Finite Group Databasebeta

← #34034840

Subgroups — #34034840

Order 18688.

642 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22627185C211yesmaximal, characteristic, complemented
2#22627184C211yesmaximal, characteristic, complemented
3#22627153C212yesmaximal, complemented
4#22627151C212yesmaximal, complemented
5#22627149C212yesmaximal, complemented
6#22627148C212yesmaximal, complemented
7#22627127C211yesmaximal, characteristic, complemented
8#22627111C211yesmaximal, characteristic, complemented
9C2×#17989687C211yesmaximal, characteristic, complemented
10#22627094C211yesmaximal, characteristic, complemented
11#22626703C211yesmaximal, characteristic
12#17989709C2212yescomplemented
13#17989708—21no—
14#17989708—21no—
15#17989708—22no—
16#17989708C2212yescomplemented
17#17989708C2212yescomplemented
18#17989708C2212yescomplemented
19C2×#14556688—21no—
20C2×#14556688C2211yescharacteristic
21#17989706—21no—
22#17989706C2211yescharacteristic
23#17989706C2211yescharacteristic
24C2×#14556687—22no—
25C2×#14556686C2211yescharacteristic
26#17989692C2212yescomplemented
27#17989691C2212yescomplemented
28#17989688C2211yescharacteristic, complemented
29#17989688C2211yescharacteristic, complemented
30#17989687C2212yescomplemented
31#17989687C2212yescomplemented
32#17989687C2211yescharacteristic, complemented
33#17989687C2211yescharacteristic, complemented
34#17989682C2212yescomplemented
35#64408⋊C16C2212yescomplemented
36D73⋊#131C2211yescharacteristic
37C2×#14556678C2211yescharacteristic
38#17989634C412yes—
39#17989632C412yes—
40#17989630C2212yes—
41#17989629C2212yes—
42#17989602C411yescharacteristic
43#17989585C411yescharacteristic
44#17989584C411yescharacteristic
45#17989553C2211yescharacteristic
46#17989383C411yescharacteristic
47#14556688—22no—
48#14556688—21no—
49#14556688D411yescharacteristic
50#14556688D411yescharacteristic
51#14556688C2312yes—
52#14556688C2311yescharacteristic
53#14556688C2311yescharacteristic
54#14556687—22no—
55#14556687—22no—
56#14556687—22no—
57#14556687—21no—
58#14556687D412yes—
59#14556686—22no—
60#14556686—21no—
61#14556686—21no—
62#14556686C2311yescharacteristic
63#14556686C2311yescharacteristic
64#14556686C2312yes—
65C2×#11924040—22no—
66C2×#11924040—22no—
67C2×#11924040—22no—
68C2×#11924040—21no—
69C2×#11924040—21no—
70#14556679C2×C411yescharacteristic
71#14556679C2311yescharacteristic
72#14556678—21no—
73#14556678—21no—
74#14556678—21no—
75#14556678—21no—
76#14556678C2×C412yes—
77#14556678C2×C411yescharacteristic
78#14556678C2312yes—
79#14556678C2311yescharacteristic
80C2×#11924038C2×C411yescharacteristic
81D73⋊M16—21no—
82D73⋊M16C2×C411yescharacteristic
83C2×#11924037—21no—
84C2×#11924036—21no—
85C2×#11924036C2311yescharacteristic
86C22×#10596833—21no—
87#14556670C2×C411yescharacteristic
88#14556662C2×C412yes—
89#14556661C2×C412yes—
90#14556656C2×C412yes—
91#14556655C2×C412yes—
92#14556655C2×C412yes—
93#14556655C2×C412yes—
94#14556645C2×C412yes—
