Finite Group Databasebeta

← #46824535

Subgroups — #46824535

Order 34432.

1492 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575501C211yesmaximal, characteristic, complemented
2C22×Q8×D269C211yesmaximal, characteristic, complemented
3#31771270C213yesmaximal, complemented
4Q8×C22⋊D269C211yesmaximal, characteristic, complemented
5#31771268C213yesmaximal, complemented
6#31771267C211yesmaximal, characteristic, complemented
7#31771266C211yesmaximal, characteristic, complemented
8#31771265C213yesmaximal, complemented
9#31771264C213yesmaximal, complemented
10#31771180C213yesmaximal, complemented
11#31771178C213yesmaximal, complemented
12#31771118C213yesmaximal, complemented
13#31771114C213yesmaximal, complemented
14#31771081C211yesmaximal, characteristic, complemented
15D4×Q8×C269C211yesmaximal, characteristic, complemented
16#22575501—28no—
17C2×#16864403C2213yes—
18C2×#16864403C2211yescharacteristic, complemented
19C2×#16864402C2213yes—
20C2×#16864402C2213yescomplemented
21C22×Dic538C2213yes—
22C2×Q8×D269—28no—
23C2×Q8×D269C2213yes—
24C2×Q8×D269C2213yes—
25C2×Q8×D269C2211yescharacteristic
26C2×Q8×D269C2211yescharacteristic
27C2×Q8×D269C2213yescomplemented
28C22×C4×D269C2213yes—
29#22575452C2216yescomplemented
30#22575451C2213yescomplemented
31#22575442C2213yescomplemented
32#22575441C2213yescomplemented
33#22575439C2216yescomplemented
34#22575438C2216yes—
35#22575438C2213yes—
36#22575435C2213yes—
37#22575433C2213yescomplemented
38#22575432C2213yescomplemented
39#22575431C2213yes—
40#22575431C2216yescomplemented
41#22575431C2216yescomplemented
42#22575430C2213yes—
43#22575429C2213yes—
44#22575429C2213yescomplemented
45#22575428C2213yes—
46#22575427C2213yes—
47#22575426C2213yes—
48#22575425C2213yes—
49#22575423C2213yes—
50#22575421C2211yescharacteristic
51Q8×Dic269C2211yescharacteristic, complemented
52C4×Dic538C2213yescomplemented
53#22575418C2213yescomplemented
54C4×D1076C2213yes—
55#22575415C2216yes—
56C4×C22⋊D269C2213yes—
57C4×C22⋊D269C2213yes—
58#22575413C2216yes—
59#22575411C2213yes—
60#22575410C2213yes—
61C22×Q8×C269C2211yescharacteristic
62C22×Q8×C269C2211yescharacteristic, complemented
63C4⋊Q8×C269C2213yescomplemented
64C22⋊Q8×C269C2213yes—
65C22⋊Q8×C269C2213yescomplemented
66C4×Q8×C269C2211yescharacteristic, complemented
67C4×D4×C269C2213yes—
68C23×D269C2311yescharacteristic
69#16864403—212no—
70#16864403—24no—
71#16864402—212no—
72#16864402—212no—
73C2×Dic538—212no—
74C2×Dic538C2313yes—
75C2×Dic538C2316yes—
76C2×Dic538C2313yes—
77C2×Dic538C2313yes—
78Q8×D269—212no—
79Q8×D269—212no—
80Q8×D269—212no—
81Q8×D269—24no—
82Q8×D269D418yescomplemented
83C2×D1076C2313yes—
84C2×D1076C2313yes—
85C2×D1076C2313yes—
86C2×C22⋊D269C2313yes—
87C2×C22⋊D269C2311yescharacteristic
88C2×C22⋊D269C2311yescharacteristic
89C2×C22⋊D269C2313yes—
90C22×Dic269C2313yes—
91C2×C4×D269—212no—
92C2×C4×D269C2316yes—
93C2×C4×D269C2313yes—
94C2×C4×D269C2313yes—
95C2×C4×D269C2313yes—
96C2×C4×D269C2313yes—
97C2×C4×D269C2313yes—
98C4⋊Dic269C2313yes—
99C4⋊Dic269C2313yes—
100#16864381C2313yes—
101#16864381C2313yes—
102#16864381C2316yes—
103#16864381C2313yes—
104#16864381C2313yes—
105C22⋊Dic269C2311yescharacteristic
106C22⋊Dic269C2313yes—
107#16864379C2311yescharacteristic
108#16864379C2313yes—
109#16864379C2316yes—
110#16864379C2313yes—
111#16864379C2313yes—
