Finite Group Databasebeta

← #29572235

Subgroups — #29572235

Order 15616.

754 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21446809C211yesmaximal, characteristic, complemented
2#21446664C211yesmaximal, characteristic, complemented
3#21444554C211yesmaximal, characteristic, complemented
4#21444411C211yesmaximal, characteristic, complemented
5#21444258C211yesmaximal, characteristic, complemented
6#21444258C211yesmaximal, characteristic, complemented
7C61×#3511C211yesmaximal, characteristic
8#16839786C2211yescharacteristic
9#16839786C2211yescharacteristic
10C2×#14519696—22no—
11C4×#11758194C2211yescharacteristic
12C4×#11758194C2211yescharacteristic
13#16839531—22no—
14#16839511—21no—
15#16839506—21no—
16#16839502—22no—
17Q8⋊#92418—21no—
18Q8⋊#92418—21no—
19D4⋊#92418—21no—
20D4⋊#92418—21no—
21C61×#556—21no—
22C61×#555—21no—
23C61×#554—21no—
24C61×#553—22no—
25C61×#548—22no—
26C61×#547—21no—
27C61×#542—22no—
28C61×#537C412yes—
29C61×#533—21no—
30C61×#532—21no—
31C61×#532C2211yescharacteristic
32C2×C4⋊Q8×C61C2211yescharacteristic
33C61×#520C412yes—
34C4×#41×C61C2211yescharacteristic
35C2×#11758194—21no—
36C2×#11758194—21no—
37C2×#11758194—21no—
38C2×#11758194—21no—
39C4⋊#92418—21no—
40C4⋊#92418—21no—
41C4⋊#92418—21no—
42C4⋊#92418—21no—
43#14519696—44no—
44#14519688—21no—
45#14519688—21no—
46C4×#92418—21no—
47C4×#92418—21no—
48C2×#41×C61—22no—
49C2×#41×C61C2×C412yes—
50C2×#41×C61C2×C411yescharacteristic
51C22×Q8×C61D412yes—
52C4⋊Q8×C61—21no—
53C4⋊Q8×C61—21no—
54C4⋊Q8×C61—21no—
55C4⋊Q8×C61—21no—
56C4⋊Q8×C61D411yescharacteristic
57#127×C61D411yescharacteristic
58#126×C61—24no—
59#125×C61—22no—
60#124×C61—22no—
61#124×C61—24no—
62#124×C61D412yes—
63#123×C61—24no—
64C22⋊Q8×C61—22no—
65C22⋊Q8×C61—24no—
66C22⋊Q8×C61—22no—
67C22⋊Q8×C61—22no—
68#121×C61—22no—
69#121×C61—22no—
70#121×C61—22no—
71#121×C61—24no—
72#121×C61—21no—
73#121×C61—21no—
74C4×Q8×C61—22no—
75C4×Q8×C61—22no—
76C4×Q8×C61D411yescharacteristic
77C4×Q8×C61D411yescharacteristic
78C4×D4×C61—22no—
79C4×D4×C61—22no—
80C4×D4×C61—24no—
81C4×D4×C61—21no—
82C4×D4×C61—21no—
83C4×D4×C61—21no—
84C4×D4×C61—21no—
85C4×D4×C61D411yescharacteristic
86C4×D4×C61D411yescharacteristic
87#117×C61—22no—
88#117×C61—22no—
89#117×C61—21no—
90#117×C61C2×C411yescharacteristic
91C2×C4⋊C4×C61—22no—
92C2×C4⋊C4×C61C2×C412yes—
93C2×C4⋊C4×C61D412yes—
94C2×C42×C61—21no—
95C2×C42×C61C2311yescharacteristic
96#41×C61—42no—
97#41×C61—44no—
98#41×C61—82no—
99#41×C61—82no—
100C2×Q8×C61—22no—
101C2×Q8×C61—22no—
102C2×Q8×C61—42no—
103C2×Q8×C61—42no—
104C2×Q8×C61C22⋊C411yescharacteristic
105C2×Q8×C61C22⋊C412yes—
106C2×D4×C61—22no—
107C2×D4×C61—21no—
108C2×D4×C61—24no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61C22⋊C411yescharacteristic
112C2×D4×C61C22⋊C412yes—
113C22×C4×C61—24no—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61C22⋊C414yes—
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C4⋊C4×C61—24no—
123C4⋊C4×C61—24no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61C22⋊C411yescharacteristic
133C4⋊C4×C61C22⋊C411yescharacteristic
134C22⋊C4×C61—24no—
135C22⋊C4×C61—24no—
136C22⋊C4×C61—24no—
137C22⋊C4×C61—21no—
138C22⋊C4×C61—21no—
139C22⋊C4×C61—21no—
140C22⋊C4×C61—22no—
141C22⋊C4×C61—22no—
142C42×C61—24no—
143C42×C61—21no—
