Finite Group Databasebeta

← #147021

Subgroups — #147021

Order 512.

734 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1813C211yesmaximal, characteristic
2C2×#1812C211yesmaximal, characteristic
3#20384C211yesmaximal, characteristic
4C2×#1395C212yesmaximal
5#12573C212yesmaximal
6C22×#131C2211yescharacteristic
7C22×#108—21no—
8C2×#427—21no—
9C2×#413C2212yes—
10C2×#410C2212yes—
11C2×#408C2211yescharacteristic
12C2×#405C411yescharacteristic, complemented
13C22×#97C411yescharacteristic, complemented
14#2297C2211yescharacteristic
15#1813—22no—
16#1813C414yescomplemented
17#1812—22no—
18#1812C414yescomplemented
19#1753—21no—
20#1753—21no—
21#1752—21no—
22#1752—21no—
23C2×#338—21no—
24C2×H(3,ℤ/4)—21no—
25#1729C411yescharacteristic, complemented
26#1728C411yescharacteristic, complemented
27#1395—24no—
28#1390—24no—
29C22×#41C2311yescharacteristic
30C2×#131—41no—
31C2×#131—41no—
32C2×#131—42no—
33C2×#131C2×C412yes—
34C2×#131C2×C412yes—
35C2×#131D412yes—
36C2×#131Q812yes—
37C22×M16—22no—
38C22×M16C2×C411yescharacteristic
39C22×M16C2×C411yescharacteristic
40C22×M16D411yescharacteristic
41C22×M16D411yescharacteristic
42C23×C8C2×C411yescharacteristic
43C23×C8C2×C411yescharacteristic
44C23×C8D411yescharacteristic
45C23×C8D411yescharacteristic
46C2×#117—22no—
47C2×#117C2×C411yescharacteristic
48C22×C42C2×C411yescharacteristic
49C2×#108—24no—
50C2×#108—41no—
51C2×#108—41no—
52C2×#108—42no—
53#427—24no—
54#427—42no—
55C2×#107—41no—
56C2×#106—41no—
57#413—22no—
58#413—22no—
59#413—22no—
60#413C2×C414yes—
61C2×#101—22no—
62C2×#100—22no—
63#410—22no—
64#410—22no—
65#410—22no—
66#410C2×C414yes—
67C2×#99—22no—
68C2×#99—22no—
69#408—21no—
70#408—21no—
71#408D412yes—
72#408Q812yes—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75#405—42no—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97—21no—
79C2×#97—41no—
80C2×#97—41no—
81C2×C4×C8—21no—
82#391—21no—
83#390—21no—
84#386D412yes—
85#386Q812yes—
86C4×C22⋊C4—21no—
87C4×C22⋊C4—21no—
88#338—24no—
89H(3,ℤ/4)—24no—
90#336—21no—
91#336—21no—
92#336—21no—
93#336—21no—
94C2×#41—41no—
95C2×#41—41no—
96C2×#41—41no—
97C2×#41—41no—
98C2×#41C22×C411yescharacteristic
99C2×#41C22×C411yescharacteristic
100C2×#41C2×D411yescharacteristic
101C2×#41C2×D411yescharacteristic
102C2×#41C2×Q811yescharacteristic
103C2×#41C2×Q811yescharacteristic
104C2×#41#4111yescharacteristic
105C2×#41#4111yescharacteristic
106C22×Q8#4111yescharacteristic
107C22×D4C2×D411yescharacteristic
108C22×D4C2×Q811yescharacteristic
109C22×D4#4111yescharacteristic
110C23×C4C4⋊C412yes—
111C23×C4C22×C411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114#131—42no—
115#131—42no—
116#131—42no—
117#131—42no—
118#131—42no—
119#131—42no—
120#131—42no—
121#131—42no—
122C2×M16—21no—
123C2×M16—24no—
124C2×M16—41no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16—41no—
128C2×M16—41no—
129C2×M16—41no—
130C2×M16—42no—
131C2×M16—41no—
132C2×M16—41no—
133C2×M16—42no—
134C2×M16—42no—
135C2×M16—42no—
136C2×M16—42no—
137C2×M16C4212yes—
138C2×M16C22⋊C412yes—
139C2×M16C22⋊C412yes—
140C2×M16C4⋊C412yes—
141C2×M16C4⋊C412yes—
142C2×M16C4⋊C412yes—
143C2×M16C4⋊C412yes—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—41no—
157C22×C8—41no—
158C22×C8—42no—
159C22×C8C4212yes—
160C22×C8C22⋊C411yescharacteristic
161C22×C8C22⋊C411yescharacteristic
162C22×C8C22⋊C412yes—
163C22×C8C22⋊C412yes—
164C22×C8C22⋊C411yescharacteristic
165C22×C8C22⋊C411yescharacteristic
166C22×C8C4⋊C412yes—
167C22×C8C4⋊C412yes—
168C22×C8C4⋊C412yes—
169C22×C8C4⋊C412yes—
170#117—24no—
171#117—42no—
172#117—41no—
173#117—41no—
174#117—42no—
175#117—42no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—42no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4C4⋊C412yes—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—41no—
186C2×C42—41no—
187C2×C42—42no—
188#108—44no—
189#108—44no—
190#107—44no—
191#106—44no—
192#101—24no—
193#101—42no—
194#100—24no—
195#100—42no—
196#99—24no—
197#99—24no—
198#99—42no—
199#99—42no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204#97—42no—
205#97—41no—
206#97—41no—
207#97—42no—
208C4×C8—42no—
209#95—22no—
210#95—22no—
