Finite Group Databasebeta

← #374798

Subgroups — #374798

Order 512.

922 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1997C211yesmaximal, characteristic
2#21324C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic
4#21170C211yesmaximal, characteristic
5#20441C211yesmaximal, characteristic
6#11303C212yesmaximal
7#11290C212yesmaximal
8#10896C212yesmaximal
9#10853C211yesmaximal, characteristic
10#10827C212yesmaximal
11C4×#343C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#428C2212yes—
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C2×#405C2212yes—
17C2×C4×M16C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19#2285C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1997C418yescomplemented
22#1993—22no—
23#1990—21no—
24#1990—21no—
25#1988—21no—
26#1988—21no—
27#1988—22no—
28#1949—22no—
29#1948—21no—
30#1947—21no—
31#1771—22no—
32#1770—21no—
33C8×C4⋊C4—21no—
34C2×#343C2211yescharacteristic
35C2×#343C2211yescharacteristic
36C2×#343C2211yescharacteristic
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#343C2211yescharacteristic
40C2×#341C2212yes—
41C2×#341C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44#1736C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47#1729C2212yes—
48C23×Q8—21no—
49C22×M16C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C23×C8C2312yes—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8C2×C414yes—
63C2×#117C2312yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70#428—24no—
71#428—42no—
72C2×#107—22no—
73C2×#106—22no—
74#423—22no—
75#423C2×C414yes—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8D411yescharacteristic
81C2×C4⋊C8D411yescharacteristic
82C2×C4⋊C8D412yes—
83#405—24no—
84#405—42no—
85C4×M16—22no—
86C4×M16—22no—
87C4×M16—22no—
88C4×M16C2×C414yes—
89C2×#97—21no—
90C2×#97—22no—
91C2×#97—21no—
92C2×C4×C8—21no—
93C2×C4×C8—22no—
94C2×C4×C8—21no—
95#384D411yescharacteristic
96#384D411yescharacteristic
97#382D412yes—
98#380—21no—
99#380—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C2×#95—21no—
107#343—22no—
108#343—22no—
109#343—22no—
110#343C2×C418yes—
111#343C2×C418yes—
112#343Q814yescomplemented
113#341—24no—
114#341—42no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#336—22no—
126#336—22no—
127C22×Q8—24no—
128C22×Q8—41no—
129C22×Q8—41no—
130C22×Q8—42no—
131C22×Q8—82no—
132C22×Q8—82no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4#4111yescharacteristic
137C23×C4#4111yescharacteristic
138C23×C4C2411yescharacteristic
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—22no—
144C2×M16—21no—
145C2×M16—21no—
146C2×M16—22no—
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—21no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—22no—
156C2×M16—21no—
157C2×M16—22no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—22no—
162C2×M16C22×C412yes—
163C2×M16C22×C414yes—
164C2×M16C22×C412yes—
165C2×M16C2×Q812yes—
166C2×M16C2×Q812yes—
167C2×M16#4112yes—
168C2×M16#4112yes—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8C2×D412yes—
181C22×C8C2×D412yes—
182C4⋊Q8—24no—
183C4⋊Q8—24no—
184C4⋊Q8—24no—
185C4⋊Q8—24no—
186C4⋊Q8—42no—
187C4⋊Q8—42no—
188C4⋊Q8—42no—
189C4⋊Q8—42no—
190#117—22no—
191#117—22no—
192#117—22no—
193#117—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C22×C414yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×Q812yes—
221C2×C4⋊C4#4112yes—
222C2×C22⋊C4—22no—
223C2×C42—22no—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42—82no—
227C2×C42C22×C412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4112yes—
233C2×C42#4112yes—
234C2×C42#4112yes—
235#107—44no—
236#106—44no—
237C4⋊C8—24no—
238C4⋊C8—24no—
239C4⋊C8—22no—
240C4⋊C8—22no—
241C4⋊C8—24no—
242#97—24no—
243#97—44no—
244#97—42no—
245C4×C8—24no—
246C4×C8—44no—
247C4×C8—42no—
248#95—22no—
249#95—22no—
250#95—41no—
251#95—41no—
252C24C2×#4111yescharacteristic
253C2×Q8—44no—
254C2×Q8—44no—
255C2×Q8—88no—
256C2×Q8—82no—
