Finite Group Databasebeta

← Q8×#137×C47 #26101066

Subgroups — Q8×#137×C47 #26101066

Order 12032.

1782 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#41×C47C211yesmaximal, characteristic
2C2×Q82×C47C211yesmaximal, characteristic, complemented
3C47×#3387C216yesmaximal, complemented
4C47×#3385C216yesmaximal, complemented
5Q8×Q16×C47C212yesmaximal, complemented
6Q8×SD16×C47C212yesmaximal, complemented
7C47×#3165C213yesmaximal, complemented
8C47×#3079C213yesmaximal
9C47×#3078C213yesmaximal, complemented
10Q8×M16×C47C211yesmaximal, characteristic, complemented
11C4×#137×C47C213yesmaximal
12C2×#137×C47C2211yescharacteristic
13Q82×C47—21no—
14Q82×C47—23no—
15Q82×C47C2212yes—
16Q82×C47C2211yescharacteristic
17Q82×C47C2216yes—
18C47×#557C2213yes—
19C47×#554—23no—
20C47×#554C2213yes—
21D4×Q8×C47—21no—
22D4×Q8×C47C2211yescharacteristic
23C47×#533—23no—
24C47×#533C2213yes—
25C2×C4⋊Q8×C47—23no—
26C2×C4⋊Q8×C47C2213yes—
27C4×#41×C47C2213yes—
28C2×C4×Q8×C47—21no—
29C2×C4×Q8×C47—21no—
30C2×C4×Q8×C47C2213yes—
31C2×C4×Q8×C47C2211yescharacteristic
32C47×#501C22112yes—
33C47×#500C2216yes—
34C47×#498C2216yes—
35C47×#479C22112yes—
36C47×#477C2216yes—
37C47×#477C2216yes—
38C47×#476C2216yes—
39C47×#475C2216yes—
40C47×#475C2216yes—
41C47×#474C2216yes—
42C47×#446C2216yes—
43C8×Q8×C47C2212yes—
44C47×#441C2216yes—
45C47×#440C2216yes—
46C4×Q16×C47C2216yes—
47C4×SD16×C47C2216yes—
48C47×#428C2216yes—
49C47×#428C2213yes—
50C47×#423C2213yes—
51C47×#419C2213yes—
52C47×#417C2213yes—
53C4×M16×C47C2213yes—
54C2×#41×C47C2311yescharacteristic
55C22×Q8×C47—21no—
56C22×Q8×C47—23no—
57C22×Q8×C47D411yescharacteristic
58C22×Q8×C47C2311yescharacteristic
59#137×C47Q818yesdirect product factor
60C2×Q16×C47C2312yes—
61C2×SD16×C47C2312yes—
62C2×M16×C47C2313yes—
63C2×M16×C47C2311yescharacteristic
64C4⋊Q8×C47—23no—
65C4⋊Q8×C47—26no—
66C4⋊Q8×C47—212no—
67C4⋊Q8×C47—26no—
68C4⋊Q8×C47—23no—
69C4⋊Q8×C47—23no—
70C4⋊Q8×C47—23no—
71C4⋊Q8×C47C2316yes—
72C4⋊Q8×C47C2313yes—
73C4⋊Q8×C47C2316yes—
74C4⋊Q8×C47C2313yes—
75#125×C47—26no—
76#125×C47C2316yes—
77C22⋊Q8×C47—23no—
78C22⋊Q8×C47—23no—
79C22⋊Q8×C47—23no—
80C22⋊Q8×C47—23no—
81C22⋊Q8×C47—26no—
82C4×Q8×C47—21no—
83C4×Q8×C47—26no—
84C4×Q8×C47—22no—
85C4×Q8×C47—23no—
86C4×Q8×C47—23no—
87C4×Q8×C47—23no—
88C4×Q8×C47—23no—
89C4×Q8×C47—23no—
90C4×Q8×C47D411yescharacteristic
91C4×Q8×C47D413yes—
92C4×Q8×C47C2313yes—
93C4×Q8×C47C2316yes—
94C4×Q8×C47C2313yes—
95C4×Q8×C47C2316yes—
96C4×Q8×C47C2311yescharacteristic
