Finite Group Databasebeta

← #216166

Subgroups — #216166

Order 512.

813 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34621C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic, complemented
3#21431C211yesmaximal, characteristic, complemented
4#9876C211yesmaximal, characteristic, complemented
5#9875C211yesmaximal, characteristic, complemented
6#9648C211yesmaximal, characteristic, complemented
7#9644C211yesmaximal, characteristic, complemented
8#8944C212yesmaximal, complemented
9#8943C212yesmaximal, complemented
10#8569C212yesmaximal, complemented
11#8567C212yesmaximal, complemented
12C2×#432C2211yescharacteristic
13C2×#423C2211yescharacteristic
14#2651—22no—
15#2648—22no—
16#2647C411yescharacteristic
17#2644C412yes—
18#2641C411yescharacteristic
19#2629C411yescharacteristic
20#2624C412yes—
21#2619C411yescharacteristic
22#2395—22no—
23#2392—22no—
24C4×#127C2211yescharacteristic
25C4×#125C2211yescharacteristic
26C4×#121C2212yes—
27#2293C2212yes—
28C4×#117C2211yescharacteristic
29#1940—22no—
30#1924—22no—
31#1917—22no—
32#1851—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1831C2211yescharacteristic
36#1831C2211yescharacteristic
37C4×C4⋊C8C2212yes—
38C4×C22⋊C8C2212yes—
39#1744C2211yescharacteristic
40C4×#97C2211yescharacteristic
41C2×D4⋊C8C2212yes—
42C2×D4⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49C22×M16—21no—
50C2×#127C2×C411yescharacteristic
51C2×#125C2×C411yescharacteristic
52C2×#121C2×C412yes—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4C2312yes—
56C2×#117—22no—
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C22×C42C2311yescharacteristic
60#432—42no—
61#423—42no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×C22⋊C8—21no—
71C2×C22⋊C8—22no—
72C2×#97—21no—
73C2×#97—22no—
74C2×C4×C8C2312yes—
75C2×C4×C8C2312yes—
76#400—22no—
77#399—22no—
78#398—22no—
79#398—22no—
80#397—22no—
81#397—22no—
82#397—22no—
83#397—22no—
84#396—22no—
85#396—22no—
86#394—22no—
87#394—22no—
88#394—22no—
89#394—22no—
90#394—22no—
91#392—22no—
92#390—22no—
93#390C2×C412yes—
94#390C2×C412yes—
95#390C2×C412yes—
96#390C2×C412yes—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2×C412yes—
100#389C2×C412yes—
101#388—22no—
102#388—22no—
103#388—22no—
104#387—22no—
105#386—22no—
106#384C2×C411yescharacteristic
107#382C2×C411yescharacteristic
108#380C2×C412yes—
109C4×C4⋊C4—22no—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4C2312yes—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4D412yes—
116C4×C22⋊C4D412yes—
117#376C2×C411yescharacteristic
118#376C2×C411yescharacteristic
119C43C2311yescharacteristic
120C43C2311yescharacteristic
121#336—22no—
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126D4⋊C8—42no—
127D4⋊C8—42no—
128D4⋊C8—42no—
129D4⋊C8—42no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4C22×C412yes—
133C23×C4—22no—
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C2×M16—42no—
138C2×M16—82no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145#127—42no—
146#125—42no—
147#121—44no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—82no—
153C4×D4—82no—
154C4×D4—82no—
155C4×D4—82no—
156#117—42no—
157#117—42no—
158#117—82no—
159#117—82no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C22⋊C412yes—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2411yescharacteristic
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—42no—
204C4⋊C8—44no—
205C4⋊C8—82no—
206C4⋊C8—82no—
207C22⋊C8—44no—
208C22⋊C8—42no—
209#97—42no—
210#97—44no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95C22⋊C412yes—
223#95C22⋊C412yes—
224#95C22⋊C412yes—
225#95C22⋊C412yes—
226C24—22no—
227C24C2×C22⋊C411yescharacteristic
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—82no—
235C2×D4—82no—
236C2×D4—82no—
237C2×D4—82no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266M16—84no—
267C2×C8—24no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—82no—
289C4⋊C4—82no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—44no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—82no—
299C22⋊C4—82no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C42—22no—
308C42—21no—
309C42—21no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—42no—
320C42—42no—
321C42—42no—
322C42—42no—
323C42H(5,𝔽2)12yes—
324C422−1+412yes—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23C22×C22⋊C411yescharacteristic
338D4—44no—
339D4—44no—
340D4—88no—
341D4—88no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4#52012yes—
381C2×C4#53511yescharacteristic
382C2×C4#53611yescharacteristic
383C2×C4#53712yes—
384C2×C4#56812yes—
385C2×C4#56812yes—
386C2×C4#57612yes—
387C2×C4#57712yes—
388C8—44no—
389C8—82no—
390C8—81no—
391C8—81no—
392C8—81no—
393C8—81no—
394C22—41no—
395C22—42no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—82no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22#231111yescharacteristic
407C22#287911yescharacteristic
408C22#287911yescharacteristic
409C22#289712yes—
410C22#290112yes—
411C4—22no—
412C4—41no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—41no—
418C4—41no—
419C4—82no—
420C4—82no—
421C4—82no—
422C4#299514yes—
423C4#322011yescharacteristic
424C4#322111yescharacteristic
425C4#324112yes—
426C2—81no—
427C2—162no—
428C2#1750512yes—
429C2#1750512yes—
430C2#2185411yescharacteristic
431C2#2185811yescharacteristic
432C2#2187211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2619C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21978C2semidirect
3. #21431C2semidirect
4. #9876C2semidirect
5. #9875C2semidirect
6. #9648C2semidirect
7. #9644C2semidirect
8. #8944C2semidirect
9. #8943C2semidirect
10. #8569C2semidirect
11. #8567C2semidirect

Maximal subgroups

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

subgroupindex
#346212
#219782
#214312
#98762
#98752
#96482
#96442
#89442
#89432
#85692
#85672