Finite Group Databasebeta

← #498015

Subgroups — #498015

Order 512.

535 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33451C211yesmaximal, characteristic
2#22802C211yesmaximal, characteristic, complemented
3#22688C211yesmaximal, characteristic, complemented
4#13152C211yesmaximal, characteristic, complemented
5#13151C211yesmaximal, characteristic, complemented
6#13113C211yesmaximal, characteristic, complemented
7#13113C211yesmaximal, characteristic, complemented
8#13011C211yesmaximal, characteristic
9#13011C211yesmaximal, characteristic
10#12892C211yesmaximal, characteristic, complemented
11#12815C211yesmaximal, characteristic, complemented
12#12815C211yesmaximal, characteristic, complemented
13#12663C211yesmaximal, characteristic, complemented
14#12642C211yesmaximal, characteristic, complemented
15#12641C211yesmaximal, characteristic, complemented
16C2×#508C2211yescharacteristic
17C2×#443C2211yescharacteristic
18C2×#443C2211yescharacteristic
19C4×#135C2211yescharacteristic
20C4×#135C2211yescharacteristic
21#2925—21no—
22#2906—21no—
23#2177C411yescharacteristic, complemented
24#2177C411yescharacteristic, complemented
25#2176C411yescharacteristic, complemented
26#2176C411yescharacteristic, complemented
27#2175C411yescharacteristic, complemented
28#2175C411yescharacteristic, complemented
29#2175C411yescharacteristic, complemented
30#2175C411yescharacteristic, complemented
31#2158C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×#359C2211yescharacteristic
34C2×#359C2211yescharacteristic
35C2×#359C2211yescharacteristic
36C2×#359C2211yescharacteristic
37#2138C2211yescharacteristic
38#2138C2211yescharacteristic
39C2×#346C2211yescharacteristic
40C2×#346C2211yescharacteristic
41C2×C4×C16C2211yescharacteristic
42#1974C2211yescharacteristic
43#1974C2211yescharacteristic
44#1945C2211yescharacteristic
45#1945C2211yescharacteristic
46#1929—21no—
47#1929—21no—
48#1866—21no—
49#1866—21no—
50#1848C2211yescharacteristic
51#1848C2211yescharacteristic
52#1806—21no—
53#1806—21no—
54#1772C2211yescharacteristic
55#1772C2211yescharacteristic
56C8×C4⋊C4C2211yescharacteristic
57#1382C2211yescharacteristic
58#1382C2211yescharacteristic
59#1382C2211yescharacteristic
60#1382C2211yescharacteristic
61#1381C2211yescharacteristic
62#1381C2211yescharacteristic
63#1381C2211yescharacteristic
64#1381C2211yescharacteristic
65#1374C2211yescharacteristic
66#1374C2211yescharacteristic
67C2×#135C2311yescharacteristic
68C2×#135C2311yescharacteristic
69C2×#131—21no—
70C2×#131—21no—
71C4×#41—21no—
72C4×#41—21no—
73#508—22no—
74#508—22no—
75C2×Q32C2×C411yescharacteristic
76C2×SD32C2×C411yescharacteristic
77C2×SD32C2×C411yescharacteristic
78C2×D16C2×C411yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82C22×C16C2311yescharacteristic
83#443—22no—
84#443—22no—
85#443—41no—
86#443—41no—
87C4×Q16C2×C411yescharacteristic
88C4×Q16C2×C411yescharacteristic
89C4×Q16C2×C411yescharacteristic
90C4×Q16C2×C411yescharacteristic
91C4×SD16—21no—
92C4×SD16—21no—
93C4×SD16—21no—
94C4×SD16—21no—
95C4×D8C2×C411yescharacteristic
96C4×D8C2×C411yescharacteristic
97C4×D8C2×C411yescharacteristic
98C4×D8C2×C411yescharacteristic
99#433—41no—
100#433—41no—
101#432—41no—
102#431—41no—
103C2×#108—21no—
104C2×#108—21no—
105C2×#108C2311yescharacteristic
106C2×#108C2311yescharacteristic
107C2×#108C2311yescharacteristic
108C2×#108C2311yescharacteristic
109#427C2311yescharacteristic
110#427C2311yescharacteristic
