Finite Group Databasebeta

← #362613

Subgroups — #362613

Order 512.

1323 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2935C211yesmaximal, characteristic
2#21218C212yesmaximal
3C2×#1950C212yesmaximal, complemented
4C2×#1813C212yesmaximal, complemented
5#11099C214yesmaximal, complemented
6C4⋊#338C211yesmaximal, characteristic, complemented
7#10760C212yesmaximal, complemented
8C4×#338C211yesmaximal, characteristic, complemented
9C22×#131C2212yes—
10C2×#532C2211yescharacteristic
11#2935—24no—
12#2935—24no—
13C2×#436C412yes—
14C2×#435C414yes—
15C2×C8×D4C412yes—
16C22×#108C2214yes—
17C2×#423C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C2×#413C2214yes—
20C2×#413C2214yes—
21C2×#408C2212yes—
22C2×C4×M16C2212yes—
23C2×C4×M16C2211yescharacteristic
24#1967—24no—
25#1966—22no—
26#1965—22no—
27#1950—24no—
28#1950—24no—
29#1813—28no—
30C2×#341C2214yes—
31C2×#338C2211yescharacteristic
32C2×#338C2212yes—
33C2×#338C2212yes—
34C2×#338C2212yes—
35C2×#338C2211yescharacteristic
36C22×#41C2312yes—
37C2×#131—24no—
38C2×#131—24no—
39C2×#131—28no—
40C2×#131—44no—
41C22×M16C2×C414yes—
42C22×M16C2314yes—
43C22×M16C2312yes—
44C22×M16C2312yes—
45C22×M16C2311yescharacteristic
46C22×M16C2311yescharacteristic
47C23×C8C2×C414yes—
48C23×C8C2312yes—
49#532—22no—
50#532—22no—
51#532—22no—
52#532—22no—
53C2×C4⋊Q8D411yescharacteristic
54C2×#127D411yescharacteristic
55C2×#124D412yes—
56C2×#121Q812yes—
57C2×#121Q812yes—
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2×C412yes—
60C22×C42C2311yescharacteristic
61#436—24no—
62#436—24no—
63#435—216no—
64C8×D4—24no—
65C8×D4—24no—
66C2×#108—28no—
67C2×#108—28no—
68C2×#108—44no—
69#423—22no—
70#423—22no—
71#423—22no—
72#423—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8C2×C411yescharacteristic
76C2×C4⋊C8C2×C412yes—
77C2×C4⋊C8C2×C411yescharacteristic
78C2×C4⋊C8D414yes—
79C2×C4⋊C8D414yes—
80#413—28no—
81#413—28no—
82#413—44no—
83#413—44no—
84C2×#101—24no—
85C2×#101—24no—
86C2×#100—24no—
87C2×#100—24no—
88#408—28no—
89C2×C22⋊C8C2×C414yes—
90C2×C22⋊C8C2×C414yes—
91C4×M16—22no—
92C4×M16—22no—
93C4×M16—22no—
94C4×M16—22no—
95C4×M16—22no—
96C4×M16—22no—
97C2×#97—22no—
98C2×#97C2×C412yes—
99C2×C4×C8—22no—
100C2×C4×C8C2×C412yes—
101#341—28no—
102#338—24no—
103#338—24no—
104#338—22no—
105#338—42no—
106#338D414yescomplemented
107C2×#41—24no—
108C2×#41—22no—
109C2×#41—24no—
110C2×#41—22no—
111C2×#41—24no—
112C2×#41—24no—
113C2×#41—22no—
114C2×#41—22no—
115C22×Q8C2×D412yes—
116C22×D4C4⋊C412yes—
117C22×D4C4⋊C412yes—
118C22×D4C2×D412yes—
119C22×D4C2×Q814yes—
120C23×C4C22×C414yes—
121C23×C4C2×Q811yescharacteristic
122C23×C4C2×Q811yescharacteristic
123C23×C4C2411yescharacteristic
124#131—416no—
125#131—416no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—21no—
133C2×M16—21no—
134C2×M16—28no—
135C2×M16—28no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—24no—
145C2×M16—24no—
146C2×M16—24no—
147C2×M16—24no—
148C2×M16—24no—
149C2×M16—24no—
150C2×M16—44no—
151C2×M16C2×D414yes—
152C2×M16C2×D412yes—
153C2×M16#4114yes—
154C2×M16#4112yes—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—24no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—28no—
161C22×C8—28no—
162C22×C8—24no—
163C22×C8—44no—
164C22×C8C22×C412yes—
165C22×C8C22×C412yes—
166C22×C8C22×C412yes—
167C22×C8C22×C412yes—
168C22×C8C2×D418yes—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171#127—22no—
172#127—22no—
173#124—24no—
174#124—24no—
175#121—24no—
176#121—24no—
177#121—24no—
178#121—24no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—24no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—24no—
185C2×C4⋊C4C4⋊C412yes—
186C2×C4⋊C4C4⋊C412yes—
187C2×C22⋊C4C4⋊C414yes—
188C2×C22⋊C4C4⋊C414yes—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42#4112yes—
196C2×C42#4112yes—
197#108—416no—
198C4⋊C8—42no—
199C4⋊C8—42no—
200C4⋊C8—42no—
201C4⋊C8—42no—
202C4⋊C8C22⋊C412yes—
203C4⋊C8C22⋊C412yes—
204C4⋊C8C22⋊C418yes—
205C4⋊C8C22⋊C412yes—
206C4⋊C8C22⋊C412yes—
207#101—48no—
208#101—48no—
209#100—48no—
210#100—48no—
211C22⋊C8—28no—
212C22⋊C8—28no—
213#97—24no—
214#97—42no—
215#97—42no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—42no—
