Finite Group Databasebeta

← #498273

Subgroups — #498273

Order 512.

502 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33626C211yesmaximal, characteristic
2#23170C211yesmaximal, characteristic, complemented
3#22688C211yesmaximal, characteristic
4#13286C211yesmaximal, characteristic, complemented
5#13179C211yesmaximal, characteristic, complemented
6#13170C211yesmaximal, characteristic, complemented
7#13169C211yesmaximal, characteristic, complemented
8#13113C211yesmaximal, characteristic, complemented
9#13113C211yesmaximal, characteristic, complemented
10#12939C211yesmaximal, characteristic
11#12939C211yesmaximal, characteristic
12#12663C211yesmaximal, characteristic, complemented
13#12642C211yesmaximal, characteristic
14#12641C211yesmaximal, characteristic
15#12640C211yesmaximal, characteristic
16#3160C2211yescharacteristic
17#2962—21no—
18C2×#443C2211yescharacteristic
19C2×#443C2211yescharacteristic
20C4×#135C2211yescharacteristic
21#2906—21no—
22#2229C411yescharacteristic, complemented
23#2229C411yescharacteristic, complemented
24#2228C411yescharacteristic, complemented
25#2228C411yescharacteristic, complemented
26#2227C411yescharacteristic, complemented
27#2226C411yescharacteristic, complemented
28#2225C411yescharacteristic, complemented
29#2224C411yescharacteristic, complemented
30C2×#368C2211yescharacteristic
31C2×#368C2211yescharacteristic
32#2158C2211yescharacteristic
33#2158C2211yescharacteristic
34C2×C4⋊C16C2211yescharacteristic
35C2×#359C2211yescharacteristic
36C2×#359C2211yescharacteristic
37#2138C2211yescharacteristic
38#2138C2211yescharacteristic
39C2×#346C2211yescharacteristic
40C2×#346C2211yescharacteristic
41C2×C4×C16C2211yescharacteristic
42#1986—21no—
43#1945C2211yescharacteristic
44#1945C2211yescharacteristic
45#1945C2211yescharacteristic
46#1945C2211yescharacteristic
47#1922—21no—
48#1922—21no—
49#1866—21no—
50#1866—21no—
51#1848C2211yescharacteristic
52#1806—21no—
53C4×#108C2211yescharacteristic
54#1772C2211yescharacteristic
55#1772C2211yescharacteristic
56C8×C4⋊C4C2211yescharacteristic
57#1382C2211yescharacteristic
58#1382C2211yescharacteristic
59#1382C2211yescharacteristic
60#1382C2211yescharacteristic
61#1382C2211yescharacteristic
62#1382C2211yescharacteristic
63#1381C2211yescharacteristic
64#1381C2211yescharacteristic
65#1374C2211yescharacteristic
66#1374C2211yescharacteristic
67C2×#135C2311yescharacteristic
68C2×#131—21no—
69C2×#131—21no—
70#533—21no—
71C4×#41—21no—
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C22×C16C2311yescharacteristic
75C22×C16C2311yescharacteristic
76#500C2×C411yescharacteristic
77#500C2×C411yescharacteristic
78#499—21no—
79#499C2×C411yescharacteristic
80#499C2×C411yescharacteristic
81#498—21no—
82#446—41no—
83#446—41no—
84#446—41no—
85C8×Q8—41no—
86#443—22no—
87#443—22no—
88#443—41no—
89#443—41no—
90C4×Q16C2×C411yescharacteristic
91C4×Q16C2×C411yescharacteristic
92C4×SD16—21no—
93C4×SD16—21no—
94C4×D8C2×C411yescharacteristic
95C4×D8C2×C411yescharacteristic
96C2×#108—21no—
97C2×#108C2311yescharacteristic
98C2×#108C2311yescharacteristic
99C2×#108C2311yescharacteristic
100C2×#108C2311yescharacteristic
101#427—21no—
102#427C2311yescharacteristic
103#427C2311yescharacteristic
104#427C2311yescharacteristic
105#424—21no—
106#424—21no—