257C2×Q8—82no—
258C2×Q8—88no—
259C2×Q8—82no—
260C2×Q8—82no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4#11712yes—
283C22×C4#11712yes—
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4#12111yescharacteristic
288C22×C4#12612yes—
289C22×C4#12612yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×Q811yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4H(5,𝔽2)11yescharacteristic
301C22×C42−1+411yescharacteristic
302C22×C42−1+411yescharacteristic
303C22×C42−1+411yescharacteristic
304M16—22no—
305M16—22no—
306M16—22no—
307M16—22no—
308M16—24no—
309M16—22no—
310M16—24no—
311M16—22no—
312M16—22no—
313M16—22no—
314M16—22no—
315M16—41no—
316M16—41no—
317M16—41no—
318M16—41no—
319M16—41no—
320M16—42no—
321M16—41no—
322M16—42no—
323M16C4×Q814yes—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—24no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—81no—
370C4⋊C4—81no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C42—42no—
374C42—42no—
375C42—82no—
376C42—82no—
377C42C4×D412yes—
378C42C4×D412yes—
379C42C4×D412yes—
380C42C4×D412yes—
381C42#12312yes—
382C42#12312yes—
383C42#12412yes—
384C42#12412yes—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23C2×#11711yescharacteristic
390C23C2×#12111yescharacteristic
391C23C2×#12611yescharacteristic
392C23#52911yescharacteristic
393C23#53311yescharacteristic
394C23#54111yescharacteristic
395C23#54211yescharacteristic
396Q8—88no—
397Q8—88no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—84no—
420C2×C4—82no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—84no—
429C2×C4#38812yes—
430C2×C4#38812yes—
431C2×C4#38812yes—
432C2×C4#38812yes—
433C2×C4#47112yes—
434C2×C4#47212yes—
435C2×C4C2×C4×D411yescharacteristic
436C2×C4C2×C4×D411yescharacteristic
437C2×C4C2×C4×Q811yescharacteristic
438C2×C4C4×#4111yescharacteristic
439C2×C4#51911yescharacteristic
440C2×C4#52011yescharacteristic
441C2×C4C2×#12311yescharacteristic
442C2×C4C2×#12411yescharacteristic
443C2×C4#52911yescharacteristic
444C2×C4#53311yescharacteristic
445C2×C4#54111yescharacteristic
446C2×C4#54211yescharacteristic
447C2×C4#54611yescharacteristic
448C2×C4#54611yescharacteristic
449C2×C4#54711yescharacteristic
450C2×C4#54711yescharacteristic
451C2×C4#54811yescharacteristic
452C2×C4#54811yescharacteristic
453C2×C4D4×Q811yescharacteristic
454C2×C4D4×Q811yescharacteristic
455C2×C4#55211yescharacteristic
456C2×C4#55211yescharacteristic
457C2×C4#55411yescharacteristic
458C2×C4#55611yescharacteristic
459C2×C4#55611yescharacteristic
460C2×C4#55611yescharacteristic
461C2×C4#55711yescharacteristic
462C2×C4#55911yescharacteristic
463C8—24no—
464C8—42no—
465C8—44no—
466C8—42no—
467C8—42no—
468C8—42no—
469C8—42no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22C2×#38811yescharacteristic
483C22#229511yescharacteristic
484C22#232011yescharacteristic
485C22#242711yescharacteristic
486C22#242911yescharacteristic
487C22#251111yescharacteristic
488C22#251411yescharacteristic
489C22C2×#47111yescharacteristic
490C22C2×#47211yescharacteristic
491C22#306911yescharacteristic
492C22#306911yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—82no—
497C4—82no—
498C4—82no—
499C4—84no—
500C4—82no—
501C4#193114yes—
502C4#236011yescharacteristic
503C4#236211yescharacteristic
504C4#236511yescharacteristic
505C4#236711yescharacteristic
506C4#244611yescharacteristic
507C4#244711yescharacteristic
508C4#248911yescharacteristic
509C4#249211yescharacteristic
510C2—21no—
511C2—21no—
512C2—21no—
513C2—21no—
514C2#1414211yescharacteristic
515C2C2×#193111yescharacteristic
516C2#2107711yescharacteristic
517C2#2535811yescharacteristic
518C2#2536111yescharacteristic
519C2#2548911yescharacteristic
520C2#2549211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1997C4semidirect
112. #343Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19972
#213242
C2×#19882
#211702
#204412
#113032
#112902
#108962
#108532
#108272
C4×#3432