219C4×C8—42no—
220C24C2×C4⋊C414yes—
221C24C22×Q811yescharacteristic
222#41—44no—
223#41—44no—
224#41—44no—
225#41—44no—
226#41—44no—
227#41—44no—
228#41—44no—
229#41—44no—
230C2×Q8—24no—
231C2×Q8—22no—
232C2×Q8—24no—
233C2×Q8—22no—
234C2×D4—28no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—28no—
238C2×D4—24no—
239C2×D4—22no—
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—44no—
246C2×D4—82no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—42no—
254C22×C4—81no—
255C22×C4C2×C4⋊C414yes—
256C22×C4C2×C4⋊C411yescharacteristic
257C22×C4C2×C4⋊C411yescharacteristic
258C22×C4C2×C4⋊C411yescharacteristic
259C22×C4C2×C4⋊C411yescharacteristic
260C22×C4C22⋊Q812yes—
261C22×C4C22⋊Q812yes—
262C22×C4C22⋊Q812yes—
263C22×C4C22⋊Q812yes—
264C22×C4C23×C411yescharacteristic
265C22×C4C22×D412yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C2×#4112yes—
269C22×C4C2×#4111yescharacteristic
270C22×C4C2×#4112yes—
271C22×C4H(5,𝔽2)12yes—
272C22×C4H(5,𝔽2)11yescharacteristic
273C22×C42−1+411yescharacteristic
274M16—24no—
275M16—24no—
276M16—24no—
277M16—24no—
278M16—24no—
279M16—24no—
280M16—416no—
281M16—42no—
282M16—42no—
283M16—42no—
284M16—42no—
285M16—42no—
286M16—42no—
287M16—44no—
288M16—44no—
289M16—44no—
290M16—44no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—28no—
298C2×C8—28no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—416no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8C2×C22⋊C414yes—
313C2×C8C2×C22⋊C414yes—
314C2×C8C4×D414yes—
315C4⋊C4—22no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—28no—
321C42C2×C22⋊C411yescharacteristic
322C42C2×C22⋊C411yescharacteristic
323C42C2×C22⋊C412yes—
324C42#11711yescharacteristic
325C42#11712yes—
326C42#11711yescharacteristic
327C42#12112yes—
328C42#12112yes—
329C42#12112yes—
330C42#12112yes—
331C23—24no—
332C23—24no—
333C23—24no—
334C23—41no—
335C23—41no—
336C23—44no—
337C23—44no—
338C23C22×C4⋊C411yescharacteristic
339C23C2×C22⋊Q812yes—
340C23D4×Q811yescharacteristic
341C23#55411yescharacteristic
342C23#55412yes—
343Q8—44no—
344Q8—44no—
345D4—44no—
346D4—48no—
347D4—48no—
348D4—44no—
349D4—44no—
350D4—44no—
351D4—84no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4#38012yes—
368C2×C4#38011yescharacteristic
369C2×C4#38011yescharacteristic
370C2×C4#38011yescharacteristic
371C2×C4#38012yes—
372C2×C4#38011yescharacteristic
373C2×C4C22×C22⋊C411yescharacteristic
374C2×C4C2×#11711yescharacteristic
375C2×C4C2×C4×D411yescharacteristic
376C2×C4C4×#4111yescharacteristic
377C2×C4#51811yescharacteristic
378C2×C4#52011yescharacteristic
379C2×C4C2×#12112yes—
380C2×C4#52912yes—
381C2×C4#53212yes—
382C2×C4#54012yes—
383C2×C4#54212yes—
384C2×C4D4212yes—
385C2×C4D4211yescharacteristic
386C2×C4#54612yes—
387C2×C4#54612yes—
388C2×C4#54711yescharacteristic
389C2×C4#54812yes—
390C2×C4#55012yes—
391C8—24no—
392C8—24no—
393C8—24no—
394C8—42no—
395C8—44no—
396C8—44no—
397C8—84no—
398C22—21no—
399C22—22no—
400C22—21no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—42no—
405C22—44no—
406C22—44no—
407C22—44no—
408C22—84no—
409C22—84no—
410C22C2×#38011yescharacteristic
411C22D4×C4⋊C411yescharacteristic
412C22#234911yescharacteristic
413C22#245212yes—
414C22#245412yes—
415C4—22no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—22no—
420C4—44no—
421C4#181312yescomplemented
422C4#229311yescharacteristic
423C4#231911yescharacteristic
424C4D4×C22⋊C411yescharacteristic
425C4#234111yescharacteristic
426C4#242712yes—
427C4#242812yes—
428C2—21no—
429C2—21no—
430C2—22no—
431C2—22no—
432C2—84no—
433C2#1392011yescharacteristic
434C2C2×#181311yescharacteristic
435C2#2067311yescharacteristic
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#380
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1950C2semidirect
4. C2×#1813C2semidirect
5. #11099C2semidirect
6. C4⋊#338C2semidirect
7. #10760C2semidirect
8. C4×#338C2semidirect
106. #338D4semidirect
421. C4#1813semidirect

Maximal subgroups

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

subgroupindex
C2×#29352
#212182
C2×#19502
C2×#18132
#110992
C4⋊#3382
#107602
C4×#3382