Finite Group Databasebeta

← #232192

Subgroups — #232192

Order 512.

500 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23807C211yesmaximal, characteristic
2#9094C211yesmaximal, characteristic
3#9092C211yesmaximal, characteristic, complemented
4#9092C211yesmaximal, characteristic, complemented
5#9092C211yesmaximal, characteristic, complemented
6#9091C211yesmaximal, characteristic
7#9091C211yesmaximal, characteristic
8#9090C211yesmaximal, characteristic
9#9088C211yesmaximal, characteristic, complemented
10#8964C211yesmaximal, characteristic
11#8962C211yesmaximal, characteristic, complemented
12#8961C211yesmaximal, characteristic
13#8958C211yesmaximal, characteristic, complemented
14#8802C211yesmaximal, characteristic, complemented
15#8802C211yesmaximal, characteristic, complemented
16#2983C2211yescharacteristic
17#2983C2211yescharacteristic
18#2982C2211yescharacteristic
19#2975C2211yescharacteristic
20C2×#441C2211yescharacteristic
21C2×#441C2211yescharacteristic
22C2×#440C2211yescharacteristic
23C2×#440C2211yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×#124C2211yescharacteristic
26#2097C411yescharacteristic
27#2097C411yescharacteristic
28#2095C411yescharacteristic
29#2094C411yescharacteristic
30#2016C411yescharacteristic
31#2015C411yescharacteristic
32#2015C411yescharacteristic
33#2013C411yescharacteristic
34#1745C2211yescharacteristic
35#1479C2211yescharacteristic
36#1479C2211yescharacteristic
37#1479C2211yescharacteristic
38#1479C2211yescharacteristic
39#1479C2211yescharacteristic
40#1479C2211yescharacteristic
41#1478C2211yescharacteristic
42#1478C2211yescharacteristic
43#1461C2211yescharacteristic
44#1461C2211yescharacteristic
45#1461C2211yescharacteristic
46#1460C2211yescharacteristic
47#1281C2211yescharacteristic
48#1281C2211yescharacteristic
49#1281C2211yescharacteristic
50#1281C2211yescharacteristic
51#1276C2211yescharacteristic
52#1276C2211yescharacteristic
53#1276C2211yescharacteristic
54#1276C2211yescharacteristic
55#1275C2211yescharacteristic
56#1275C2211yescharacteristic
57#1275C2211yescharacteristic
58#1275C2211yescharacteristic
59C22×Q16C2×C411yescharacteristic
60C22×Q16C2×C411yescharacteristic
61C22×SD16C2×C411yescharacteristic
62C22×SD16C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×#124C2×C411yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×Q8C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69#446—41no—
70#446—41no—
71#446—41no—
72#446—41no—
73C8×Q8—41no—
74C8×Q8—41no—
75#441—41no—
76#441—41no—
77#441—41no—
78#441—41no—
79#440—41no—
80#440—41no—
81#440—41no—
82#440—41no—
83#436—41no—
84#436—41no—
85#432—41no—
86#432—41no—
87#431—41no—
88#431C2311yescharacteristic
89#431C2311yescharacteristic
90#431C2311yescharacteristic
91#431C2311yescharacteristic
92#431C2311yescharacteristic
93#431C2311yescharacteristic
94#431C2311yescharacteristic
95#431C2311yescharacteristic
96C2×#107C2×C411yescharacteristic
97C2×#107C2×C411yescharacteristic
98C2×#106C2×C411yescharacteristic
99C2×#106C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C411yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108#407—41no—
109C2×#97C2311yescharacteristic
110C2×#97C2311yescharacteristic
111#391C2×C411yescharacteristic
112#391C2×C411yescharacteristic
113#391C2×C411yescharacteristic
114#390C2×C411yescharacteristic
115#389—41no—
116#388—41no—
117#384C2×C411yescharacteristic
118#382C2×C411yescharacteristic
119C4×C4⋊C4—41no—
120C4×C22⋊C4—41no—
121C43C2311yescharacteristic
122#336C2×C411yescharacteristic
123#336C2×C411yescharacteristic
124#336C2×C411yescharacteristic
125#336C2×C411yescharacteristic
126Q8⋊C8—41no—
127Q8⋊C8—41no—
128Q8⋊C8—41no—
129Q8⋊C8—41no—
130Q8⋊C8—41no—
131Q8⋊C8—41no—
132D4⋊C8—41no—
133D4⋊C8—41no—
134#324—41no—
135#324—41no—
136#324—41no—
137#323—41no—