107#424—21no—
108#424—21no—
109#424—41no—
110#424—41no—
111#423—21no—
112#423—41no—
113C2×C4⋊C8—41no—
114C2×C4⋊C8D411yescharacteristic
115C2×#104—21no—
116C2×#104—21no—
117#416—21no—
118#405—21no—
119#405—21no—
120C4×M16—21no—
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125#368—41no—
126#368—41no—
127#367C2×C411yescharacteristic
128#367C2×C411yescharacteristic
129#367C2×C411yescharacteristic
130#367C2×C411yescharacteristic
131#366C2×C411yescharacteristic
132#366C2×C411yescharacteristic
133#366C2×C411yescharacteristic
134#366C2×C411yescharacteristic
135C4⋊C16C2×C411yescharacteristic
136C4⋊C16C2×C411yescharacteristic
137C4⋊C16C2×C411yescharacteristic
138C4⋊C16C2×C411yescharacteristic
139#359—22no—
140#359Q814yescomplemented
141#358C2×C411yescharacteristic
142#358C2×C411yescharacteristic
143#358C2×C411yescharacteristic
144#358C2×C411yescharacteristic
145#357C2×C411yescharacteristic
146#357C2×C411yescharacteristic
147#357C2×C411yescharacteristic
148#357C2×C411yescharacteristic
149#346—21no—
150#346—21no—
151#346—41no—
152C4×C16—41no—
153#341—21no—
154#339—21no—
155#339—21no—
156#339—21no—
157#339—21no—
158#339—21no—
159#339—21no—
160#336D411yescharacteristic
161#336D411yescharacteristic
162C2×#41—21no—
163#135—21no—
164#135—21no—
165#135C2×Q812yes—
166#135#4112yes—
167C2×Q16C22×C411yescharacteristic
168C2×SD16—21no—
169C2×D8C22×C411yescharacteristic
170#131—42no—
171#131—42no—
172#131—81no—
173#131—81no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—41no—
179C2×M16—41no—
180C22×C8—41no—
181C22×C8—41no—
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2411yescharacteristic
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188#125—21no—
189#125—21no—
190C22⋊Q8—41no—
191C22⋊Q8—41no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—41no—
199C4×D4—41no—
200#117—21no—
201#117—21no—
202#117—21no—
203#117—41no—
204C2×C4⋊C4—21no—
205C2×C42—41no—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16C22×C411yescharacteristic
219C2×C16C22×C411yescharacteristic
220C2×C16C22×C411yescharacteristic
221C2×C16C22×C411yescharacteristic
222C2×C16C2×Q812yes—
223#108—21no—
224#108—21no—
225#108—21no—
226#108—21no—
227#108—21no—
228#108—21no—
229#108—21no—
230#108—81no—
231#108C2×Q812yes—
232#107—21no—
233#107—21no—
234#107C22×C411yescharacteristic
235#107C22×C411yescharacteristic
236#107C22×C411yescharacteristic
237#107C22×C411yescharacteristic
238#107C22×C411yescharacteristic
239#107C22×C411yescharacteristic
240#106—21no—
241#106—21no—
242#106—21no—
243#106—21no—
244#106—21no—
245C4⋊C8—41no—
246C4⋊C8—41no—
247C4⋊C8—41no—
248C4⋊C8—41no—
249C4⋊C8—41no—
250C4⋊C8—41no—
251C4⋊C8—41no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254C4⋊C8—41no—
255C4⋊C8D811yescharacteristic
256C4⋊C8D811yescharacteristic
257C4⋊C8D811yescharacteristic
258C4⋊C8D811yescharacteristic
259#104—42no—
260#104—42no—
261#103—21no—
262#103—21no—
263#102—21no—
264#102—21no—
265#97—41no—
266#97—41no—
267#97—41no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—41no—
273C4×C8—41no—
274C4×C8—41no—
275C4×C8C22×C411yescharacteristic
276C4×C8C22×C411yescharacteristic
277C4×C8#4111yescharacteristic
278C4×C8#4111yescharacteristic
279#95—21no—
280#41—22no—
281#41—22no—
282#41—41no—
283#41—41no—
284C2×Q8—21no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×D4—21no—
