Finite Group Databasebeta

← #72321447

Subgroups — #72321447

Order 2048.

769 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72321129C211yesmaximal, characteristic, complemented
2#72321124C211yesmaximal, characteristic, complemented
3#72320271C211yesmaximal, characteristic, complemented
4C2×#94614C211yesmaximal, characteristic, complemented
5C2×#153598C211yesmaximal, characteristic
6#72234379C211yesmaximal, characteristic, complemented
7#72232009C211yesmaximal, characteristic, complemented
8#72224652C218yesmaximal, complemented
9C2×#13742C411yescharacteristic
10C2×#13656C2211yescharacteristic
11C2×#13654C2211yescharacteristic
12#483513—21no—
13#483504—21no—
14#153598C414yes—
15#153598C2214yes—
16C2×#7548C411yescharacteristic
17C2×#7541C411yescharacteristic
18C2×#7540C411yescharacteristic
19C2×#7511C2211yescharacteristic
20C2×#7511C2211yescharacteristic
21#151086—22no—
22#151086C2214yescomplemented
23#151085—22no—
24#151085C2214yescomplemented
25C2×#7464C2211yescharacteristic
26#151037C2211yescharacteristic
27#150069—21no—
28#150064—21no—
29#149496C2211yescharacteristic
30#94754C2214yescomplemented
31#94614C2214yescomplemented
32#94613C2214yescomplemented
33#94506C2214yescomplemented
34C2×#2262C2×C411yescharacteristic
35C22×D32C2×C411yescharacteristic
36C2×#2178—21no—
37C2×C4×D16—21no—
38#33443—41no—
39#17854—41no—
40#13742C814yescomplemented
41#13742C2×C414yes—
42C2×SD128C812yescomplemented
43C2×D64C812yescomplemented
44C2×M128—21no—
45C2×M128C2×C411yescharacteristic
46C2×M128C2×C411yescharacteristic
47#13656—21no—
48#13656—21no—
49#13656C2×C412yes—
50#13656C2312yes—
51#13654—21no—
52#13654—21no—
53#13654C2×C412yes—
54#13654C2312yes—
55C2×#1425C2×C411yescharacteristic
56C2×#1424C2×C411yescharacteristic
57C2×#1417—21no—
58C2×#1415—21no—
59C4×M64D411yescharacteristic
60C2×#1397C2311yescharacteristic
61#12047—24no—
62#12046—42no—
63#12045—24no—
64C8×D16—42no—
65C2×#1373—21no—
66#12008—21no—
67#11602—41no—
68#11593—41no—
69#11399D411yescharacteristic
70#7548C2×C414yes—
71#7541C2×C414yes—
72#7540C2×C414yes—
73#7511—21no—
74#7511C2×C412yes—
75#7511C2312yes—
76#7511C2312yes—
77#7464—21no—
78#7464C2312yes—
79#7463—21no—
80#7463C2312yes—
81#7388—24no—
82#7385—24no—
83#7330D412yes—
84#7330C2312yes—
85C2×#508—21no—
86C22×D16—21no—
87#3407—81no—
88#2972—81no—
89C2×#443—41no—
90C2×C4×D8—41no—
91#2262—21no—
92#2262—21no—
93#2262C2×C812yescomplemented
94#2262C22×C412yes—
95C2×Q64C2×C811yescharacteristic, complemented
96C2×SD64—21no—
97C2×D32—21no—
98C2×D32—21no—
99C2×D32—21no—
100C2×D32C2×C812yes—
101C2×D32C2×C812yes—
102C2×D32C2×C811yescharacteristic, complemented
103C2×D32C22×C412yes—
104C2×M64—21no—
105C22×C32C22×C411yescharacteristic
106C22×C32#4111yescharacteristic
107#2178—22no—
108#2178—22no—
109#2178—41no—
110#2178—41no—
111C4×D16—22no—
112C4×D16—22no—
113C4×D16—41no—
114C4×D16—41no—
115#2171—82no—
116#2170—44no—
117C2×#368—21no—
118C2×#366—21no—
119C2×#364—41no—
120C2×#359—41no—
121C2×#357—41no—
122#2117—81no—
123C4×M32D811yescharacteristic
124C2×C4×C16C2×D411yescharacteristic
125#1580—82no—
126C8×D8—44no—
127#1564—41no—
128#1478—81no—
129#1450D811yescharacteristic
130SD128C2×C818yescomplemented
131D64C2×C818yescomplemented
132M128—22no—
133M128—21no—
134M128—22no—
135M128C2×C812yescomplemented
136M128C22×C412yes—
137M128C22×C412yes—
138C2×C64—22no—
139C2×C64—21no—
140C2×C64C2×C812yescomplemented
141#1425—21no—
142#1425C22×C412yes—
143#1424—21no—
144#1424C22×C412yes—
145#1417—22no—
146#1417—22no—
147#1415—22no—
148#1415—22no—
149#1397C22×C411yescharacteristic
150#1397C2×D412yes—