144C42×C61—21no—
145C42×C61—21no—
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150#11758194—22no—
151#11758194—22no—
152#11758194—41no—
153#11758194—41no—
154#11758194—42no—
155#11758194—42no—
156C2×#92418—41no—
157C2×#92418—41no—
158C2×#92418—41no—
159C2×#92418—41no—
160C23×C61—21no—
161C23×C61—22no—
162C23×C61C2×C22⋊C411yescharacteristic
163Q8×C61—22no—
164Q8×C61—42no—
165Q8×C61—42no—
166Q8×C61—42no—
167Q8×C61—82no—
168D4×C61—22no—
169D4×C61—42no—
170D4×C61—42no—
171D4×C61—44no—
172D4×C61—41no—
173D4×C61—41no—
174D4×C61—41no—
175D4×C61—41no—
176D4×C61—84no—
177C2×C4×C61—22no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—22no—
185C2×C4×C61—21no—
186C2×C4×C61—24no—
187C2×C4×C61—42no—
188C2×C4×C61—44no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—44no—
194C2×C4×C61—84no—
195C2×C4×C61C2×C22⋊C411yescharacteristic
196C2×C4×C61C2×C22⋊C411yescharacteristic
197C2×C4×C61#12011yescharacteristic
198C2×C4×C61#12011yescharacteristic
199C2×C4×C61#12011yescharacteristic
200C2×C4×C61#12011yescharacteristic
201#92418—42no—
202#92418—42no—
203#92418—81no—
204#92418—81no—
205#8470—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—41no—
210C22×C61—42no—
211C22×C61—41no—
212C22×C61—42no—
213C22×C61—41no—
214C22×C61—42no—
215C22×C61#37911yescharacteristic
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—41no—
220C4×C61—44no—
221C4×C61—41no—
222C4×C61—41no—
223C4×C61—42no—
224C4×C61#41312yes—
225C4×C61#45311yescharacteristic
226C4×C61#45411yescharacteristic
227C4×C61#45511yescharacteristic
228C4×C61#45611yescharacteristic
229#3511D6111yescharacteristic
230#1997—611no—
231#1966—611no—
232#1560—611no—
233#1525—611no—
234#1489—611no—
235#1489—611no—
236C2×C61—41no—
237C2×C61—81no—
238C2×C61—82no—
239C2×C61#178511yescharacteristic
240C2×C61#179211yescharacteristic
241C2×C61#179311yescharacteristic
242#556—21no—
243#555—21no—
244#554—21no—
245#553—22no—
246#548—22no—
247#547—21no—
248#542—22no—
249#537Dic6112yes—
250#533—21no—
251#532—21no—
252#532C2×D6111yescharacteristic
253C2×C4⋊Q8C2×D6111yescharacteristic
254#520Dic6112yes—
255C4×#41C2×D6111yescharacteristic
256#423—611no—
257#423—611no—
258C2×#101—1222no—
259C4×M16—611no—
260C4×M16—611no—
261#343—1222no—
262#333—1221no—
263#331—1221no—
264#329—1222no—
265Q8⋊C8—1221no—
266Q8⋊C8—1221no—
267D4⋊C8—1221no—
268D4⋊C8—1221no—
269C61#847011yescharacteristic, complemented, Sylow
270C2×#41—22no—
271C2×#41C2×Dic6111yescharacteristic
272C2×#41C2×Dic6112yes—
273C22×Q8C22⋊D6112yes—
274C2×M16—1221no—
275C2×M16—1221no—
276C2×M16—1221no—
277C2×M16—1221no—
278C4⋊Q8—21no—
279C4⋊Q8—21no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8C22⋊D6111yescharacteristic
283#127C22⋊D6111yescharacteristic
284#126—24no—
285#125—22no—
286#124—22no—
287#124—24no—
288#124C22⋊D6112yes—
289#123—24no—
290C22⋊Q8—22no—
291C22⋊Q8—24no—
292C22⋊Q8—22no—
293C22⋊Q8—22no—
294#121—22no—
295#121—22no—
296#121—22no—
297#121—24no—
298#121—21no—
299#121—21no—
300C4×Q8—22no—
301C4×Q8—22no—
302C4×Q8C22⋊D6111yescharacteristic
303C4×Q8C22⋊D6111yescharacteristic
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—24no—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4—21no—
310C4×D4—21no—
311C4×D4C22⋊D6111yescharacteristic
312C4×D4C22⋊D6111yescharacteristic