95C8×#64408C2312yes—
96C2×#11924027C2×C411yescharacteristic
97#41×D73C2×C411yescharacteristic
98#14556624C811yescharacteristic
99#14556621C811yescharacteristic
100#14556620C812yes—
101#14556617C811yescharacteristic
102#14556615C812yes—
103#14556612C2×C412yes—
104#14556610C2×C412yes—
105M16×D73C2×C411yescharacteristic
106AGL(1,ℤ/8)×C73C811yescharacteristic
107#11924040—21no—
108#11924040—21no—
109#11924040—22no—
110#11924040—21no—
111#11924040—21no—
112#11924040C2×D412yes—
113#11924040C2×D412yes—
114#11924038—21no—
115#11924038—21no—
116#11924038—21no—
117#11924038C22×C411yescharacteristic
118#11924038C22×C411yescharacteristic
119#11924038C22×C412yes—
120#11924037—21no—
121#11924037—22no—
122#11924037—21no—
123#11924037—22no—
124#11924037—21no—
125#11924036—21no—
126#11924036—21no—
127#11924036—21no—
128#11924036C22×C411yescharacteristic
129#11924036C2×D411yescharacteristic
130#11924036#4111yescharacteristic
131#11924036C2411yescharacteristic
132C2×#10596833—21no—
133C2×#10596833—22no—
134C2×#10596833—21no—
135C2×#10596833—22no—
136C2×#10596833—21no—
137C2×#10596833C2×D411yescharacteristic
138C2×#10596833#4111yescharacteristic
139C4⋊#64408—21no—
140C22⋊#64408—21no—
141C4×#64408C22×C412yes—
142#11924029C22×C411yescharacteristic
143#11924028—21no—
144#11924028C2×C811yescharacteristic
145#11924027—21no—
146#11924027—21no—
147#11924027—21no—
148#11924027C2×C812yes—
149#11924027C22×C411yescharacteristic
150#11924027C22×C412yes—
151C2×Dic146C2×C811yescharacteristic
152Q8×D73C2×C811yescharacteristic
153C2×C22⋊D73—21no—
154D4×D73—21no—
155D4×D73C2×C811yescharacteristic
156C22×Dic73—21no—
157C2×C4×D73—21no—
158C2×C4×D73C22×C411yescharacteristic
159Dic292C2×C812yes—
160#11924018C2×C812yes—
161#11924017C2×C812yes—
162#11924016C2×C812yes—
163#11924016C2×C812yes—
164#11924013C2×C812yes—
165#11924012C2×C811yescharacteristic
166#11924011C2×C812yes—
167C8×D73C22×C412yes—
168#41×C73C2×C811yescharacteristic
169C2×D4×C73C2×C811yescharacteristic
170SD16×C73C2×C812yes—
171D8×C73C2×C812yes—
172M16×C73C2×C811yescharacteristic
173C23×C73—21no—
174C22×D73—21no—
175C22×D73#13111yescharacteristic
176C2×#64408—22no—
177Q8×C73C22×C811yescharacteristic
178D4×C73—21no—
179D4×C73—21no—
180D4×C73C22×C811yescharacteristic
181D4×C73C22×C812yes—
182C2×C4×C73—21no—
183C2×C4×C73C22×C811yescharacteristic
184C22⋊D73—21no—
185C22⋊D73—21no—
186C22⋊D73—22no—
187C22⋊D73—22no—
188C22⋊D73—21no—
189C2×Dic73—21no—
190C2×Dic73—21no—
191C2×Dic73—22no—
192C2×Dic73—22no—
193C2×Dic73C4×D411yescharacteristic
194D292—21no—
195D292C22×C811yescharacteristic
196C4×D73—21no—
197C4×D73—21no—
198C4×D73—21no—
199C4×D73C4×D411yescharacteristic
200C4×D73C22×C811yescharacteristic
201C4×D73#13111yescharacteristic
202C4×D73C23×C411yescharacteristic
203Dic146—21no—
204Dic146—21no—
205Dic146C22×C811yescharacteristic
206Dic146C22×C812yes—
207#10596833—22no—
208#10596833—21no—
209#10596833—21no—
210#10596833C4×D411yescharacteristic
211#10596833C4×D411yescharacteristic