151#1397C2×D411yescharacteristic
152#1397#4111yescharacteristic
153#1397C2411yescharacteristic
154C4×C32—21no—
155#1373—22no—
156#1373—41no—
157#1372—22no—
158#1371—41no—
159#1335—44no—
160#1331—44no—
161#1311C2×D412yes—
162C8×C16D812yes—
163C2×#135—41no—
164C22×D8—41no—
165C2×#131—81no—
166C2×C4×D4—81no—
167#508—22no—
168#508—22no—
169#508—41no—
170#508—41no—
171C2×Q32—21no—
172C2×SD32—41no—
173C2×D16—22no—
174C2×D16—22no—
175C2×D16—22no—
176C2×D16—21no—
177C2×D16—41no—
178C2×D16—41no—
179C2×D16—41no—
180#504—84no—
181#504—162no—
182C2×M32—21no—
183C2×M32—81no—
184C2×M32—162no—
185C22×C16C4×D411yescharacteristic
186C22×C16#13511yescharacteristic
187#443—42no—
188#443—42no—
189#443—81no—
190#443—81no—
191C4×D8—42no—
192C4×D8—42no—
193C4×D8—81no—
194C4×D8—81no—
195C8×D4—84no—
196C8×D4—162no—
197#431—21no—
198#431—81no—
199C2×#108—41no—
200C2×#107—41no—
201C2×#104—81no—
202C2×#102—81no—
203C2×C22⋊C8—162no—
204C2×C4×C8C2×D811yescharacteristic
205Q64—21no—
206Q64C22×C812yes—
207SD64—22no—
208SD64—22no—
209D32—22no—
210D32—22no—
211D32—22no—
212D32—21no—
213D32C22×C812yes—
214D32C22×C814yes—
215M64—41no—
216M64—41no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—22no—
220C2×C32C4×D411yescharacteristic
221C2×C32C4×D411yescharacteristic
222C2×C32C22×C811yescharacteristic
223C2×C32C22×C811yescharacteristic
224C2×C32#13111yescharacteristic
225C2×C32#13111yescharacteristic
226C2×C32C23×C411yescharacteristic
227C2×C32C2×#4111yescharacteristic
228#368—22no—
229#368—41no—
230#366—22no—
231#366—41no—
232#364—42no—
233#364—81no—
234#359—42no—
235#359—42no—
236#357—42no—
237#357—42no—
238#350—84no—
239#346—21no—
240C4×C16C4×D411yescharacteristic
241C4×C16C2×D812yes—
242C4×C16C2×D811yescharacteristic
243C4×C16#13511yescharacteristic
244C4×C16C22×D411yescharacteristic
245#335—42no—
246#334—81no—
247D4⋊C8—84no—
248#322—21no—
249C82C2×D812yes—
250C64—24no—
251C64—22no—
252C64C22×C814yes—
253C2×#41—81no—
254C22×D4—81no—
255C23×C4—161no—
256#135—42no—
257#135—42no—
258#135—81no—
259#135—81no—
260C2×Q16—41no—
261C2×SD16—81no—
262C2×D8—42no—
263C2×D8—42no—
264C2×D8—41no—
265C2×D8—42no—
266C2×D8—81no—
267C2×D8—81no—
268C2×D8—81no—
269#131—82no—
270#131—82no—
271#131—161no—
272#131—161no—
273C2×M16—81no—
274C2×M16—161no—
275C22×C8—161no—
276C22×C8—164no—
277C22×C8C4×D811yescharacteristic
278C22×C8#50911yescharacteristic
279C4×D4—82no—
280C4×D4—82no—
281C4×D4—161no—
282C4×D4—161no—
283C2×C4⋊C4—81no—
284C2×C22⋊C4—161no—
285C2×C42#50911yescharacteristic
286Q32—22no—
287Q32—41no—
288SD32—42no—
289SD32—42no—
290D16—24no—
291D16—22no—
292D16—42no—
293D16—42no—
294D16—41no—
295D16—42no—
296M32—41no—
297M32—41no—
298M32—82no—
299M32—164no—
300M32—161no—
301C2×C16—21no—
302C2×C16—21no—
303C2×C16—22no—
304C2×C16—164no—
305C2×C16—161no—
306C2×C16C8×D411yescharacteristic
307C2×C16C8×D411yescharacteristic
308C2×C16C4×D811yescharacteristic
309C2×C16C4×D811yescharacteristic
310C2×C16#44311yescharacteristic
311C2×C16#44311yescharacteristic
312C2×C16C2×C4×D411yescharacteristic
313C2×C16C2×#13511yescharacteristic
314#108—42no—
315#108—81no—
316#107—42no—
317#107—81no—
318C4⋊C8—41no—
319C4⋊C8—82no—
320#104—82no—
321#104—82no—
322#102—82no—
323#102—82no—
324C22⋊C8—41no—
325C22⋊C8—164no—
326C22⋊C8—161no—
327C4×C8—22no—
328C4×C8—161no—
329C4×C8C4×D811yescharacteristic
330C4×C8#50911yescharacteristic
331C4×C8#50911yescharacteristic
332C4×C8C22×D811yescharacteristic
333C32—21no—
334C32—21no—
335C32—42no—
336C32C8×D411yescharacteristic
337C32C8×D411yescharacteristic
338C32C23×C811yescharacteristic
339C32C2×#13111yescharacteristic
