Finite Group Databasebeta

← #370068

Subgroups — #370068

Order 512.

860 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1952C211yesmaximal, characteristic
2#21033C212yesmaximal
3C2×C4×#107C211yesmaximal, characteristic
4#20435C211yesmaximal, characteristic
5#11303C212yesmaximal
6#11216C212yesmaximal
7#10870C211yesmaximal, characteristic
8#10870C211yesmaximal, characteristic
9#10870C211yesmaximal, characteristic
10#10864C211yesmaximal, characteristic
11#10802C212yesmaximal
12C22×#108C2211yescharacteristic
13C22×#108C2211yescharacteristic
14C2×#428C2212yes—
15C22×#107C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#405C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2296C2211yescharacteristic
22C2×C4×C4⋊C4C2212yes—
23#1952C418yescomplemented
24#1921—22no—
25#1920—22no—
26#1919—22no—
27#1918—22no—
28C4×#107—21no—
29C4×#107—21no—
30C4×#107—21no—
31C4×#107—21no—
32#1771—21no—
33#1771—21no—
34C8×C4⋊C4—21no—
35C8×C4⋊C4—21no—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#343C2212yes—
40C2×#341C2212yes—
41#1736C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44C2×#340C2212yes—
45#1729C2211yescharacteristic
46#1728C2212yes—
47#1728C2211yescharacteristic
48C22×M16C2312yes—
49C22×M16C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C42—22no—
59C22×C42C2311yescharacteristic
60C2×#108—22no—
61C2×#108—22no—
62C2×#108—22no—
63C2×#108—22no—
64C2×#108—22no—
65C2×#108C2×C414yes—
66#428—24no—
67#428C2×C418yes—
68C2×#107—21no—
69C2×#107—21no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#107C2×C414yes—
76C2×#107Q814yes—
77C2×#106—22no—
78#424—22no—
79#424—22no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82#405—22no—
83#405—22no—
84#405—22no—
85#405C2×C414yes—
86C2×#97—21no—
87C2×#97—21no—
88C2×C4×C8—21no—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—21no—
92C2×C4×C8D411yescharacteristic
93C2×C4×C8D411yescharacteristic
94C2×C4×C8D411yescharacteristic
95C2×C4×C8D411yescharacteristic
96#389D411yescharacteristic
97#389D411yescharacteristic
98#389D411yescharacteristic
99#389D411yescharacteristic
100#385—21no—
101#385—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C2×#95—22no—
109#343—24no—
110#343—42no—
111#343—42no—
112#343C2×C418yes—
113#341—42no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#336—21no—
123#336—21no—
124#336—22no—
125#336—22no—
126#336—21no—
127#336—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4#4111yescharacteristic
131C23×C4#4111yescharacteristic
132C23×C4C2411yescharacteristic
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—22no—
136C2×M16—21no—
137C2×M16—21no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—21no—
146C2×M16—21no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16C22×C412yes—
155C2×M16C22×C414yes—
156C2×M16#4112yes—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8C22×C412yes—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×Q812yes—
173C22×C8#4112yes—
174C22×C8#4112yes—
175C22×C8#4112yes—
176C22×C8#4112yes—
177#117—21no—
178#117—21no—
179#117—21no—
180#117—21no—
181#117—21no—
182#117—21no—
183#117C22×C412yes—
184#117#4112yes—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4C22×C414yes—
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×Q814yes—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42—84no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220#108—28no—
221#108—44no—
222#107—24no—
223#107—24no—
224#107—24no—
225#107—24no—
226#107—28no—
227#106—28no—
228C4⋊C8—24no—
229C4⋊C8—24no—
230#97—24no—
231#97—42no—
232C4×C8—24no—
233C4×C8—22no—
234C4×C8—22no—
235C4×C8—22no—
236C4×C8—22no—
237C4×C8—42no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—42no—
243#95—42no—
244C24C2×#4111yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C22×Q811yescharacteristic
275C22×C4C2×#4111yescharacteristic
276C22×C4C2×#4111yescharacteristic
