Finite Group Databasebeta

← #6854474

Subgroups — #6854474

Order 512.

1502 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic
2Q8×C4⋊Q8C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic
4#31524C218yesmaximal
5#30304C212yesmaximal
6#29193C212yesmaximal
7#29141C211yesmaximal, characteristic
8#29141C212yesmaximal
9#28511C211yesmaximal, characteristic
10#24817C214yesmaximal
11#24333C214yesmaximal
12#22270C212yesmaximal
13C4×#494C211yesmaximal, characteristic
14#21339C211yesmaximal, characteristic
15C2×Q82C2212yes—
16C2×#494C2211yescharacteristic
17C2×#477C2214yes—
18C2×C4×Q16C2214yes—
19#2868C2211yescharacteristic
20#2868C2211yescharacteristic
21#2868C2212yes—
22#2859C2212yes—
23#2582C2212yes—
24#2567—21no—
25#2567C2214yes—
26#2567C2212yes—
27#2552—21no—
28#2551—21no—
29#2551—22no—
30#2551—22no—
31#2551—21no—
32#2550—22no—
33#2507—22no—
34#2451C2214yes—
35#2369—21no—
36#2355—21no—
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4C2212yes—
40C4×C4⋊Q8C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C42×Q8C2211yescharacteristic
45#2095C2214yes—
46#2083C2218yes—
47#2068C2218yes—
48#2057C2218yes—
49#2024C2218yes—
50#2024C2214yes—
51#1990C2214yes—
52#1990C2214yes—
53#1984C2212yes—
54#1969C2214yes—
55#1956C2211yescharacteristic
56#1956C2212yes—
57#1956C2212yes—
58#1956C2211yescharacteristic
59#1939C2212yes—
60#1916C2214yes—
61#1878C2218yes—
62#1878C2218yes—
63#1863C2218yes—
64#1845C2212yes—
65#1845C2211yescharacteristic
66#1834C2212yes—
67#1834C2212yes—
68#1834C2212yes—
69#1803C2214yes—
70#1803C2218yes—
71C4×C4⋊C8C2212yes—
72C4×C4⋊C8C2211yescharacteristic
73#1744C2212yes—
74C42×C8C2211yescharacteristic
75C22×Q16C2314yes—
76Q82—48no—
77C2×C4⋊Q8—22no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—22no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2314yes—
89C2×C4⋊Q8C2312yes—
90C2×#125—22no—
91C2×C4×Q8—22no—
92C2×C4×Q8D411yescharacteristic
93C2×C4×Q8D411yescharacteristic
94C2×C4×Q8C2312yes—
95C2×C4×Q8C2312yes—
96C2×C4×Q8C2314yes—
97C2×C4×Q8C2314yes—
98C2×C4×Q8C2311yescharacteristic
99#494Q818yescomplemented
100#477—48no—
101C4×Q16—48no—
102C2×#107C2318yes—
103C2×#107C2314yes—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2314yes—
106C2×#103C2318yes—
107C2×#103C2314yes—
108C2×#103C2318yes—
109C2×#103C2314yes—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2312yes—
112C2×C4×C8C2312yes—
113C2×C4×C8C2312yes—
114C2×C4×C8C2311yescharacteristic
115#398—24no—
116#398—24no—
117#395—24no—
118#395—24no—
119#395—24no—
120#395—24no—
121#391—22no—
122#391—24no—
123#391C2314yes—
124#391C2312yes—
125#391C2314yes—
126#391C2312yes—
127#389—22no—
128#389—24no—
129#389—22no—
130#389—24no—
131#389—22no—
132#389—22no—
133#389—21no—
134#389—21no—
135#389—22no—
136#389—22no—
137#389—21no—
138#389—22no—
139#389—21no—
140#389—22no—
141#389—22no—
142#389—24no—
143#389C2314yes—
144#389C2314yes—
145#387—24no—
146#384—22no—
147#384—22no—
148#384—22no—
149#384—21no—
150#384—21no—
151#384—21no—
152#384—21no—
153#384C2312yes—
154#384C2312yes—
155#384C2311yescharacteristic
156#384C2311yescharacteristic
157#384C2314yes—
158#384C2312yes—
159#384C2312yes—
160#382—22no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4D411yescharacteristic
171C4×C4⋊C4D412yes—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4D411yescharacteristic
174C4×C4⋊C4C2311yescharacteristic
175C4×C4⋊C4C2312yes—
176C43C2312yes—
177C43C2311yescharacteristic
178#340C2318yes—
179#340C2318yes—
180#340C2318yes—
181#336C2314yes—
182#336C2314yes—
183#336C2312yes—
184#336C2314yes—
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8C2×D411yescharacteristic
189C22×Q8C2412yes—
190C22×Q8C2412yes—
191C2×Q16—44no—
192C2×Q16—44no—
193C2×Q16C2×Q8116yes—
194C22×C8C2414yes—
195C22×C8C2412yes—
196C22×C8C2412yes—
197C4⋊Q8—28no—
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200C4⋊Q8—48no—
201C4⋊Q8—44no—
202C4⋊Q8—416no—
