61—41no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61C2×C22⋊C411yescharacteristic
140C23×C61#12112yes—
141C23×C61#12411yescharacteristic
142C23×C61#12711yescharacteristic
143Q8×C61—82no—
144D4×C61—42no—
145D4×C61—42no—
146D4×C61—82no—
147D4×C61—84no—
148D4×C61—82no—
149D4×C61—82no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—22no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—81no—
171C2×C4×C61—81no—
172C2×C4×C61#9911yescharacteristic
173C2×C4×C61#9911yescharacteristic
174C2×C4×C61C4×D412yes—
175C2×Dic61—44no—
176C2×Dic61—82no—
177C2×Dic61—82no—
178C2×Dic61—82no—
179C2×Dic61—82no—
180C2×Dic61—82no—
181#13225—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—22no—
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—42no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61—41no—
191C22×C61—41no—
192C22×C61—41no—
193C22×C61—41no—
194C22×C61—81no—
195C22×C61—81no—
196C22×C61—81no—
197C22×C61—81no—
198C22×C61—81no—
199C22×C61—81no—
200C22×C61—81no—
201C22×C61—81no—
202C22×C61—81no—
203C22×C61#39011yescharacteristic
204C22×C61C2×#9911yescharacteristic
205C22×C61#41011yescharacteristic
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—22no—
209C4×C61—42no—
210C4×C61—41no—
211C4×C61—41no—
212C4×C61—41no—
213C4×C61—81no—
214C4×C61—81no—
215Dic61—84no—
216Dic61—162no—
217#2544D6111yescharacteristic
218C2×#354—612no—
219C2×#353—612no—
220#1799—611no—
221C4×#99—611no—
222C2×C61—21no—
223C2×C61—21no—
224C2×C61—42no—
225C2×C61—41no—
226C2×C61—41no—
227C2×C61—161no—
228C2×C61#196411yescharacteristic
229C2×C61#212611yescharacteristic
230C2×C61#212711yescharacteristic
231C2×#127C2×D6111yescharacteristic
232C2×#124Dic6111yescharacteristic, complemented
233C2×#124C2×D6111yescharacteristic
234C2×C4×D4C2×D6111yescharacteristic
235C2×#99—612no—
236C2×#99—612no—
237#394—21no—
238#394—21no—
239#391Dic6111yescharacteristic, complemented
240#390Dic6111yescharacteristic, complemented
241#386—21no—
242#386—21no—
243#380—1221no—
244C4×C4⋊C4Dic6111yescharacteristic, complemented
245C4×C22⋊C4—1221no—
246#354—1224no—
247#353—1224no—
248#342—1221no—
249#342—1221no—
250C61#1322511yescharacteristic, complemented, Sylow
251C22×Q8C2×Dic6111yescharacteristic
252C22×D4—21no—
253C22×D4—21no—
254C22×D4C2×Dic6111yescharacteristic
255C22×D4C22×D6111yescharacteristic
256C23×C4C22⋊D6111yescharacteristic
257C23×C4C22⋊D6111yescharacteristic
258#127—22no—
259#127—22no—
260#124—22no—
261#124—22no—
262#124—41no—
263#124—41no—
264C4×D4—41no—
265C4×D4—41no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—2441no—
268C2×C4⋊C4C2×Dic6111yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—1222no—
276C2×C22⋊C4—1222no—
277C2×C22⋊C4C22⋊D6111yescharacteristic
278C2×C22⋊C4C22⋊D6111yescharacteristic
279C2×C42—2441no—
280C2×C42C2×Dic6111yescharacteristic
281C2×C42C2×Dic6111yescharacteristic
282C2×C42C2×Dic6111yescharacteristic
283#99—614no—
284#99—1222no—
285#99—1222no—
286#99—1222no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—2441no—
291#95—2441no—
292C24—21no—
293C24C2×C22⋊D6111yescharacteristic
294C24C2×C22⋊D6111yescharacteristic
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×D4—22no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—81no—
309C2×D4—81no—
310C2×D4D4×D6112yes—
311C2×D4#1175822412yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—2442no—
322C22×C4—2442no—
323C22×C4C22⋊Dic6111yescharacteristic
324C22×C4C22⋊Dic6111yescharacteristic
325C22×C4C22⋊Dic6111yescharacteristic
326C22×C4C22⋊Dic6111yescharacteristic
327C22×C4C22×Dic6111yescharacteristic
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—4882no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—81no—
340C22⋊C4—81no—
341C22⋊C4—1224no—
342C22⋊C4—2442no—
343C22⋊C4—2442no—
344C22⋊C4—2442no—
345C42—21no—
346C42—21no—
347C42—22no—
348C42—22no—
349C42—4882no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23C2×C22⋊Dic6111yescharacteristic
366C23#1451974812yes—
367C23#1451976411yescharacteristic
368C23#1451977311yescharacteristic
369Q8—82no—
370D4—42no—
371D4—42no—
372D4—82no—
373D4—84no—
374D4—82no—
375D4—82no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—4882no—
399C2×C4—4882no—
400C2×C4—4882no—
401C2×C4—4882no—
402C2×C4—4882no—
403C2×C4—2444no—
404C2×C4#1451969211yescharacteristic
405C2×C4#1451969211yescharacteristic
406C2×C4D4×Dic6112yes—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—21no—
412C22—42no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22#1683966511yescharacteristic
429C22C2×#1451969211yescharacteristic
430C22#1683973611yescharacteristic
431C4—22no—
432C4—21no—
433C4—21no—
434C4—42no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—81no—
439C4—81no—
440C4—9762no—
441C4—4884no—
442C2—21no—
443C2—21no—
444C2—41no—
445C2—42no—
446C2—41no—
447C2—161no—
448C2#2144665611yescharacteristic
449C2#2144748411yescharacteristic
450C2#2144748811yescharacteristic
subgroupG/H
centerC22C2×#14519692
commutatorC22×C4×C61C22×C4
FrattiniC22×D4C22×D61
FittingC61×#2544C2
socleC22×C61C2×#99
prime radicalC2×#127×C61C22
cosocleC22×D4×C61C23
hypercenter#2544D61
hypocenterC61#13225
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2544

Complements

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

NH ≅ G/Ntype
1. C2×#16839557C2semidirect
3. #21445807C2semidirect
4. C4×#14519692C2semidirect
15. C2×#124×C61C4semidirect
20. C61×#391C4semidirect
21. C61×#390C4semidirect
24. C4×C4⋊C4×C61C4semidirect
232. C2×#124Dic61semidirect
239. #391Dic61semidirect
240. #390Dic61semidirect
244. C4×C4⋊C4Dic61semidirect
250. C61#13225semidirect

Maximal subgroups

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

subgroupindex
C2×#168395572
C2×#168395552
#214458072
C4×#145196922
C61×#25442
#1322561