97C4×Q8×C47C2313yes—
98C4×D4×C47—23no—
99C4×D4×C47—23no—
100C4×D4×C47C2313yes—
101C4×D4×C47C2313yes—
102#117×C47—23no—
103#117×C47C2313yes—
104C2×C4⋊C4×C47—23no—
105C2×C4⋊C4×C47—23no—
106C2×C4⋊C4×C47—23no—
107C2×C4⋊C4×C47D413yes—
108C2×C4⋊C4×C47C2313yes—
109C2×C42×C47—23no—
110C2×C42×C47—23no—
111C2×C42×C47C2313yes—
112#107×C47C23112yes—
113#107×C47C2316yes—
114#106×C47C23112yes—
115#106×C47C2316yes—
116C4⋊C8×C47C2316yes—
117C4⋊C8×C47C2316yes—
118#103×C47C2316yes—
119#103×C47C2316yes—
120#103×C47C2316yes—
121#102×C47C2316yes—
122#97×C47C2316yes—
123C4×C8×C47C2316yes—
124#41×C47—22no—
125#41×C47C2×Q814yes—
126C2×Q8×C47—24no—
127C2×Q8×C47—21no—
128C2×Q8×C47—23no—
129C2×Q8×C47—23no—
130C2×Q8×C47—26no—
131C2×Q8×C47—26no—
132C2×Q8×C47—22no—
133C2×Q8×C47—44no—
134C2×Q8×C47—86no—
135C2×Q8×C47C2×D413yes—
136C2×Q8×C47C2×D411yescharacteristic
137C2×Q8×C47C2×Q814yescomplemented
138C2×Q8×C47C2412yes—
139C2×Q8×C47C2411yescharacteristic
140C2×D4×C47—21no—
141C2×D4×C47C2411yescharacteristic
142C22×C4×C47—21no—
143C22×C4×C47—21no—
144C22×C4×C47—23no—
145C22×C4×C47—23no—
146C22×C4×C47—23no—
147C22×C4×C47C2×D413yes—
148C22×C4×C47C2×D413yes—
149C22×C4×C47C2411yescharacteristic
150Q16×C47C2×Q818yescomplemented
151SD16×C47C2×Q818yescomplemented
152M16×C47—26no—
153M16×C47C2×Q814yescomplemented
154C2×C8×C47C2416yes—
155C2×C8×C47C2412yes—
156C4⋊C4×C47—26no—
157C4⋊C4×C47—23no—
158C4⋊C4×C47—26no—
159C4⋊C4×C47—26no—
160C4⋊C4×C47—26no—
161C4⋊C4×C47—26no—
162C4⋊C4×C47—23no—
163C4⋊C4×C47—23no—
164C4⋊C4×C47—23no—
165C4⋊C4×C47—23no—
166C4⋊C4×C47—26no—
167C4⋊C4×C47C2×D413yes—
168C4⋊C4×C47C2×D416yes—
169C4⋊C4×C47C2×D413yes—
170C4⋊C4×C47C2413yes—
171C4⋊C4×C47C2413yes—
172C4⋊C4×C47C2416yes—
173C22⋊C4×C47—23no—
174C22⋊C4×C47—23no—
175C22⋊C4×C47—23no—
176C42×C47—23no—
177C42×C47—23no—
178C42×C47—23no—
179C42×C47—26no—
180C42×C47—26no—
181C42×C47C2×D413yes—
182C42×C47C2×D413yes—
183C42×C47C2413yes—
184C42×C47C2413yes—
185C23×C47—21no—
186C23×C47C22×D411yescharacteristic
187Q8×C47—24no—
188Q8×C47—26no—
189Q8×C47—22no—
190Q8×C47—21no—
191Q8×C47—46no—
192Q8×C47—83no—
193Q8×C47—86no—
194Q8×C47—86no—
195Q8×C47#13714yesdirect product factor
196Q8×C47C22×Q814yes—
197Q8×C47C22×Q812yes—
198D4×C47—24no—
199D4×C47—21no—
200D4×C47C22×Q812yes—
201C2×C4×C47—24no—
202C2×C4×C47—21no—
203C2×C4×C47—22no—
204C2×C4×C47—24no—
205C2×C4×C47—26no—
206C2×C4×C47—23no—
207C2×C4×C47—23no—
208C2×C4×C47—26no—
209C2×C4×C47—43no—
210C2×C4×C47—46no—