111#424—21no—
112#424—21no—
113#424—41no—
114#424—41no—
115#423—21no—
116#423—21no—
117#423—41no—
118C2×C4⋊C8—41no—
119C2×C4⋊C8D411yescharacteristic
120C2×#104—21no—
121C2×#104—21no—
122C2×#104—21no—
123C2×#104—21no—
124#416—21no—
125#416—21no—
126#408—21no—
127#408—21no—
128#405—21no—
129#405—21no—
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134#367C2×C411yescharacteristic
135#367C2×C411yescharacteristic
136#366C2×C411yescharacteristic
137#366C2×C411yescharacteristic
138C4⋊C16—21no—
139C4⋊C16—21no—
140#359—22no—
141#359—22no—
142#359—41no—
143#359—41no—
144#358C2×C411yescharacteristic
145#358C2×C411yescharacteristic
146#358C2×C411yescharacteristic
147#358C2×C411yescharacteristic
148#357C2×C411yescharacteristic
149#357C2×C411yescharacteristic
150#357C2×C411yescharacteristic
151#357C2×C411yescharacteristic
152#346—41no—
153#346C2×C411yescharacteristic
154#346C2×C411yescharacteristic
155#346C2×C411yescharacteristic
156#346C2×C411yescharacteristic
157C4×C16—41no—
158#339—21no—
159#339—21no—
160#339—21no—
161#339—21no—
162#336D411yescharacteristic
163#336D411yescharacteristic
164C2×#41—21no—
165C2×#41—21no—
166#135—21no—
167#135—21no—
168#135—21no—
169#135—21no—
170#135—21no—
171#135—21no—
172#135#4112yes—
173#135#4112yes—
174C2×Q16C22×C411yescharacteristic
175C2×Q16C22×C411yescharacteristic
176C2×SD16—21no—
177C2×SD16—21no—
178C2×D8C22×C411yescharacteristic
179C2×D8C22×C411yescharacteristic
180#131—42no—
181#131—42no—
182#131—81no—
183#131—81no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—41no—
189C2×M16—41no—
190C22×C8—41no—
191C22×C8—41no—
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2411yescharacteristic
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C4×D4—41no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—41no—
208#117—21no—
209#117—21no—
210#117—41no—
211#117—41no—
212C2×C4⋊C4—21no—
213C2×C42—41no—
214C2×C42—41no—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218Q32—22no—
219SD32—22no—
220SD32—22no—
221D16—22no—
222C2×C16—21no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—21no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16C22×C411yescharacteristic
233C2×C16C22×C411yescharacteristic
234C2×C16C22×C411yescharacteristic
235C2×C16C22×C411yescharacteristic
236#108—21no—
237#108—21no—
238#108—21no—
239#108—21no—
240#108—21no—
241#108—21no—
242#108—21no—
243#108—21no—
244#108—81no—
245#108—81no—
246#107C22×C411yescharacteristic
247#107C22×C411yescharacteristic
248#107C22×C411yescharacteristic
249#107C22×C411yescharacteristic
250#106—21no—
251#106—21no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4⋊C8—41no—
257C4⋊C8—41no—
258C4⋊C8—41no—
259C4⋊C8—41no—
260C4⋊C8D811yescharacteristic
261C4⋊C8D811yescharacteristic
262C4⋊C8D811yescharacteristic
263C4⋊C8D811yescharacteristic
264#104—42no—
265#104—42no—
266#104—81no—
267#104—81no—
268#103—21no—
269#103—21no—
270#103—21no—
271#103—21no—
272#102—21no—
273#102—21no—
274#102—21no—
275#102—21no—
276C22⋊C8—41no—
277C22⋊C8—41no—
278C22⋊C8—41no—
279C22⋊C8—41no—
280#97—41no—
281#97—41no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—41no—
287C4×C8—41no—
288C4×C8C22×C411yescharacteristic
289C4×C8C22×C411yescharacteristic
290C4×C8#4111yescharacteristic
291C4×C8#4111yescharacteristic
292#95—21no—
293#41—22no—
294#41—22no—
295#41—41no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—41no—
300#41—41no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—41no—
