Finite Group Databasebeta

← #506435

Subgroups — #506435

Order 512.

560 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2183C211yesmaximal, characteristic, complemented
2C2×#2175C211yesmaximal, characteristic
3#28268C211yesmaximal, characteristic, complemented
4#28263C211yesmaximal, characteristic
5#13477C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic
7#13475C211yesmaximal, characteristic, complemented
8#13474C211yesmaximal, characteristic
9#13238C211yesmaximal, characteristic
10#13237C211yesmaximal, characteristic
11#13167C211yesmaximal, characteristic, complemented
12#13165C211yesmaximal, characteristic, complemented
13#12908C211yesmaximal, characteristic, complemented
14#12813C211yesmaximal, characteristic
15#12812C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C2×C4×D8C2211yescharacteristic
20#2581—21no—
21#2538—21no—
22#2183—24no—
23#2175—24no—
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#366C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×C4⋊C16C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#346C2211yescharacteristic
39C2×#346C2211yescharacteristic
40C2×C4×C16C2211yescharacteristic
41#2067—21no—
42#2067—21no—
43#2066—21no—
44#2066—21no—
45#1970C2211yescharacteristic
46#1970C2211yescharacteristic
47#1969C2211yescharacteristic
48#1969C2211yescharacteristic
49#1942C2211yescharacteristic
50#1942C2211yescharacteristic
51#1942C2211yescharacteristic
52#1942C2211yescharacteristic
53#1846C2211yescharacteristic
54#1845C2211yescharacteristic
55#1805—21no—
56#1802—21no—
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×D8C2311yescharacteristic
63C2×C22⋊Q8—41no—
64C2×C4×Q8—21no—
65C2×C4×Q8D411yescharacteristic
66C2×C4×D4—21no—
67C2×SD32—22no—
68C2×SD32—22no—
69C2×SD32Q814yescomplemented
70C22×C16C2311yescharacteristic
71C22×C16C2311yescharacteristic
72C22×C16C2311yescharacteristic
73C22×C16C2311yescharacteristic
74C8×Q8—22no—
75C8×Q8—22no—
76C4×Q16—22no—
77C4×Q16—22no—
78C4×D8—22no—
79C4×D8—22no—
80C2×#107—21no—
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#107C2311yescharacteristic
87C2×#106—21no—
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8D411yescharacteristic
91C2×#103—21no—
92C2×#103—21no—
93C2×#103—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#400—41no—
101#396—41no—
102#396—41no—
103#395—41no—
104#391—21no—
105#391—21no—
106#390—21no—
107#390—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#387—41no—
113#387—41no—
114#384—21no—
115#382—21no—
116#380—41no—
117#367—22no—
118#367—41no—
119#366—22no—
120#366—22no—
121#366—41no—
122C4⋊C16—22no—
123C4⋊C16—22no—
124C4⋊C16—22no—
125#358—22no—
126#358—41no—
127#358—41no—
128#357—22no—
129#357—41no—
130#357—41no—
131#346—22no—
132#346D414yes—
133C4×C16—22no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138C22×Q8—21no—
139C22×Q8D811yescharacteristic
140C22×D4—21no—
141C23×C4—41no—
142C2×Q16—21no—
143C2×Q16—21no—
144C2×Q16—21no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16C2×Q812yes—
148C2×D8—21no—
149C2×D8—21no—
150C2×D8—21no—
151C2×D8—21no—
152C2×D8—21no—
153C2×D8C2×Q812yes—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2411yescharacteristic
158C22⋊Q8—84no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×D4—42no—
164C4×D4—42no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4D811yescharacteristic
179C2×C4⋊C4D811yescharacteristic
180C2×C4⋊C4D811yescharacteristic
181C2×C22⋊C4—41no—
182C2×C22⋊C4—41no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C42—41no—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190SD32—28no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—21no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—21no—