211C2×C4×C47—46no—
212C2×C4×C47—83no—
213C2×C4×C47C22×D413yes—
214C2×C4×C47C22×D413yes—
215C2×C4×C47C22×D413yes—
216C2×C4×C47C22×D411yescharacteristic
217C2×C4×C47C22×D413yes—
218C2×C4×C47C22×Q812yes—
219C2×C4×C472−1+412yes—
220C2×C4×C47C2511yescharacteristic
221C8×C47—26no—
222C8×C47C22×Q814yes—
223Q8×#137C4711yesmaximal, characteristic, direct product factor, Sylow
224C22×C47—21no—
225C22×C47—21no—
226C22×C47—22no—
227C22×C47—41no—
228C22×C47—41no—
229C22×C47D4×Q812yes—
230C22×C47C23×D411yescharacteristic
231C4×C47—23no—
232C4×C47—23no—
233C4×C47—22no—
234C4×C47—24no—
235C4×C47—23no—
236C4×C47—46no—
237C4×C47—43no—
238C4×C47—43no—
239C4×C47D4×Q812yes—
240C4×C47C2×#13716yes—
241C4×C47C23×Q811yescharacteristic
242C4×C47C2×2−1+411yescharacteristic
243Q8×#41C2×C4711yescharacteristic
244C2×Q82C2×C4711yescharacteristic, complemented
245#3387C2×C4716yescomplemented
246#3385C2×C4716yescomplemented
247Q8×Q16C2×C4712yescomplemented
248Q8×SD16C2×C4712yescomplemented
249#3165C2×C4713yescomplemented
250#3079C2×C4713yes—
251#3078C2×C4713yescomplemented
252Q8×M16C2×C4711yescharacteristic, complemented
253C4×#137C2×C4713yes—
254C2×C47—22no—
255C2×C47—42no—
256C2×C47C2×D4×Q811yescharacteristic
257C2×C47C22×#13711yescharacteristic
258C2×C47#358511yescharacteristic
259C2×#137C22×C4711yescharacteristic
260Q82—21no—
261Q82—23no—
262Q82C22×C4712yes—
263Q82C22×C4711yescharacteristic
264Q82C22×C4716yes—
265#557C22×C4713yes—
266#554—23no—
267#554C22×C4713yes—
268D4×Q8—21no—
269D4×Q8C22×C4711yescharacteristic
270#533—23no—
271#533C22×C4713yes—
272C2×C4⋊Q8—23no—
273C2×C4⋊Q8C22×C4713yes—
274C4×#41C22×C4713yes—
275C2×C4×Q8—21no—
276C2×C4×Q8—21no—
277C2×C4×Q8C22×C4713yes—
278C2×C4×Q8C22×C4711yescharacteristic
279#501C22×C47112yes—
280#500C22×C4716yes—
281#498C22×C4716yes—
282#479C22×C47112yes—
283#477C22×C4716yes—
284#477C22×C4716yes—
285#476C22×C4716yes—
286#475C22×C4716yes—
287#475C22×C4716yes—
288#474C22×C4716yes—
289#446C22×C4716yes—
290C8×Q8C22×C4712yes—
291#441C22×C4716yes—
292#440C22×C4716yes—
293C4×Q16C22×C4716yes—
294C4×SD16C22×C4716yes—
295#428C22×C4716yes—
296#428C22×C4713yes—
297#423C22×C4713yes—
298#419C22×C4713yes—
299#417C22×C4713yes—
300C4×M16C22×C4713yes—
301C47Q8×#13711yescharacteristic, direct product factor, Sylow
302C2×#41C23×C4711yescharacteristic
303C22×Q8—21no—
304C22×Q8—23no—
305C22×Q8D4×C4711yescharacteristic
306C22×Q8C23×C4711yescharacteristic
307#137Q8×C4718yesdirect product factor