203C4⋊Q8—48no—
204C4⋊Q8—44no—
205C4⋊Q8—84no—
206C4⋊Q8—82no—
207C4⋊Q8—82no—
208C4⋊Q8—82no—
209C4⋊Q8—84no—
210C4⋊Q8C2×Q814yes—
211C4⋊Q8C2×Q814yes—
212#125—84no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—28no—
216C4×Q8—28no—
217C4×Q8—48no—
218C4×Q8—48no—
219C4×Q8—48no—
220C4×Q8Q1618yescomplemented
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2414yes—
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2412yes—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D412yes—
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D412yes—
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2412yes—
271C2×C42C2412yes—
272C2×C42C2411yescharacteristic
273C2×C42C2411yescharacteristic
274C2×C42C2411yescharacteristic
275#107—44no—
276#107—48no—
277C4⋊C8—24no—
278C4⋊C8—44no—
279#103—48no—
280#103—44no—
281#103—44no—
282#103—48no—
283C4×C8—24no—
284C4×C8—22no—
285C4×C8—24no—
286C4×C8—42no—
287C4×C8C2×Q814yes—
288#95—24no—
289#95—22no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—22no—
297#95—24no—
298#95C2×D412yes—
299#95C2×D412yes—
300#95C2×D412yes—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—48no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—82no—
313C2×Q8—82no—
314C2×Q8—82no—
315C2×Q8—82no—
316C2×Q8—84no—
317C2×Q8—84no—
318C2×Q8C2×Q1614yes—
319C2×Q8C22×Q814yes—
320C2×Q8C22×Q814yes—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4C22×D412yes—
328C22×C4C22×D412yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D412yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D412yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C2511yescharacteristic
338Q16—416no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—24no—
347C2×C8C22×Q814yes—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—28no—
359C4⋊C4—28no—
360C4⋊C4—48no—
361C4⋊C4—48no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—44no—
371C4⋊C4—42no—
372C4⋊C4—44no—
373C4⋊C4—42no—
374C4⋊C4—84no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4—84no—
381C4⋊C4—82no—
382C4⋊C4—84no—
383C4⋊C4C2×Q1618yes—
384C4⋊C4C2×Q1614yes—
385C42—24no—
386C42—24no—
387C42—22no—
388C42—22no—
389C42—24no—
390C42—24no—
391C42—22no—
392C42—24no—
393C42—48no—
394C42—44no—
395C42—44no—
396C42—84no—
397C42C2×Q1618yes—
398C42C2×Q1614yes—
399C42#13714yes—
400C42C22×Q812yes—
401C42H(5,𝔽2)14yes—
402C422−1+412yes—
403C23C23×D411yescharacteristic
404Q8—28no—
405Q8—28no—
406Q8—28no—
407Q8—44no—
408Q8—44no—
409Q8—164no—
410Q8—164no—
411Q8—168no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4#53512yes—
447C2×C4#53512yes—
448C2×C4D4×Q812yes—
449C2×C4D4×Q812yes—
450C2×C4C22×Q1614yes—
451C2×C4C22×Q1612yes—
452C2×C4C22×Q1614yes—
453C2×C4C22×Q1612yes—
454C2×C4C22×Q1612yes—
455C2×C4C2×#13712yes—
456C2×C4C2×#13712yes—
457C2×C4C2×#13712yes—
458C2×C4#57612yes—
459C2×C4#57712yes—
460C2×C4C23×Q811yescharacteristic
461C2×C4C2×H(5,𝔽2)12yes—
462C2×C4C2×2−1+411yescharacteristic
463C2×C4#58512yes—
464C8—24no—
465C8—42no—
466C8—44no—
467C22C2×#53511yescharacteristic
468C22C2×D4×Q811yescharacteristic
469C22#350011yescharacteristic
470C22C23×Q1611yescharacteristic
471C22C22×#13711yescharacteristic
472C22C2×#57611yescharacteristic
473C22C2×#57711yescharacteristic
474C4—24no—
475C4—22no—
476C4—22no—
477C4—24no—
478C4—22no—
479C4—42no—
480C4—42no—
481C4—44no—
482C4—44no—
483C4—84no—
484C4—82no—
485C4—84no—
486C4—82no—
487C4#320914yes—
488C4Q8×Q1614yes—
489C4#338714yes—
490C4#350112yes—
491C4#354712yes—
492C2#3961811yescharacteristic
493C2C2×#320911yescharacteristic
494C2#6082211yescharacteristic
495C2C2×Q8×Q1611yescharacteristic
496C2C2×#338711yescharacteristic
497C2#6253011yescharacteristic
498C2#6253711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
99. #494Q8semidirect
220. C4×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#365632
Q8×C4⋊Q82
#332892
#315242
#303042
#291932
#291412
#285112
#248172
#243332
#222702
C4×#4942
#213392