Finite Group Databasebeta

← #226015

Subgroups — #226015

Order 512.

631 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#443C211yesmaximal, characteristic
2#17881C212yesmaximal
3#17799C211yesmaximal, characteristic, complemented
4#17789C211yesmaximal, characteristic, complemented
5#9610C212yesmaximal, complemented
6#8841C212yesmaximal, complemented
7#8839C212yesmaximal, complemented
8#8806C211yesmaximal, characteristic, complemented
9#8605C211yesmaximal, characteristic, complemented
10#8605C211yesmaximal, characteristic, complemented
11#8267C211yesmaximal, characteristic, complemented
12#2989C2212yes—
13#2988C2212yes—
14C2×#443C411yescharacteristic
15C2×#443C411yescharacteristic
16C4×#135C2211yescharacteristic
17C4×#131C2211yescharacteristic
18C4×#131C2211yescharacteristic
19#1842C2212yes—
20C4×#108C2211yescharacteristic
21#1772C411yescharacteristic
22#1772C411yescharacteristic
23#1764C411yescharacteristic
24#1764C411yescharacteristic
25#1764C411yescharacteristic
26#1764C411yescharacteristic
27C4×#104C2211yescharacteristic
28C4×#104C2211yescharacteristic
29C42×C8C2211yescharacteristic
30#1588—22no—
31#1585—22no—
32#1584—22no—
33#1581—22no—
34#1569—22no—
35#1568—21no—
36#1567—21no—
37#1565—22no—
38#1565C2212yes—
39#1565C2212yes—
40#1564—21no—
41#1564C2211yescharacteristic
42#1564C2211yescharacteristic
43C2×#335—21no—
44C2×#335C2211yescharacteristic
45C2×#335C2211yescharacteristic
46C2×#335C2212yes—
47#1477C2212yes—
48#1477C2212yes—
49#1452C2212yes—
50#1283C2212yes—
51#1283C2212yes—
52#1276C2212yes—
53C2×#135C2×C411yescharacteristic
54C2×#131C2×C411yescharacteristic
55C2×#131C2×C411yescharacteristic
56C2×#131C2×C411yescharacteristic
57C2×#131C2×C411yescharacteristic
58C4×#41C2311yescharacteristic
59C4×#41C2311yescharacteristic
60#446—22no—
61C8×Q8—22no—
62#443—24no—
63#443—24no—
64C4×Q16D411yescharacteristic
65C4×Q16Q811yescharacteristic
66C4×SD16D411yescharacteristic
67C4×SD16D411yescharacteristic
68C4×SD16Q811yescharacteristic
69C4×SD16Q811yescharacteristic
70C4×D8D411yescharacteristic
71C4×D8Q811yescharacteristic
72#436—22no—
73#435—42no—
74C8×D4—22no—
75C8×D4—42no—
76#432—22no—
77#431—22no—
78#431C2312yes—
79#431C2312yes—
80C2×#108C2×C411yescharacteristic
81C2×#108C2×C411yescharacteristic
82#427C2×C411yescharacteristic
83#423—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—42no—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88C2×#104C2×C411yescharacteristic
89C2×#104C2×C411yescharacteristic
90C2×#104C2×C411yescharacteristic
91C2×#104C2×C411yescharacteristic
92#416C2×C411yescharacteristic
93#416C2×C411yescharacteristic
94#405—21no—
95#405—21no—
96#405—21no—
97#405—21no—
98#405C2×C411yescharacteristic
99#405C2×C411yescharacteristic
100#405C2×C411yescharacteristic
101#405C2×C411yescharacteristic
102C4×M16—21no—
103C4×M16—21no—
104C4×M16C2311yescharacteristic
105C4×M16C2311yescharacteristic
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2×C411yescharacteristic
109C2×C4×C8C2×C411yescharacteristic
110C2×C4×C8C2×C411yescharacteristic
111C2×C4×C8C2×C411yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C43C2311yescharacteristic
115#341C2×C411yescharacteristic
116#341C2×C411yescharacteristic
117#339C2×C411yescharacteristic
118#339C2×C411yescharacteristic