288C2×D4—41no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4#13511yescharacteristic
295C22×C4#13511yescharacteristic
296C22×C4C22×D411yescharacteristic
297Q16—21no—
298Q16C4×Q812yes—
299SD16—22no—
300SD16—22no—
301D8—21no—
302D8C4×Q812yes—
303M16—22no—
304M16—41no—
305M16—41no—
306M16—41no—
307M16—41no—
308M16—41no—
309M16—41no—
310M16—41no—
311M16—82no—
312M16—82no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—82no—
324C2×C8—82no—
325C2×C8C4×D411yescharacteristic
326C2×C8C4×D411yescharacteristic
327C2×C8C2×D811yescharacteristic
328C2×C8C2×D811yescharacteristic
329C2×C8C2×D811yescharacteristic
330C2×C8C2×D811yescharacteristic
331C2×C8C23×C411yescharacteristic
332C2×C8C22×Q811yescharacteristic
333C2×C8C2×#4111yescharacteristic
334C2×C8C2×#4111yescharacteristic
335C2×C8H(5,𝔽2)11yescharacteristic
336C2×C82−1+411yescharacteristic
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—42no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42C4×D411yescharacteristic
367C42C4×D411yescharacteristic
368C42C2×D811yescharacteristic
369C42C2×D811yescharacteristic
370C16—21no—
371C16—21no—
372C16—21no—
373C16—21no—
374C16—22no—
375C16—41no—
376C23—41no—
377C23C2×#13511yescharacteristic
378Q8—22no—
379Q8—41no—
380Q8—82no—
381Q8—161no—
382D4—22no—
383D4—42no—
384D4—42no—
385D4—41no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
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—82no—
404C2×C4—161no—
405C2×C4C4×D811yescharacteristic
406C2×C4C4×D811yescharacteristic
407C2×C4C4×D811yescharacteristic
408C2×C4C4×D811yescharacteristic
409C2×C4C2×C4×D411yescharacteristic
410C2×C4#54611yescharacteristic
411C2×C4D4×Q811yescharacteristic
412C2×C4C22×D811yescharacteristic
413C2×C4#57711yescharacteristic
414C2×C4#57811yescharacteristic
415C8—21no—
416C8—21no—
417C8—21no—
418C8—21no—
419C8—21no—
420C8—21no—
421C8—42no—
422C8—81no—
423C8—81no—
424C8—81no—
425C8C2×C4×Q811yescharacteristic
426C8#52011yescharacteristic
427C8#55211yescharacteristic
428C8#55411yescharacteristic
429C22—21no—
430C22—21no—
431C22—42no—
432C22—81no—
433C22—81no—
434C22C2×C4×D811yescharacteristic
435C22#330211yescharacteristic
436C22#338111yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—42no—
443C4—81no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—81no—
449C4—81no—
450C4#236011yescharacteristic
451C4#294711yescharacteristic
452C4#329411yescharacteristic
453C4Q8×D811yescharacteristic
454C2—21no—
455C2—21no—
456C2—82no—
457C2#2296211yescharacteristic
458C2#3346311yescharacteristic
459C2#3346511yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×C4×D8
prime radicalC2×#135C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23170C2semidirect
4. #13286C2semidirect
5. #13179C2semidirect
6. #13170C2semidirect
7. #13169C2semidirect
8. #13113C2semidirect
9. #13113C2semidirect
12. #12663C2semidirect
22. #2229C4semidirect
23. #2229C4semidirect
24. #2228C4semidirect
25. #2228C4semidirect
26. #2227C4semidirect
27. #2226C4semidirect
28. #2225C4semidirect
29. #2224C4semidirect
140. #359Q8semidirect

Maximal subgroups

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

subgroupindex
#336262
#231702
#226882
#132862
#131792
#131702
#131692
#131132
#129392
#126632
#126422
#126412
#126402