308C2×Q16C23×C4712yes—
309C2×SD16C23×C4712yes—
310C2×M16C23×C4713yes—
311C2×M16C23×C4711yescharacteristic
312C4⋊Q8—23no—
313C4⋊Q8—26no—
314C4⋊Q8—212no—
315C4⋊Q8—26no—
316C4⋊Q8—23no—
317C4⋊Q8—23no—
318C4⋊Q8—23no—
319C4⋊Q8C23×C4716yes—
320C4⋊Q8C23×C4716yes—
321C4⋊Q8C23×C4713yes—
322C4⋊Q8C23×C4713yes—
323#125—26no—
324#125C23×C4716yes—
325C22⋊Q8—23no—
326C22⋊Q8—23no—
327C22⋊Q8—23no—
328C22⋊Q8—23no—
329C22⋊Q8—26no—
330C4×Q8—21no—
331C4×Q8—26no—
332C4×Q8—22no—
333C4×Q8—23no—
334C4×Q8—23no—
335C4×Q8—23no—
336C4×Q8—23no—
337C4×Q8—23no—
338C4×Q8D4×C4711yescharacteristic
339C4×Q8D4×C4713yes—
340C4×Q8C23×C4716yes—
341C4×Q8C23×C4716yes—
342C4×Q8C23×C4713yes—
343C4×Q8C23×C4713yes—
344C4×Q8C23×C4711yescharacteristic
345C4×Q8C23×C4713yes—
346C4×D4—23no—
347C4×D4—23no—
348C4×D4C23×C4713yes—
349C4×D4C23×C4713yes—
350#117—23no—
351#117C23×C4713yes—
352C2×C4⋊C4—23no—
353C2×C4⋊C4—23no—
354C2×C4⋊C4—23no—
355C2×C4⋊C4D4×C4713yes—
356C2×C4⋊C4C23×C4713yes—
357C2×C42—23no—
358C2×C42—23no—
359C2×C42C23×C4713yes—
360#107C23×C4716yes—
361#107C23×C47112yes—
362#106C23×C4716yes—
363#106C23×C47112yes—
364C4⋊C8C23×C4716yes—
365C4⋊C8C23×C4716yes—
366#103C23×C4716yes—
367#103C23×C4716yes—
368#103C23×C4716yes—
369#102C23×C4716yes—
370#97C23×C4716yes—
371C4×C8C23×C4716yes—
372#41—22no—
373#41C2×Q8×C4714yes—
374C2×Q8—24no—
375C2×Q8—21no—
376C2×Q8—23no—
377C2×Q8—23no—
378C2×Q8—26no—
379C2×Q8—26no—
380C2×Q8—22no—
381C2×Q8—44no—
382C2×Q8—86no—
383C2×Q8C2×D4×C4713yes—
384C2×Q8C2×D4×C4711yescharacteristic
385C2×Q8C2×Q8×C4714yescomplemented
386C2×Q8C24×C4712yes—
387C2×Q8C24×C4711yescharacteristic
388C2×D4—21no—
389C2×D4C24×C4711yescharacteristic
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—23no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4C2×D4×C4713yes—
396C22×C4C2×D4×C4713yes—
397C22×C4C24×C4711yescharacteristic
398Q16C2×Q8×C4718yescomplemented
399SD16C2×Q8×C4718yescomplemented
400M16—26no—
401M16C2×Q8×C4714yescomplemented
402C2×C8C24×C4716yes—
403C2×C8C24×C4712yes—
404C4⋊C4—26no—
405C4⋊C4—23no—
406C4⋊C4—26no—
407C4⋊C4—26no—
408C4⋊C4—26no—
409C4⋊C4—26no—
410C4⋊C4—23no—
411C4⋊C4—23no—
412C4⋊C4—23no—
413C4⋊C4—23no—
414C4⋊C4—26no—
415C4⋊C4C2×D4×C4713yes—
416C4⋊C4C2×D4×C4716yes—
417C4⋊C4C2×D4×C4713yes—
418C4⋊C4C24×C4716yes—
419C4⋊C4C24×C4713yes—
420C4⋊C4C24×C4713yes—
421C22⋊C4—23no—
422C22⋊C4—23no—
423C22⋊C4—23no—
424C42—23no—
425C42—23no—
426C42—23no—
427C42—26no—
428C42—26no—
429C42C2×D4×C4713yes—
430C42C2×D4×C4713yes—