119#339C2×C411yescharacteristic
120#339C2×C411yescharacteristic
121#335—22no—
122#335—22no—
123#335—21no—
124#335—21no—
125#335—21no—
126#335—21no—
127#335—22no—
128#335—22no—
129#335—21no—
130#335D411yescharacteristic
131#335D411yescharacteristic
132#334—22no—
133#334—22no—
134#334—21no—
135#334D412yes—
136Q8⋊C8—22no—
137Q8⋊C8—22no—
138D4⋊C8—22no—
139D4⋊C8—22no—
140#322—22no—
141#322—22no—
142C2×#41C22×C411yescharacteristic
143C2×#41C22×C411yescharacteristic
144#135—22no—
145#135—22no—
146C2×Q16C4⋊C411yescharacteristic
147C2×SD16C4⋊C411yescharacteristic
148C2×SD16C4⋊C411yescharacteristic
149C2×D8C4⋊C411yescharacteristic
150#131—22no—
151#131—22no—
152#131—22no—
153#131—22no—
154#131—22no—
155#131—41no—
156#131—41no—
157#131M1614yescomplemented
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—42no—
163C2×M16C22×C411yescharacteristic
164C2×M16C22×C411yescharacteristic
165C2×M16C22×C411yescharacteristic
166C2×M16C22×C411yescharacteristic
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—42no—
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C4×Q8C2×D411yescharacteristic
179C4×Q8C2×D411yescharacteristic
180C4×Q8C2×Q811yescharacteristic
181C4×Q8C2×Q811yescharacteristic
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4C2×D411yescharacteristic
187C4×D4C2×D411yescharacteristic
188C4×D4C2×Q811yescharacteristic
189C4×D4C2×Q811yescharacteristic
190#117—21no—
191#117—21no—
192#117C22×C411yescharacteristic
193#117C22×C411yescharacteristic
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C2411yescharacteristic
201#108—22no—
202#108—22no—
203#107C4⋊C411yescharacteristic
204#107C4⋊C411yescharacteristic
205#106C4⋊C411yescharacteristic
206#106C4⋊C411yescharacteristic
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—22no—
212C4⋊C8—42no—
213C4⋊C8—42no—
214C4⋊C8—84no—
215C4⋊C8C2×D412yes—
216C4⋊C8C2×D412yes—
217#104—22no—
218#104—22no—
219#104—22no—
220#104M1614yescomplemented
221#103C4⋊C411yescharacteristic
222#103C4⋊C411yescharacteristic
223#103C4⋊C411yescharacteristic
224#103C4⋊C411yescharacteristic
225#102C4⋊C411yescharacteristic
226#102C4⋊C411yescharacteristic
227#102C4⋊C411yescharacteristic
228#102C4⋊C411yescharacteristic
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—42no—
232C22⋊C8—42no—
233#97—21no—
234#97—21no—
235#97—21no—
236#97—21no—
237#97—21no—
238#97—21no—
239#97—42no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—22no—
245C4×C8—22no—
246C4×C8—22no—
247C4×C8—22no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—22no—
251C4×C8—41no—
252C4×C8—42no—
253C4×C8C2×D411yescharacteristic
254C4×C8C2×D411yescharacteristic
255C4×C8C2×D411yescharacteristic
256C4×C8C2×D411yescharacteristic
257C4×C8C2×Q811yescharacteristic
258C4×C8C2×Q811yescharacteristic
259C4×C8#4111yescharacteristic
260C4×C8#4111yescharacteristic
261#41—21no—
262#41—21no—
263#41—21no—
264#41—21no—
265#41—21no—
266#41—21no—
267#41C2×M1612yes—
268#41C2×M1612yes—
269C2×Q8C2×C4⋊C411yescharacteristic
270C2×Q8C2×C4⋊C411yescharacteristic
271C2×D4—21no—
272C2×D4—21no—
273C2×D4C2×C4⋊C411yescharacteristic
274C2×D4C2×C4⋊C411yescharacteristic
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4C23×C411yescharacteristic
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C42−1+411yescharacteristic