306C2×D4—41no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4#13511yescharacteristic
316C22×C4#13511yescharacteristic
317C22×C4C22×D411yescharacteristic
318Q16—21no—
319Q16—21no—
320Q16—21no—
321Q16#11712yes—
322SD16—22no—
323SD16—22no—
324SD16—41no—
325SD16—41no—
326D8—21no—
327D8—21no—
328D8—21no—
329D8#11712yes—
330M16—41no—
331M16—41no—
332M16—41no—
333M16—41no—
334M16—41no—
335M16—41no—
336M16—41no—
337M16—41no—
338M16—82no—
339M16—82no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—82no—
351C2×C8—82no—
352C2×C8C4×D411yescharacteristic
353C2×C8C4×D411yescharacteristic
354C2×C8C2×D811yescharacteristic
355C2×C8C2×D811yescharacteristic
356C2×C8C2×D811yescharacteristic
357C2×C8C2×D811yescharacteristic
358C2×C8C23×C411yescharacteristic
359C2×C8C2×#4111yescharacteristic
360C2×C8C2×#4111yescharacteristic
361C2×C8C2×#4111yescharacteristic
362C2×C8H(5,𝔽2)11yescharacteristic
363C2×C82−1+411yescharacteristic
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—42no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42C4×D411yescharacteristic
390C42C4×D411yescharacteristic
391C42C2×D811yescharacteristic
392C42C2×D811yescharacteristic
393C16—21no—
394C16—21no—
395C16—21no—
396C16—21no—
397C16—41no—
398C16—41no—
399C23—41no—
400C23—41no—
401C23C2×#13511yescharacteristic
402Q8—22no—
403Q8—41no—
404Q8—41no—
405Q8—41no—
406D4—22no—
407D4—42no—
408D4—42no—
409D4—41no—
410D4—41no—
411D4—41no—
412D4—81no—
413D4—81no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—161no—
435C2×C4—161no—
436C2×C4C4×D811yescharacteristic
437C2×C4C4×D811yescharacteristic
438C2×C4C4×D811yescharacteristic
439C2×C4C4×D811yescharacteristic
440C2×C4C2×C4×D411yescharacteristic
441C2×C4#54611yescharacteristic
442C2×C4#54811yescharacteristic
443C2×C4C22×D811yescharacteristic
444C2×C4#57711yescharacteristic
445C2×C4#57711yescharacteristic
446C8—21no—
447C8—21no—
448C8—21no—
449C8—21no—
450C8—21no—
451C8—21no—
452C8—81no—
453C8—81no—
454C8—81no—
455C8—81no—
456C8C2×#11711yescharacteristic
457C8#52011yescharacteristic
458C8#55211yescharacteristic
459C8#55311yescharacteristic
460C22—21no—
461C22—21no—
462C22—42no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22C2×C4×D811yescharacteristic
469C22#330311yescharacteristic
470C22#336011yescharacteristic
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—42no—
476C4—42no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4#235211yescharacteristic
484C4#295211yescharacteristic
485C4#329411yescharacteristic
486C4#335811yescharacteristic
487C2—21no—
488C2—21no—
489C2—82no—
490C2—161no—
491C2#2276611yescharacteristic
492C2#3346311yescharacteristic
493C2#3346411yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×C4×D8
prime radicalC2×#508C22
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. #22802C2semidirect
3. #22688C2semidirect
4. #13152C2semidirect
5. #13151C2semidirect
6. #13113C2semidirect
7. #13113C2semidirect
10. #12892C2semidirect
11. #12815C2semidirect
12. #12815C2semidirect
13. #12663C2semidirect
14. #12642C2semidirect
15. #12641C2semidirect
23. #2177C4semidirect
24. #2177C4semidirect
25. #2176C4semidirect
26. #2176C4semidirect
27. #2175C4semidirect
28. #2175C4semidirect
29. #2175C4semidirect
30. #2175C4semidirect

Maximal subgroups

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

subgroupindex
#334512
#228022
#226882
#131522
#131512
#131132
#130112
#128922
#128152
#126632
#126422
#126412