138C22×Q8C22×C411yescharacteristic
139C22×Q8C22×C411yescharacteristic
140C22×Q8C22×C411yescharacteristic
141C22×D4C22×C411yescharacteristic
142C23×C4—21no—
143C2×Q16—22no—
144C2×Q16—41no—
145C2×Q16—41no—
146C2×Q16—41no—
147C2×Q16—41no—
148C2×Q16—41no—
149C2×SD16—22no—
150C2×SD16—41no—
151C2×SD16—41no—
152C2×SD16—41no—
153C2×SD16—41no—
154C2×SD16—41no—
155C2×M16—42no—
156C22×C8C22×C411yescharacteristic
157C22×C8C22×C411yescharacteristic
158C22×C8C22×C411yescharacteristic
159C22×C8C22×C411yescharacteristic
160C4⋊Q8—22no—
161C4⋊Q8—41no—
162#124—41no—
163#124—41no—
164C4×Q8—21no—
165C4×Q8—21no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C2411yescharacteristic
200#107—41no—
201#107—41no—
202#106—41no—
203#106—41no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—42no—
213C4⋊C8—42no—
214C4⋊C8—42no—
215#103—41no—
216#103—41no—
217#103—41no—
218#103—41no—
219#103—41no—
220#103—41no—
221#102—41no—
222#102—41no—
223C22⋊C8—21no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231C22⋊C8—42no—
232#97—21no—
233#97—21no—
234#97—21no—
235#97—21no—
236#97—42no—
237#97—42no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—42no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251C24—21no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8#13112yes—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—82no—
288C22×C4C23×C411yescharacteristic
289C22×C4H(5,𝔽2)11yescharacteristic
290C22×C4H(5,𝔽2)11yescharacteristic
291Q16—44no—
292Q16—44no—
293SD16—44no—
294SD16—44no—
295M16—162no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—44no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—82no—
332C22⋊C4—44no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—82no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—42no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—41no—
346C42—41no—
347C42—81no—
348C42H(5,𝔽2)11yescharacteristic
349C42H(5,𝔽2)11yescharacteristic
350C42H(5,𝔽2)11yescharacteristic
351C42H(5,𝔽2)11yescharacteristic
352C23—41no—
353C23—41no—
354C23—41no—
355C23—81no—
356C23—81no—
357C23#51811yescharacteristic
358Q8—22no—
359Q8—22no—
360Q8—41no—
361Q8—41no—
362Q8—41no—
363Q8—41no—
364Q8—41no—
365Q8—41no—
366Q8—41no—
367Q8—41no—
368Q8—41no—
369Q8—41no—
370D4—41no—
371D4—41no—
372D4—41no—
373D4—41no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—161no—
401C2×C4#51811yescharacteristic
402C2×C4#51811yescharacteristic
403C2×C4#56011yescharacteristic
404C2×C4#56011yescharacteristic
405C2×C4#56011yescharacteristic
406C2×C4#56011yescharacteristic
407C2×C4C2×#13111yescharacteristic
408C2×C4#56811yescharacteristic
409C2×C4#56811yescharacteristic
410C2×C4#56811yescharacteristic
411C8—41no—
412C8—41no—
413C8—41no—
414C8—41no—
415C8—81no—
416C8—81no—
417C8—81no—
418C8—81no—
419C8—81no—
420C8—81no—
421C8—81no—
422C8—81no—
423C22—21no—
424C22—21no—
425C22—82no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—161no—
430C22#238011yescharacteristic
431C22#297611yescharacteristic
432C22#297811yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—42no—
438C4—42no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4#297611yescharacteristic
447C4#297611yescharacteristic
448C4#297811yescharacteristic
449C4#297811yescharacteristic
450C2—21no—
451C2—21no—
452C2—161no—
453C2#2335911yescharacteristic
454C2#2336011yescharacteristic
455C2#2336211yescharacteristic
subgroupG/H
center, socleC22#2380
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9092C2semidirect
4. #9092C2semidirect
5. #9092C2semidirect
9. #9088C2semidirect
11. #8962C2semidirect
13. #8958C2semidirect
14. #8802C2semidirect
15. #8802C2semidirect

Maximal subgroups

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

subgroupindex
#238072
#90942
#90922
#90912
#90902
#90882
#89642
#89622
#89612
#89582
#88022