Finite Group Databasebeta

← C2×#19194 #6353676

Subgroups — C2×#19194 #6353676

Order 512.

1840 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3246C212yesmaximal
2C2×#3160C211yesmaximal, characteristic, complemented
3C2×#2931C211yesmaximal, characteristic
4#19194C2116yesmaximal, direct product factor
5C2×#1695C212yesmaximal, complemented
6C2×#1693C212yesmaximal, complemented
7C2×#1671C214yesmaximal, complemented
8C22×#334C211yesmaximal, characteristic, complemented
9C2×#1550C212yesmaximal, complemented
10C22×#131—21no—
11C2×#533C2212yes—
12C2×#532C2211yescharacteristic
13#3246—28no—
14#3246C22116yes—
15#3160—24no—
16#3160C2218yescomplemented
17C2×#500C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#499—21no—
20C2×#499—21no—
21C2×#498C2212yes—
22C2×#475—22no—
23C2×#476C2214yes—
24C2×#465—24no—
25C2×#460—22no—
26C2×#461C2214yes—
27#2931—24no—
28#2931C2218yes—
29C2×#436—21no—
30C2×#436C2212yes—
31C2×C8×D4—21no—
32C2×C8×D4C2212yes—
33C2×#427—21no—
34C2×#427—21no—
35C2×#423C2211yescharacteristic
36C22×C4⋊C8C2212yes—
37C2×#408—21no—
38C22×C4×C8C2211yescharacteristic
39#1695C22116yescomplemented
40#1693C22116yescomplemented
41#1671C22132yescomplemented
42C2×#334C2212yes—
43C2×#334C2212yes—
44C2×#334C2218yescomplemented
45#1550C22116yescomplemented
46C2×#333C2214yes—
47C2×#332C2214yes—
48C22×#41—21no—
49C22×SD16—24no—
50C2×#131—28no—
51C2×#131—48no—
52C22×M16—21no—
53C22×M16—22no—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8D411yescharacteristic
57C23×C8D411yescharacteristic
58#533—24no—
59#533—24no—
60#533—24no—
61#533C2318yes—
62#532—22no—
63#532—22no—
64#532D414yes—
65#532C2314yes—
66C2×C4⋊Q8D411yescharacteristic
67C2×C4⋊Q8C2312yes—
68C2×C4⋊Q8C2312yes—
69C2×#127D411yescharacteristic
70C2×#125—22no—
71C2×#124—21no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×#121—21no—
75C2×#121D412yes—
76C2×C4×Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×D4—21no—
79C2×C4×D4C2312yes—
80C2×#117—22no—
81C22×C42C2311yescharacteristic
82#500—22no—
83#500—22no—
84#500C2314yes—
85#500C2314yes—
86#499—28no—
87#499—28no—
88#498—24no—
89#498C2318yes—
90#476—28no—
91#476C23116yes—
92#475—216no—
93#465—232no—
94#461—28no—
95#461C23116yes—
96#460—216no—
97#436—28no—
98#436—24no—
99#436C2318yes—
100C8×D4—28no—
101C8×D4—24no—
102C8×D4C2318yes—
103#427—28no—
104#427—28no—
105C2×#107—21no—
106C2×#107—21no—
107C2×#107—21no—
108C2×#107—21no—
109C2×#106—22no—
110C2×#106—22no—
111C2×#106—24no—
112#423—22no—
113#423C2314yes—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—24no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2314yes—
119C2×C4⋊C8C2318yes—
120C2×#103—24no—
121C2×#102—24no—
122#408—28no—
123C2×C22⋊C8—22no—
124C2×C22⋊C8—22no—
125C2×C4×C8D411yescharacteristic
126C2×C4×C8D411yescharacteristic
127C2×C4×C8D414yes—
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2314yes—
131#334C2318yes—
132#334C2318yes—
133#333C23116yes—
134#332C23116yes—
135C2×#41—24no—
136C2×#41—24no—
137C2×#41—44no—
138C2×#41—44no—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4C2×D412yes—
146C23×C4—22no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4#4111yescharacteristic
150C2×SD16—216no—
151C2×SD16—88no—
152#131—432no—
153C2×M16—24no—
154C2×M16—28no—
155C2×M16—44no—
156C2×M16—48no—
157C22×C8—24no—
158C22×C8—22no—
159C22×C8—24no—
160C22×C8—28no—
161C22×C8—28no—
162C22×C8—48no—
163C22×C8—84no—
164C22×C8SD1618yescomplemented
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D414yes—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D414yes—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8C2×D414yes—
179C4⋊Q8C2414yes—
180C4⋊Q8C2414yes—
181#127—22no—
182#127C2×D414yes—
183#125—28no—
184#125—44no—
185#124—24no—
186#124—24no—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C22⋊Q8—44no—
190C22⋊Q8—44no—
191#121—24no—
192#121—24no—
193#121—24no—
194#121C2×D418yes—
195C4×Q8—28no—
196C4×Q8—28no—
197C4×Q8—44no—
198C4×Q8—44no—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4C2×D414yes—
204C4×D4C2414yes—
205#117—28no—
206#117—44no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C2×D412yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—42no—