199C2×C16C2×D412yes—
200C2×C16C2×D412yes—
201C2×C16C2×Q812yes—
202C2×C16#4112yes—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—21no—
209#107—21no—
210#107—21no—
211#107—21no—
212#107—21no—
213#107—21no—
214#107—21no—
215#107—81no—
216#106—81no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220#103—42no—
221#103—81no—
222#103—81no—
223#102—42no—
224#102—81no—
225#102—81no—
226C4×C8—21no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8C2×D412yes—
231C4×C8#4112yes—
232#95—41no—
233#95—41no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—41no—
239#95—41no—
240#95—41no—
241#95—41no—
242C24—41no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×D4—22no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4C2×D811yescharacteristic
269C22×C4C2×D811yescharacteristic
270C22×C4C2×D811yescharacteristic
271C22×C4C2×D811yescharacteristic
272C22×C4C2×D811yescharacteristic
273C22×C4C2×D811yescharacteristic
274C22×C4C22×D411yescharacteristic
275Q16—22no—
276Q16—22no—
277Q16—41no—
278Q16—41no—
279D8—22no—
280D8—22no—
281D8—41no—
282D8—41no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8#12112yes—
292C2×C8C22×D411yescharacteristic
293C2×C8C22×Q811yescharacteristic
294C2×C8C2×#4111yescharacteristic
295C2×C8C2×#4111yescharacteristic
296C2×C8H(5,𝔽2)11yescharacteristic
297C2×C82−1+411yescharacteristic
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—161no—
320C22⋊C4—82no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—161no—
324C22⋊C4—161no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—82no—
330C42#12112yes—
331C42#13512yes—
332C16—22no—
333C16—41no—
334C16—41no—
335C16C22⋊Q814yescomplemented
336C23—42no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23C22×D811yescharacteristic
344Q8—44no—
345Q8—42no—
346Q8—42no—
347Q8—81no—
348Q8—81no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4—81no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—42no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—164no—
390C2×C4#46612yes—
391C2×C4#46612yes—
392C2×C4#50912yes—
393C2×C4C2×#12111yescharacteristic
394C2×C4D4×Q811yescharacteristic
395C2×C4#55011yescharacteristic
396C2×C4C2×#13511yescharacteristic
397C2×C4#57611yescharacteristic
398C2×C4#57811yescharacteristic
399C8—22no—
400C8—41no—
401C8—41no—
402C8C2×C22⋊Q811yescharacteristic
403C8#53711yescharacteristic
404C8#55411yescharacteristic
405C8#55611yescharacteristic
406C22—82no—
407C22—82no—
408C22—82no—
409C22—161no—
410C22—161no—
411C22—161no—
412C22—161no—
413C22C2×#46611yescharacteristic
414C22Q8×D811yescharacteristic
415C22#339111yescharacteristic
416C22C2×#50911yescharacteristic
417C22#341411yescharacteristic
418C22#341611yescharacteristic
419C22#341711yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—82no—
427C4—161no—
428C4—161no—
429C4—161no—
430C4—161no—
431C4—161no—
432C4—161no—
433C4#253811yescharacteristic
434C4#306111yescharacteristic
435C4#338011yescharacteristic
436C4#339311yescharacteristic
437C2—162no—
438C2#2826311yescharacteristic
439C2#3360511yescharacteristic
440C2#3361311yescharacteristic
441C2#3395311yescharacteristic
442C2#3395511yescharacteristic
443C2#3396411yescharacteristic
444C2#3396911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2183C2semidirect
3. #28268C2semidirect
5. #13477C2semidirect
7. #13475C2semidirect
11. #13167C2semidirect
12. #13165C2semidirect
13. #12908C2semidirect
15. #12812C2semidirect
69. C2×SD32Q8semidirect
335. C16C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#21832
C2×#21752
#282682
#282632
#134772
#134762
#134752
#134742
#132382
#132372
#131672
#131652
#129082
#128132
#128122