212#10596833C22×D411yescharacteristic
213#10596833C2×#4111yescharacteristic
214C8×C73C22×C812yes—
215#10596831C22×C812yes—
216C22×C73—22no—
217C22×C73—21no—
218C22×C73—81no—
219C22×C73C8×D411yescharacteristic
220C2×D73—21no—
221C2×D73—41no—
222C2×D73—81no—
223C2×D73C8×D411yescharacteristic
224C2×D73C2×#13111yescharacteristic
225#64408—42no—
226C4×C73—21no—
227C4×C73C8×D411yescharacteristic
228C4×C73C23×C811yescharacteristic
229Dic73—21no—
230Dic73—22no—
231Dic73C8×D411yescharacteristic
232Dic73C2×C4×D411yescharacteristic
233#33442—731nomaximal, Sylow
234C2×C73—21no—
235C2×C73—41no—
236C2×C73—42no—
237C2×C73C2×C8×D411yescharacteristic
238D73—21no—
239D73—21no—
240D73—41no—
241#3407—731no—
242#3407—731no—
243#2955—731no—
244#2171—732no—
245#2171—732no—
246#2170—732no—
247#2170—732no—
248#2153—731no—
249C2×#350—731no—
250C2×#350—731no—
251#2108—731no—
252C73#3344211yescharacteristic, complemented, Sylow
253#575—731no—
254C2×#131—731no—
255C2×#131—731no—
256#504—732no—
257#504—732no—
258#504—732no—
259#504—732no—
260#504—1461no—
261#504—1461no—
262#504—1462no—
263C2×M32—731no—
264C2×M32—731no—
265C2×M32—731no—
266C2×M32—1462no—
267C2×M32—1461no—
268C2×M32—1461no—
269C22×C16—731no—
270#444—732no—
271#444—732no—
272#443—732no—
273#443—732no—
274#430—731no—
275#421—731no—
276#421—731no—
277#405—731no—
278#364—732no—
279#364—732no—
280#350—732no—
281#350—731no—
282#350—732no—
283#350—731no—
284#350—731no—
285#350—731no—
286#346—732no—
287C4×C16—732no—
288C2×#41—731no—
289C2×#41—731no—
290#137—731no—
291#137—732no—
292#137—731no—
293AGL(1,ℤ/8)—731no—
294AGL(1,ℤ/8)—732no—
295AGL(1,ℤ/8)#1059684211yescharacteristic
296#135—732no—
297#135—732no—
298#131—732no—
299#131—731no—
300#131—732no—
301#131—731no—
302#131—731no—
303#131—731no—
304#131—1461no—
305#131—1461no—
306#131—1461no—
307#131—1461no—
308C2×M16—731no—
309C2×M16—731no—
310C2×M16—731no—
311C2×M16—1461no—
312C2×M16—1461no—
313C22×C8—731no—
314C22×C8—1461no—
315C22×C8—1461no—
316#117—731no—
317M32—732no—
318M32—731no—
319M32—731no—
320M32—732no—
321M32—731no—
322M32—731no—
323M32—1462no—
324M32—1462no—
325M32—1462no—
326M32—1462no—
327M32—1461no—
328M32—1461no—
329C2×C16—732no—
330C2×C16—731no—
331C2×C16—731no—
332C2×C16—1462no—
333C2×C16—1462no—
334C2×C16—1462no—
335C2×C16—1462no—
336C2×C16—1461no—
337C2×C16—1461no—
338C2×C16—1461no—
339C2×C16—1461no—
340#108—732no—
341#108—732no—
342#104—732no—
343#104—732no—
344#104—732no—
345#104—732no—
346#97—732no—
347C4×C8—732no—
348#41—731no—
349#41—732no—
350#41—732no—
351#41—731no—
352#41—731no—
353#41—1461no—
354#41—1461no—
355#41—1461no—
356#41—1461no—
357#41C2×#1059684211yescharacteristic
358C2×Q8—731no—
359C2×Q8—731no—
360C2×D4—731no—
361C2×D4—1461no—
362C2×D4—1461no—
363C2×D4C2×#1059684211yescharacteristic
364C22×C4—731no—
365C22×C4—1461no—
366C22×C4—1461no—
367Q16—732no—
368Q16—732no—
369SD16—732no—
370SD16—732no—
371SD16—732no—
372SD16C2×#1059684212yes—
373D8—732no—
374D8C2×#1059684212yes—