340C24—161no—
341#41—82no—
342#41—82no—
343#41—161no—
344#41—161no—
345C2×Q8—81no—
346C2×D4—82no—
347C2×D4—82no—
348C2×D4—81no—
349C2×D4—82no—
350C2×D4—161no—
351C2×D4—161no—
352C2×D4—161no—
353C2×D4—161no—
354C22×C4—21no—
355C22×C4—162no—
356C22×C4—162no—
357C22×C4—161no—
358C22×C4—161no—
359C22×C4—642no—
360C22×C4#217711yescharacteristic
361Q16—42no—
362Q16—81no—
363SD16—82no—
364SD16—82no—
365D8—44no—
366D8—42no—
367D8—82no—
368D8—82no—
369D8—81no—
370D8—82no—
371M16—82no—
372M16—162no—
373M16—162no—
374M16—161no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—42no—
378C2×C8—162no—
379C2×C8—162no—
380C2×C8—161no—
381C2×C8—162no—
382C2×C8—644no—
383C2×C8C8×D811yescharacteristic
384C2×C8C8×D811yescharacteristic
385C2×C8#217711yescharacteristic
386C2×C8#217711yescharacteristic
387C2×C8#217911yescharacteristic
388C2×C8#217911yescharacteristic
389C2×C8C2×C4×D811yescharacteristic
390C2×C8C2×#50911yescharacteristic
391C4⋊C4—82no—
392C4⋊C4—161no—
393C22⋊C4—162no—
394C22⋊C4—162no—
395C42—21no—
396C42#217711yescharacteristic
397C42C2×#50911yescharacteristic
398C16—21no—
399C16—21no—
400C16—42no—
401C16—162no—
402C16C8×D811yescharacteristic
403C16C8×D811yescharacteristic
404C16C2×C8×D411yescharacteristic
405C16C2×#44311yescharacteristic
406C23—162no—
407C23—161no—
408C23—162no—
409C23—162no—
410C23—641no—
411C23—641no—
412C23#1365811yescharacteristic
413Q8—82no—
414Q8—161no—
415D4—84no—
416D4—82no—
417D4—162no—
418D4—162no—
419D4—162no—
420D4—162no—
421D4—162no—
422D4—161no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—81no—
426C2×C4—161no—
427C2×C4—162no—
428C2×C4—162no—
429C2×C4—162no—
430C2×C4—324no—
431C2×C4—642no—
432C2×C4—641no—
433C2×C4#1204511yescharacteristic
434C2×C4#1204511yescharacteristic
435C2×C4#1365811yescharacteristic
436C2×C4#1365811yescharacteristic
437C2×C4#1365811yescharacteristic
438C2×C4C2×#217711yescharacteristic
439C8—21no—
440C8—21no—
441C8—82no—
442C8—162no—
443C8—322no—
444C8#1204511yescharacteristic
445C8#1204511yescharacteristic
446C8C2×C8×D811yescharacteristic
447C8C2×#217911yescharacteristic
448C22—41no—
449C22—162no—
450C22—164no—
451C22—322no—
452C22—322no—
453C22—641no—
454C22—642no—
455C22—642no—
456C22#15108511yescharacteristic
457C22#15108511yescharacteristic
458C22C2×#1365811yescharacteristic
459C4—41no—
460C4—41no—
461C4—162no—
462C4—322no—
463C4#15108511yescharacteristic
464C4#15108511yescharacteristic
465C4C2×#1204511yescharacteristic
466C4C2×#1365811yescharacteristic
467C2—21no—
468C2—21no—
469C2—21no—
470C2—324no—
471C2—322no—
472C2C2×#15108511yescharacteristic
subgroupG/H
center, socleC2C2×#151085
commutatorC32C23×C8
Frattini, cosocle#1397C24
prime radicalC2×#13742C4
LCSG ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ #11399 ◁ #149496 ◁ G
LECSG ▷ #1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72321129C2semidirect
2. #72321124C2semidirect
3. #72320271C2semidirect
4. C2×#94614C2semidirect
6. #72234379C2semidirect
7. #72232009C2semidirect
8. #72224652C2semidirect
22. #151086C22semidirect
24. #151085C22semidirect
30. #94754C22semidirect
31. #94614C22semidirect
32. #94613C22semidirect
33. #94506C22semidirect
40. #13742C8semidirect
42. C2×SD128C8semidirect
43. C2×D64C8semidirect
93. #2262C2×C8semidirect
95. C2×Q64C2×C8semidirect
102. C2×D32C2×C8semidirect
130. SD128C2×C8semidirect
131. D64C2×C8semidirect
135. M128C2×C8semidirect
140. C2×C64C2×C8semidirect

Maximal subgroups

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

subgroupindex
#723211292
#723211242
#723202712
C2×#946142
C2×#1535982
#722343792
#722320092
#722246522