Finite Group Databasebeta

← #131908

Subgroups — #131908

Order 512.

736 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3334C211yesmaximal, characteristic
2#13282C211yesmaximal, characteristic, complemented
3#12811C211yesmaximal, characteristic, complemented
4C2×#1364C211yesmaximal, characteristic, complemented
5C2×#1331C211yesmaximal, characteristic, complemented
6C2×D4⋊C16C211yesmaximal, characteristic, complemented
7#8644C211yesmaximal, characteristic, complemented
8C2×#547—21no—
9#3334—22no—
10#3334—22no—
11#3334—22no—
12#3334C414yescomplemented
13C2×#500C2211yescharacteristic
14C2×#480—22no—
15C2×#474—21no—
16C2×#466—21no—
17C2×C4×D8C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C22×#107—21no—
20C2×C4⋊C16C2211yescharacteristic
21C2×#357—21no—
22C2×C4×C16C2211yescharacteristic
23#1845C2211yescharacteristic
24#1802—21no—
25C2×#335C2211yescharacteristic
26#1380—21no—
27#1380—22no—
28#1364—22no—
29#1331—22no—
30D4⋊C16—22no—
31#1296—21no—
32C22×#41—41no—
33C22×D8D411yescharacteristic
34C23×C8—21no—
35#547—24no—
36#547—24no—
37#547—42no—
38#547—42no—
39C2×C4⋊Q8—21no—
40C2×#123—42no—
41C2×C22⋊Q8—41no—
42C2×#121—22no—
43C2×C4×D4—21no—
44C2×C4×D4D411yescharacteristic
45C22×C4⋊C4—22no—
46C22×C16—21no—
47C22×C16—21no—
48#500—21no—
49#500—21no—
50#500C2×C412yes—
51#500D412yes—
52#480—28no—
53#480—28no—
54#474—24no—
55#474—24no—
56#466—24no—
57#466—24no—
58C4×D8—21no—
59C4×D8—21no—
60C4×D8C2×C412yes—
61C4×D8D412yes—
62C8×D4—21no—
63C8×D4—21no—
64C8×D4C2×C412yes—
65C8×D4D412yes—
66C2×#107—24no—
67C2×#107—22no—
68C2×#107—21no—
69C2×#107—24no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107D412yes—
73C2×#107D411yescharacteristic
74C2×C4⋊C8—21no—
75C2×C4⋊C8D411yescharacteristic
76C2×#102—22no—
77C2×#102—21no—
78C2×#102—41no—
79C2×C22⋊C8—21no—
80C2×C4×C8C2311yescharacteristic
81#384—21no—
82C4⋊C16—21no—
83C4⋊C16—21no—
84#357—42no—
85C4×C16—21no—
86C4×C16—21no—
87#340—41no—
88#335—21no—
89#335—21no—
90C2×#41—44no—
91C2×#41—44no—
92C2×#41—82no—
93C2×#41—82no—
94C2×#41—82no—
95C2×#41—82no—
96C22×Q8—41no—
97C22×D4—21no—
98C22×D4—42no—
99C22×D4D811yescharacteristic
100C23×C4—21no—
101C23×C4—42no—
102C23×C4—41no—
103C2×D8—21no—
104C2×D8—21no—
105C2×D8—21no—
106C2×D8—21no—
107C2×D8—21no—
108C2×D8C22⋊C412yes—
109C22×C8—21no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8—41no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8C2×D411yescharacteristic
118C22×C8C2×D411yescharacteristic
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123#123—48no—
124#123—48no—
125C22⋊Q8—44no—
126C22⋊Q8—44no—
127#121—28no—
128#121—44no—
129C4×D4—22no—
130C4×D4—21no—
131C4×D4—22no—
132C4×D4—21no—
133C4×D4—41no—
134C4×D4—41no—
135C4×D4C22⋊C412yes—
136C4×D4SD1612yes—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4D811yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—41no—
150C2×C42C2×D411yescharacteristic
151C2×C16—22no—
152C2×C16—41no—
153C2×C16—41no—
154C2×C16—41no—
155C2×C16—41no—
156C2×C16—41no—
157#107—28no—
158#107—22no—
159#107—21no—
160#107—22no—
161#107—22no—
162#107—42no—
163#107—41no—
164#107—41no—
165#107C22⋊C412yes—
166#107C22⋊C414yes—
167C4⋊C8—21no—
168C4⋊C8—41no—
169C4⋊C8—41no—
170C4⋊C8C22⋊C412yes—
171#102—28no—
172#102—22no—
173#102—22no—
174#102—82no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C4×C8C22×C411yescharacteristic
178C4×C8C2×D411yescharacteristic
179C4×C8C2×D411yescharacteristic
180C4×C8C2×D411yescharacteristic
181C24—21no—
182C24—41no—
183#41—88no—
184#41—88no—
185#41—88no—
186#41—88no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—81no—
190C2×Q8—81no—
191C2×Q8—81no—
192C2×Q8—81no—
193C2×D4—22no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—48no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—84no—
204C2×D4—84no—
205C2×D4#10212yes—
206C2×D4D1612yes—