375M16—731no—
376M16—732no—
377M16—731no—
378M16—731no—
379M16—732no—
380M16—1462no—
381M16—1461no—
382M16—1462no—
383M16—1461no—
384M16—1461no—
385M16—1461no—
386M16—1461no—
387M16—1461no—
388M16C2×#1059684211yescharacteristic
389C2×C8—731no—
390C2×C8—731no—
391C2×C8—731no—
392C2×C8—731no—
393C2×C8—731no—
394C2×C8—731no—
395C2×C8—732no—
396C2×C8—1462no—
397C2×C8—1461no—
398C2×C8—1462no—
399C2×C8—1461no—
400C2×C8—1461no—
401C2×C8—1461no—
402C2×C8—1461no—
403C2×C8—1461no—
404C4⋊C4—1461no—
405C22⋊C4—1461no—
406C42—732no—
407C16—732no—
408C16—732no—
409C16—1461no—
410C16—1461no—
411C16—1461no—
412C16—1461no—
413C16—1462no—
414C23—21no—
415C23—731no—
416C23—1461no—
417Q8—731no—
418Q8—732no—
419Q8—1461no—
420Q8—1461no—
421Q8C22×#1059684211yescharacteristic
422D4—21no—
423D4—21no—
424D4—731no—
425D4—1461no—
426D4—1461no—
427D4—1461no—
428D4—1462no—
429D4—1462no—
430D4—1461no—
431D4C22×#1059684212yes—
432D4C22×#1059684211yescharacteristic
433C2×C4—21no—
434C2×C4—731no—
435C2×C4—731no—
436C2×C4—731no—
437C2×C4—731no—
438C2×C4—731no—
439C2×C4—1461no—
440C2×C4—1461no—
441C2×C4—1462no—
442C2×C4—1461no—
443C2×C4—1462no—
444C2×C4—1462no—
445C2×C4—1461no—
446C2×C4—1461no—
447C2×C4C22×#1059684211yescharacteristic
448C8—731no—
449C8—731no—
450C8—731no—
451C8—731no—
452C8—732no—
453C8—1462no—
454C8—1461no—
455C8—1461no—
456C8C22×#1059684212yes—
457C22—21no—
458C22—22no—
459C22—81no—
460C22—2921no—
461C22—5841no—
462C22—731no—
463C22—731no—
464C22—1461no—
465C22D4×#1059684211yescharacteristic
466C4—21no—
467C4—2922no—
468C4—731no—
469C4—731no—
470C4—1461no—
471C4—1462no—
472C4D4×#1059684211yescharacteristic
473C4C23×#1059684211yescharacteristic
474C2—21no—
475C2—42no—
476C2—41no—
477C2—2921no—
478C2—1461no—
479C2—1461no—
480C2C2×D4×#1059684211yescharacteristic
subgroupG/H
centerC2C2×D4×#10596842
commutatorC4×C73C23×C8
FrattiniC4C23×#10596842
FittingAGL(1,ℤ/8)×C73C8
socleC2×C73C2×C8×D4
prime radical#17989383C4
cosocle#11924036C24
hypercenterAGL(1,ℤ/8)#10596842
hypocenterC73#33442
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #22627185C2semidirect
2. #22627184C2semidirect
3. #22627153C2semidirect
4. #22627151C2semidirect
5. #22627149C2semidirect
6. #22627148C2semidirect
7. #22627127C2semidirect
8. #22627111C2semidirect
9. C2×#17989687C2semidirect
10. #22627094C2semidirect
12. #17989709C22semidirect
16. #17989708C22semidirect
17. #17989708C22semidirect
18. #17989708C22semidirect
26. #17989692C22semidirect
27. #17989691C22semidirect
28. #17989688C22semidirect
29. #17989688C22semidirect
30. #17989687C22semidirect
31. #17989687C22semidirect
32. #17989687C22semidirect
33. #17989687C22semidirect
34. #17989682C22semidirect
35. #64408⋊C16C22semidirect
252. C73#33442semidirect

Maximal subgroups

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

subgroupindex
#226271852
#226271842
#226271532
#226271512
#226271492
#226271482
#226271272
#226271112
C2×#179896872
#226270942
#226267032
#3344273