313#117—22no—
314#117—22no—
315#117—21no—
316#117C2×Dic6111yescharacteristic
317C2×C4⋊C4—22no—
318C2×C4⋊C4C2×Dic6112yes—
319C2×C4⋊C4C22⋊D6112yes—
320C2×C42—21no—
321C2×C42C22×D6111yescharacteristic
322C4⋊C8—1221no—
323C4⋊C8—1221no—
324C4⋊C8—1221no—
325C4⋊C8—1221no—
326#101—2444no—
327#97—1221no—
328#97—1221no—
329C4×C8—1221no—
330C4×C8—1221no—
331#41—42no—
332#41—44no—
333#41—82no—
334#41—82no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8C22⋊Dic6111yescharacteristic
340C2×Q8C22⋊Dic6112yes—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—24no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4C22⋊Dic6111yescharacteristic
347C2×D4C22⋊Dic6112yes—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4C22⋊Dic6114yes—
354C22×C4C22×Dic6111yescharacteristic
355C22×C4C2×C22⋊D6111yescharacteristic
356C22×C4C2×C22⋊D6111yescharacteristic
357M16—1222no—
358M16—1222no—
359M16—2442no—
360M16—2442no—
361M16—2441no—
362M16—2441no—
363C2×C8—2441no—
364C2×C8—2441no—
365C2×C8—2441no—
366C2×C8—2441no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4C22⋊Dic6111yescharacteristic
378C4⋊C4C22⋊Dic6111yescharacteristic
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—21no—
383C22⋊C4—21no—
384C22⋊C4—21no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C42—24no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42C2×C22⋊D6111yescharacteristic
392C42C2×C22⋊D6111yescharacteristic
393C42C2×C22⋊D6111yescharacteristic
394C42C2×C22⋊D6111yescharacteristic
395C23—21no—
396C23—22no—
397C23C2×C22⋊Dic6111yescharacteristic
398Q8—22no—
399Q8—42no—
400Q8—42no—
401Q8—42no—
402Q8—82no—
403D4—22no—
404D4—42no—
405D4—42no—
406D4—44no—
407D4—41no—
408D4—41no—
409D4—41no—
410D4—41no—
411D4—84no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—44no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—44no—
429C2×C4—84no—
430C2×C4C2×C22⋊Dic6111yescharacteristic
431C2×C4C2×C22⋊Dic6111yescharacteristic
432C2×C4#1451974611yescharacteristic
433C2×C4#1451974611yescharacteristic
434C2×C4#1451974611yescharacteristic
435C2×C4#1451974611yescharacteristic
436C8—4881no—
437C8—4881no—
438C8—2442no—
439C8—2442no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—41no—
444C22—42no—
445C22—41no—
446C22—42no—
447C22—41no—
448C22—42no—
449C22#1683961811yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—41no—
454C4—42no—
455C4—41no—
456C4—44no—
457C4—41no—
458C4#1683974612yes—
459C4#1683993011yescharacteristic
460C4#1683993511yescharacteristic
461C4#1683994011yescharacteristic
462C4#1683994511yescharacteristic
463C2—41no—
464C2—81no—
465C2—82no—
466C2#2144574711yescharacteristic
467C2#2144577311yescharacteristic
468C2#2144577811yescharacteristic
subgroupG/H
centerC22#16839618
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#3511C2
socleC22×C61#379
cosocleC2×C42×C61C23
hypercenter#3511D61
hypocenterC61#8470
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. #21446809C2semidirect
2. #21446664C2semidirect
3. #21444554C2semidirect
4. #21444411C2semidirect
5. #21444258C2semidirect
6. #21444258C2semidirect
269. C61#8470semidirect

Maximal subgroups

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

subgroupindex
#214468092
#214466642
#214445542
#214444112
#214442582
C61×#35112
#847061