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C22×C4H(5,𝔽2)11yescharacteristic
282C22×C42−1+411yescharacteristic
283C22×C42−1+411yescharacteristic
284C22×C42−1+411yescharacteristic
285M16—24no—
286M16—22no—
287M16—22no—
288M16—24no—
289M16—22no—
290M16—22no—
291M16—41no—
292M16—41no—
293M16—42no—
294M16—41no—
295M16—42no—
296M16—41no—
297M16—42no—
298M16—42no—
299M16—42no—
300M16—42no—
301C2×C8—22no—
302C2×C8—24no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8#11714yes—
324C2×C8C4×D412yes—
325C2×C8C4×D412yes—
326C2×C8C4×Q814yes—
327C2×C8C22⋊Q812yes—
328C2×C8C22⋊Q812yes—
329C2×C8#12312yes—
330C2×C8#12312yes—
331C2×C8#12514yes—
332C2×C8#12614yes—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—24no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4C4×D412yes—
352C4⋊C4C4×D412yes—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C42—22no—
358C42—21no—
359C42—21no—
360C42—22no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—41no—
368C42—41no—
369C42—84no—
370C42—84no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23C4×#4111yescharacteristic
376C23C2×#12111yescharacteristic
377C23#52911yescharacteristic
378C23#55211yescharacteristic
379C23#55211yescharacteristic
380C23#55411yescharacteristic
381C23#55611yescharacteristic
382C2×C4—24no—
383C2×C4—24no—
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—22no—
395C2×C4—22no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—44no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—44no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—84no—
417C2×C4—84no—
418C2×C4#47212yes—
419C2×C4#47312yes—
420C2×C4C2×#11711yescharacteristic
421C2×C4C2×C4×D411yescharacteristic
422C2×C4C2×C4×D411yescharacteristic
423C2×C4C2×C4×Q811yescharacteristic
424C2×C4#51911yescharacteristic
425C2×C4#52011yescharacteristic
426C2×C4C2×C22⋊Q811yescharacteristic
427C2×C4C2×#12311yescharacteristic
428C2×C4#52511yescharacteristic
429C2×C4C2×#12511yescharacteristic
430C2×C4C2×#12611yescharacteristic
431C2×C4#53211yescharacteristic
432C2×C4#53611yescharacteristic
433C2×C4#53711yescharacteristic
434C2×C4#54111yescharacteristic
435C2×C4#54211yescharacteristic
436C2×C4#54711yescharacteristic
437C2×C4#54711yescharacteristic
438C2×C4D4×Q811yescharacteristic
439C2×C4#55011yescharacteristic
440C2×C4#55211yescharacteristic
441C2×C4#55311yescharacteristic
442C2×C4#55511yescharacteristic
443C2×C4#55611yescharacteristic
444C2×C4#55611yescharacteristic
445C2×C4#55611yescharacteristic
446C2×C4#55711yescharacteristic
447C2×C4#55711yescharacteristic
448C8—42no—
449C8—44no—
450C8—41no—
451C8—41no—
452C8—42no—
453C8—82no—
454C8#38714yes—
455C8#38714yes—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22C4×#12111yescharacteristic
469C22#235611yescharacteristic
470C22#237011yescharacteristic
471C22#248611yescharacteristic
472C22#250811yescharacteristic
473C22#251411yescharacteristic
474C22#253811yescharacteristic
475C22C2×#47211yescharacteristic
476C22C2×#47311yescharacteristic
477C22#306711yescharacteristic
478C22#306711yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—42no—
484C4—42no—
485C4—81no—
486C4—81no—
487C4—84no—
488C4—84no—
489C4C2×#38711yescharacteristic
490C4#232311yescharacteristic
491C4#235411yescharacteristic
492C4#236511yescharacteristic
493C4#248511yescharacteristic
494C4#250611yescharacteristic
495C4#251211yescharacteristic
496C4#258111yescharacteristic
497C2—21no—
498C2—21no—
499C2—21no—
500C2—21no—
501C2#1404211yescharacteristic
502C2C4×#47211yescharacteristic
503C2C4×#47311yescharacteristic
504C2#2767411yescharacteristic
505C2#2767511yescharacteristic
506C2#2829511yescharacteristic
507C2#2829711yescharacteristic
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
23. #1952C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19522
#210332
C2×C4×#1072
#204352
#113032
#112162
#108702
#108642
#108022