286Q16—22no—
287SD16—22no—
288SD16—22no—
289D8—22no—
290M16—21no—
291M16—21no—
292M16—24no—
293M16—21no—
294M16—21no—
295M16—21no—
296M16—21no—
297M16—42no—
298M16—42no—
299M16—42no—
300M16—162no—
301M16C2×M1612yes—
302M16C2×M1612yes—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—24no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—82no—
323C2×C8—82no—
324C2×C8—162no—
325C2×C8C2×C4⋊C411yescharacteristic
326C2×C8C2×C4⋊C411yescharacteristic
327C2×C8C2×C4⋊C411yescharacteristic
328C2×C8C2×C4⋊C411yescharacteristic
329C2×C8C4×D411yescharacteristic
330C2×C8C4×D411yescharacteristic
331C2×C8C4×D411yescharacteristic
332C2×C8C4×D411yescharacteristic
333C2×C8C2×M1612yes—
334C2×C8#13112yes—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4C2×C4⋊C411yescharacteristic
340C4⋊C4C2×C4⋊C411yescharacteristic
341C4⋊C4C2×C4⋊C411yescharacteristic
342C4⋊C4C2×C4⋊C411yescharacteristic
343C22⋊C4—21no—
344C22⋊C4—21no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—21no—
349C42—22no—
350C42—22no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—42no—
361C42C2×M1612yes—
362C42C22×D411yescharacteristic
363C42C22×D411yescharacteristic
364C42C22×Q811yescharacteristic
365C42C2×#4111yescharacteristic
366C23—21no—
367C23—21no—
368C23#52011yescharacteristic
369Q8—21no—
370Q8—21no—
371Q8—21no—
372Q8#42312yes—
373D4—22no—
374D4—22no—
375D4—21no—
376D4—21no—
377D4—21no—
378D4—41no—
379D4—41no—
380D4#42312yes—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4C22×C4⋊C411yescharacteristic
407C2×C4C2×C4×D411yescharacteristic
408C2×C4D4211yescharacteristic
409C2×C4#54711yescharacteristic
410C2×C4D4×Q811yescharacteristic
411C2×C4#55411yescharacteristic
412C2×C4C22×M1611yescharacteristic
413C2×C4C22×M1611yescharacteristic
414C2×C4C2×#13111yescharacteristic
415C2×C4#56811yescharacteristic
416C8—21no—
417C8—21no—
418C8—21no—
419C8—21no—
420C8—22no—
421C8—21no—
422C8—21no—
423C8—22no—
424C8—41no—
425C8—41no—
426C8—42no—
427C8—42no—
428C8—82no—
429C8—82no—
430C8—82no—
431C8#43512yes—
432C8#43512yes—
433C22—21no—
434C22—21no—
435C22—22no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22D4×C4⋊C411yescharacteristic
442C22#298911yescharacteristic
443C22#299011yescharacteristic
444C4—24no—
445C4—22no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—42no—
450C4—41no—
451C4—41no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4C2×#42311yescharacteristic
456C4#290611yescharacteristic
457C4C2×#43511yescharacteristic
458C4D4×M1611yescharacteristic
459C4#329211yescharacteristic
460C4#329311yescharacteristic
461C2—21no—
462C2—21no—
463C2—42no—
464C2—81no—
465C2#2264811yescharacteristic
466C2#2269211yescharacteristic
467C2#2269411yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22D4×C4⋊C4
prime radicalC2×#135C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17799C2semidirect
4. #17789C2semidirect
5. #9610C2semidirect
6. #8841C2semidirect
7. #8839C2semidirect
8. #8806C2semidirect
9. #8605C2semidirect
10. #8605C2semidirect
11. #8267C2semidirect
157. #131M16semidirect
220. #104M16semidirect

Maximal subgroups

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

subgroupindex
C4×#4432
#178812
#177992
#177892
#96102
#88412
#88392
#88062
#86052
#82672