207C2×D4SD3212yes—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—48no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—84no—
221C22×C4—84no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4#12011yescharacteristic
228C22×C4C2×D811yescharacteristic
229C22×C4AGL(1,ℤ/8)11yescharacteristic
230D8—22no—
231D8—22no—
232D8—41no—
233D8—41no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—22no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—81no—
244C2×C8—81no—
245C2×C8—81no—
246C2×C8C2×C22⋊C411yescharacteristic
247C2×C8C2×C22⋊C411yescharacteristic
248C2×C8#12011yescharacteristic
249C2×C8#12011yescharacteristic
250C2×C8#13511yescharacteristic
251C2×C8#13511yescharacteristic
252C2×C8AGL(1,ℤ/8)11yescharacteristic
253C2×C8#13711yescharacteristic
254C4⋊C4—21no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—48no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—81no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C4⋊C4—81no—
266C4⋊C4—161no—
267C4⋊C4#10212yes—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—48no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C42C2×C22⋊C411yescharacteristic
274C42#12011yescharacteristic
275C42C2×SD1611yescharacteristic
276C42AGL(1,ℤ/8)11yescharacteristic
277C16—42no—
278C16—81no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—42no—
287C23—81no—
288C23—81no—
289C23—81no—
290C23—81no—
291C23—81no—
292C23—81no—
293C23#44711yescharacteristic
294Q8—82no—
295Q8—82no—
296Q8—82no—
297Q8—82no—
298D4—22no—
299D4—22no—
300D4—22no—
301D4—42no—
302D4—42no—
303D4—816no—
304D4—81no—
305D4—81no—
306D4#35714yescomplemented
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—816no—
319C2×C4—81no—
320C2×C4—82no—
321C2×C4—81no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—161no—
328C2×C4—161no—
329C2×C4—161no—
330C2×C4#37911yescharacteristic
331C2×C4C2×#10211yescharacteristic
332C2×C4#41811yescharacteristic
333C2×C4#44711yescharacteristic
334C2×C4#44911yescharacteristic
335C2×C4#45011yescharacteristic
336C2×C4#45011yescharacteristic
337C2×C4#45211yescharacteristic
338C2×C4#45411yescharacteristic
339C2×C4#45611yescharacteristic
340C2×C4C2×D1611yescharacteristic
341C2×C4C2×SD3211yescharacteristic
342C2×C4#50911yescharacteristic
343C2×C4#50911yescharacteristic
344C8—21no—
345C8—21no—
346C8—42no—
347C8—161no—
348C8#41611yescharacteristic
349C8#41711yescharacteristic
350C8#44911yescharacteristic
351C8#45211yescharacteristic
352C22—24no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—82no—
360C22—82no—
361C22—82no—
362C22—161no—
363C22—161no—
364C22—161no—
365C22—161no—
366C22#162511yescharacteristic
367C22#164711yescharacteristic
368C22#178611yescharacteristic
369C22#218411yescharacteristic
370C22#218711yescharacteristic
371C22#219311yescharacteristic
372C22#219511yescharacteristic
373C4—21no—
374C4—21no—
375C4—42no—
376C4—82no—
377C4—84no—
378C4—321no—
379C4#163511yescharacteristic
380C4#165411yescharacteristic
381C4#178811yescharacteristic
382C4#179311yescharacteristic
383C4C2×#35711yescharacteristic
384C4#214011yescharacteristic
385C4#218411yescharacteristic
386C4#218711yescharacteristic
387C2—44no—
388C2—162no—
389C2#844611yescharacteristic
390C2#1234011yescharacteristic
391C2#1234411yescharacteristic
392C2#1239211yescharacteristic
393C2#1239611yescharacteristic
394C2#1273111yescharacteristic
395C2#1274911yescharacteristic
subgroupG/H
center, socleC23#447
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#3334C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13282C2semidirect
3. #12811C2semidirect
4. C2×#1364C2semidirect
5. C2×#1331C2semidirect
6. C2×D4⋊C16C2semidirect
7. #8644C2semidirect
12. #3334C4semidirect
306. D4#357semidirect

Maximal subgroups

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

subgroupindex
C2×#33342
#132822
#128112
C2×#13642
C2×#13312
C2×D4⋊C162
#86442