218C2×C42—42no—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42#4111yescharacteristic
224C2×C42#4112yes—
225C2×C42C2411yescharacteristic
226C2×C42C2412yes—
227#107—24no—
228#107—24no—
229#107—24no—
230#107—24no—
231#106—28no—
232#106—28no—
233#106—216no—
234C4⋊C8—28no—
235C4⋊C8—24no—
236C4⋊C8—24no—
237C4⋊C8—22no—
238C4⋊C8C2414yes—
239C4⋊C8C2418yes—
240#103—216no—
241#102—216no—
242C22⋊C8—28no—
243C22⋊C8—28no—
244C4×C8C2×D412yes—
245C4×C8C2×D414yes—
246C4×C8C2×D414yes—
247C4×C8C2×D412yes—
248C4×C8C2412yes—
249C4×C8C2412yes—
250C24—22no—
251C24#12111yescharacteristic
252#41—48no—
253#41—48no—
254#41—416no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—84no—
264C2×D4—28no—
265C2×D4—22no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—28no—
269C2×D4—44no—
270C2×D4—48no—
271C2×D4—44no—
272C2×D4C22×D414yes—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4#12112yes—
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4#12112yes—
286C22×C4#12112yes—
287C22×C4#12111yescharacteristic
288C22×C4C2×SD1614yes—
289C22×C4#13714yes—
290C22×C4C22×D412yes—
291C22×C4C22×D412yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4112yes—
296SD16—816no—
297M16—48no—
298M16—416no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—28no—
302C2×C8—24no—
303C2×C8—28no—
304C2×C8—44no—
305C2×C8—48no—
306C2×C8—416no—
307C2×C8—88no—
308C2×C8C2×SD1614yes—
309C2×C8C2×SD1614yes—
310C2×C8C2×SD16116yescomplemented
311C2×C8C22×D412yes—
312C2×C8C22×D412yes—
313C2×C8C22×D412yes—
314C2×C8C22×D412yes—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—48no—
326C4⋊C4—48no—
327C4⋊C4C22×D414yes—
328C22⋊C4—28no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—48no—
332C42—48no—
333C42#12112yes—
334C42#12112yes—
335C42#12112yes—
336C42C22×D412yes—
337C42C22×D411yescharacteristic
338C42C22×D411yescharacteristic
339C42C22×D412yes—
340C42C2×#4111yescharacteristic
341C42C2×#4112yes—
342C42C2511yescharacteristic
343C23—22no—
344C23—22no—
345C23—24no—
346C23—44no—
347C23—44no—
348C23—44no—
349C23#46114yes—
350C23C2×#12112yes—
351C23C2×#12111yescharacteristic
352Q8—48no—
353Q8—44no—
354Q8—44no—
355Q8—84no—
356Q8—84no—
357Q8—84no—
358D4—28no—
359D4—48no—
360D4—416no—
361D4—48no—
362D4—44no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—48no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4#46114yes—
380C2×C4#46118yes—
381C2×C4C2×#12111yescharacteristic
382C2×C4C2×#12112yes—
383C2×C4C2×#12112yes—
384C2×C4C2×#12112yes—
385C2×C4C2×#12111yescharacteristic
386C2×C4C2×#12111yescharacteristic
387C2×C4C22×SD1612yes—
388C2×C4C22×SD1614yes—
389C2×C4C2×#13712yes—
390C2×C4C2×#13714yes—
391C2×C4#57714yes—
392C2×C4C23×D411yescharacteristic
393C2×C4C23×D411yescharacteristic
394C2×C4C22×#4111yescharacteristic
395C8—22no—
396C8—22no—
397C8—44no—
398C8—48no—
399C8C22×SD1614yes—
400C8C22×SD1614yes—
401C22—22no—
402C22—22no—
403C22—28no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22—88no—
408C22C2×#46114yes—
409C22C2×#46112yes—
410C22#306012yes—
411C22#306712yes—
412C22C22×#12111yescharacteristic
413C4—22no—
414C4—22no—
415C4—21no—
416C4—21no—
417C4—44no—
418C4—84no—
419C4—84no—
420C4C2×#46112yes—
421C4C2×#46114yes—
422C4#306012yes—
423C4C23×SD1611yescharacteristic
424C4C22×#13711yescharacteristic
425C4C2×#57712yes—
426C2—24no—
427C2—84no—
428C2#1919414yesdirect product factor
429C2C22×#46111yescharacteristic
430C2C2×#306011yescharacteristic
431C2C2×#306711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC42C25
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3160C2semidirect
4. #19194C2direct (N × H)
5. C2×#1695C2semidirect
6. C2×#1693C2semidirect
7. C2×#1671C2semidirect
8. C22×#334C2semidirect
9. C2×#1550C2semidirect
16. #3160C22semidirect
39. #1695C22semidirect
40. #1693C22semidirect
41. #1671C22semidirect
44. C2×#334C22semidirect
45. #1550C22semidirect
164. C22×C8SD16semidirect
310. C2×C8C2×SD16semidirect
428. C2#19194direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32462
C2×#31602
C2×#29312
#191942
C2×#16952
C2×#16932
C2×#16712
C22×#3342
C2×#15502