211C24C2×C4⋊C411yescharacteristic
212C24#12111yescharacteristic
213C24C22⋊Q811yescharacteristic
214#41—41no—
215#41—41no—
216#41—41no—
217#41—41no—
218#41—41no—
219#41—41no—
220#41—41no—
221#41—41no—
222#41—41no—
223#41—41no—
224#41—41no—
225#41—41no—
226#41—41no—
227#41—41no—
228#41—41no—
229#41—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8C4×D411yescharacteristic
233C2×Q8C4×Q811yescharacteristic
234C2×Q8#12411yescharacteristic
235C2×Q8#12511yescharacteristic
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4C4×D411yescharacteristic
243C2×D4C4×D411yescharacteristic
244C2×D4C4×D411yescharacteristic
245C2×D4C4×Q811yescharacteristic
246C2×D4C4×Q811yescharacteristic
247C2×D4C4×Q811yescharacteristic
248C2×D4#12411yescharacteristic
249C2×D4#12511yescharacteristic
250C2×D4#12711yescharacteristic
251C2×D4C4⋊Q811yescharacteristic
252C2×D4C4⋊Q811yescharacteristic
253C2×D4C4⋊Q811yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—42no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4C2×C4211yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C4⋊C411yescharacteristic
276C22×C4C2×C4⋊C411yescharacteristic
277C22×C4C2×C4⋊C411yescharacteristic
278C22×C4C2×C4⋊C411yescharacteristic
279C22×C4#11711yescharacteristic
280C22×C4#11711yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4#12111yescharacteristic
283C22×C4#12111yescharacteristic
284C22×C4C22⋊Q811yescharacteristic
285C22×C4C22⋊Q811yescharacteristic
286C22×C4C22⋊Q811yescharacteristic
287M16—42no—
288M16—42no—
289M16—42no—
290M16—42no—
291M16—42no—
292M16—42no—
293M16—42no—
294M16—42no—
295M16—44no—
296M16—44no—
297M16—82no—
298M16—82no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—42no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—44no—
326C2×C8—82no—
327C2×C8—82no—
328C2×C8#9512yes—
329C2×C8#9512yes—
330C2×C8#9512yes—
331C2×C8#9512yes—
332C2×C8#9911yescharacteristic
333C2×C8#9911yescharacteristic
334C2×C8#9912yes—
335C4⋊C4—42no—
336C4⋊C4—44no—
337C4⋊C4—82no—
338C22⋊C4—42no—
339C22⋊C4—44no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C42—42no—
344C42—44no—
345C42—41no—
346C42—41no—
347C42—41no—
348C42—41no—
349C42—82no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23C4×C4⋊C411yescharacteristic
360C23#38011yescharacteristic
361C23#38411yescharacteristic
362C23#38911yescharacteristic
363C23#38911yescharacteristic
364C23#39011yescharacteristic
365C23#39111yescharacteristic
366Q8—41no—
367Q8—41no—
368Q8—41no—
369Q8—41no—
370D4—41no—
371D4—41no—
372D4—41no—
373D4—41no—
374D4—41no—
375D4—41no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—41no—
380D4—41no—
381D4—41no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4—164no—
421C2×C4C2×#9511yescharacteristic
422C2×C4C4×C4⋊C411yescharacteristic
423C2×C4#38011yescharacteristic
424C2×C4#38211yescharacteristic
425C2×C4#38911yescharacteristic
426C2×C4#38911yescharacteristic
427C2×C4#39011yescharacteristic
428C2×C4#39111yescharacteristic
429C2×C4C2×#9911yescharacteristic
430C2×C4C2×#9911yescharacteristic
431C2×C4#41011yescharacteristic
432C2×C4#41011yescharacteristic
433C8—22no—
434C8—22no—
435C8—22no—
436C8—42no—
437C8—42no—
438C8—84no—
439C8#34212yes—
440C8#34212yes—
441C22—22no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—81no—
459C22—81no—
460C22C4⋊#9511yescharacteristic
461C22#179911yescharacteristic
462C22#179911yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—82no—
470C4—82no—
471C4—164no—
472C4C2×#34211yescharacteristic
473C4#174011yescharacteristic
474C4#179911yescharacteristic
475C4#179911yescharacteristic
476C2—21no—
477C2—21no—
478C2—41no—
479C2—41no—
480C2—41no—
481C2—41no—
482C2—41no—
483C2—41no—
484C2#1084611yescharacteristic
485C2#1280612yes—
subgroupG/H
centerC2×C4#410
commutatorC22×C4C2×C42
Frattini, prime radical, cosocleC22×#41C23
socleC22#1799
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G
LECSG ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
12. C2×#405C4semidirect
13. C22×#97C4semidirect
16. #1813C4semidirect
18. #1812C4semidirect
25. #1729C4semidirect
26. #1728C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18132
C2×#18122
#203842
C2×#13952
#125732