431C42C24×C4713yes—
432C42C24×C4713yes—
433C23—21no—
434C23C22×D4×C4711yescharacteristic
435Q8—24no—
436Q8—26no—
437Q8—22no—
438Q8—21no—
439Q8—46no—
440Q8—83no—
441Q8—86no—
442Q8—86no—
443Q8#137×C4714yesdirect product factor
444Q8C22×Q8×C4714yes—
445Q8C22×Q8×C4712yes—
446D4—24no—
447D4—21no—
448D4C22×Q8×C4712yes—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—26no—
454C2×C4—23no—
455C2×C4—23no—
456C2×C4—26no—
457C2×C4—43no—
458C2×C4—46no—
459C2×C4—46no—
460C2×C4—83no—
461C2×C4C22×D4×C4713yes—
462C2×C4C22×D4×C4713yes—
463C2×C4C22×D4×C4713yes—
464C2×C4C22×D4×C4711yescharacteristic
465C2×C4C22×D4×C4713yes—
466C2×C4C22×Q8×C4712yes—
467C2×C42−1+4×C4712yes—
468C2×C4C25×C4711yescharacteristic
469C8—26no—
470C8C22×Q8×C4714yes—
471C22—21no—
472C22—21no—
473C22—22no—
474C22—41no—
475C22—41no—
476C22D4×Q8×C4712yes—
477C22C23×D4×C4711yescharacteristic
478C4—23no—
479C4—23no—
480C4—23no—
481C4—22no—
482C4—24no—
483C4—43no—
484C4—46no—
485C4—43no—
486C4D4×Q8×C4712yes—
487C4C2×#137×C4716yes—
488C4C23×Q8×C4711yescharacteristic
489C4C2×2−1+4×C4711yescharacteristic
490C2—22no—
491C2—42no—
492C2C2×D4×Q8×C4711yescharacteristic
493C2C22×#137×C4711yescharacteristic
494C2C47×#358511yescharacteristic
subgroupG/H
center, socleC22×C47C23×D4
commutator, Frattini, cosocleC2×C4C25×C47
prime radicalC2×#41×C47C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C47 ◁ C2×C4×Q8×C47 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×Q82×C47C2semidirect
3. C47×#3387C2semidirect
4. C47×#3385C2semidirect
5. Q8×Q16×C47C2semidirect
6. Q8×SD16×C47C2semidirect
7. C47×#3165C2semidirect
9. C47×#3078C2semidirect
10. Q8×M16×C47C2semidirect
59. #137×C47Q8direct (N × H)
137. C2×Q8×C47C2×Q8semidirect
150. Q16×C47C2×Q8semidirect
151. SD16×C47C2×Q8semidirect
153. M16×C47C2×Q8semidirect
195. Q8×C47#137direct (N × H)
223. Q8×#137C47direct (N × H)
244. C2×Q82C2×C47semidirect
245. #3387C2×C47semidirect
246. #3385C2×C47semidirect
247. Q8×Q16C2×C47semidirect
248. Q8×SD16C2×C47semidirect
249. #3165C2×C47semidirect
251. #3078C2×C47semidirect
252. Q8×M16C2×C47semidirect
301. C47Q8×#137direct (N × H)
307. #137Q8×C47direct (N × H)
385. C2×Q8C2×Q8×C47semidirect
398. Q16C2×Q8×C47semidirect
399. SD16C2×Q8×C47semidirect
401. M16C2×Q8×C47semidirect
443. Q8#137×C47direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×#41×C472
C2×Q82×C472
C47×#33872
C47×#33852
Q8×Q16×C472
Q8×SD16×C472
C47×#31652
C47×#30792
C47×#30782
Q8×M16×C472
C4×#137×C472
Q8×#13747