112#16864379C2316yes—
113#16864379C2311yescharacteristic
114#16864379C2313yes—
115C4×Dic269C2313yes—
116C4×Dic269C2316yes—
117C2×Q8×C269—24no—
118C2×Q8×C269—24no—
119C2×Q8×C269C2313yes—
120C2×Q8×C269C2313yes—
121C2×Q8×C269C2311yescharacteristic
122C2×D4×C269C2311yescharacteristic
123C22×C4×C269C2313yes—
124C22×C4×C269C2313yes—
125C4⋊C4×C269C2313yes—
126C4⋊C4×C269C2316yes—
127C4⋊C4×C269C2313yes—
128C22⋊C4×C269C2313yes—
129C22⋊C4×C269C2313yes—
130C42×C269C2313yes—
131C22×D269—22no—
132C22×D269—22no—
133C22×D269—22no—
134C22×D269C2413yes—
135C22×D269C2411yescharacteristic
136C22×D269C2411yescharacteristic
137Dic538—26no—
138Dic538—212no—
139Dic538—26no—
140Dic538C2×D4112yes—
141D1076—26no—
142D1076—26no—
143D1076—26no—
144C22⋊D269—26no—
145C22⋊D269—22no—
146C22⋊D269—22no—
147C22⋊D269—26no—
148C2×Dic269—26no—
149C2×Dic269C2411yescharacteristic
150C2×Dic269C2413yes—
151C2×Dic269C2413yes—
152C2×Dic269C2411yescharacteristic
153C2×Dic269C2413yes—
154C4×D269—26no—
155C4×D269—26no—
156C4×D269—26no—
157C4×D269—26no—
158C4×D269—212no—
159C4×D269C2×D4112yes—
160C23×C269C2411yescharacteristic
161C23×C269C2411yescharacteristic
162Q8×C269—26no—
163Q8×C269—26no—
164Q8×C269C2×D414yes—
165D4×C269—22no—
166C2×C4×C269—26no—
167C2×C4×C269—26no—
168C2×C4×C269C2411yescharacteristic
169C2×C4×C269C2413yes—
170C2×C4×C269C2413yes—
171C2×C4×C269C2413yes—
172C2×C4×C269C2413yes—
173C22×C269—21no—
174C22×C269—21no—
175C22×C269—21no—
176C22×C269—21no—
177C22×C269—21no—
178C22×C2692−1+412yescomplemented
179C22×C269C2511yescharacteristic
180C2×D269—28no—
181C2×D269—21no—
182C2×D269—21no—
183C2×D269—21no—
184C2×D269—21no—
185C2×D269—23no—
186C2×D269—23no—
187C2×D269—23no—
188C2×D269C22×D412yes—
189C2×D2692−1+412yescomplemented
190C4×C269—23no—
191C4×C269—23no—
192C4×C269—21no—
193C4×C269—23no—
194C4×C269C22×D416yes—
195Dic269—21no—
196Dic269—23no—
197Dic269—21no—
198Dic269—23no—
199Dic269C22×D416yes—
200C2×C269—22no—
201C2×C269—41no—
202C2×C269C23×D411yescharacteristic
203C2×C269C2×2−1+411yescharacteristic
204C2×C269C2×2−1+411yescharacteristic
205D269—22no—
206D269—22no—
207D269—43no—
208C269#348611yescharacteristic, complemented, Sylow
209#3486—2691nomaximal, Sylow
210C2×2−1+4—2691no—
211C23×Q8—2691no—
212D4×Q8—2691no—
213D4×Q8—2693no—
214D4×Q8—2693no—
215D4×Q8D26911yescharacteristic, complemented
216#548—2691no—
217#548—2693no—
218#548—2691no—
219#548—2693no—
220#536—2693no—
221#536—2693no—
222#525—2693no—
223#525—2693no—
224#519—2691no—
2252−1+4—5388no—
226C2×#41—2693no—
227C2×#41—2691no—
228C2×#41—2693no—
229C2×#41—2693no—
230C22×Q8—2691no—
231C22×Q8—2693no—
232C22×Q8—2693no—
233C22×Q8—2691no—
234C22×Q8—2693no—
235C22×Q8—2693no—
236C22×Q8—5388no—
237C22×Q8C2×D26911yescharacteristic
238C22×Q8C2×D26911yescharacteristic, complemented
239C23×C4—2693no—
240C4⋊Q8—2693no—
241C4⋊Q8—2696no—
242C4⋊Q8C2×D26913yescomplemented
243#124—2693no—
244#124—2696no—
245#124—2693no—
246#123—2696no—
247#123—2693no—
248#123—2693no—
249C22⋊Q8—2693no—
250C22⋊Q8—2693no—
251C22⋊Q8—2693no—
252C22⋊Q8—2693no—
253C22⋊Q8—2696no—
254C22⋊Q8—2696no—
255C22⋊Q8—2693no—
256C22⋊Q8—2693no—
257C22⋊Q8C2×D26913yes—
258C22⋊Q8C2×D26913yescomplemented
259#121—2693no—
260#121—2693no—
261#121—2693no—
262#121—2693no—
263#120—2691no—
264#120—2693no—
265C4×Q8—2691no—
266C4×Q8—2693no—
267C4×Q8—2693no—
268C4×Q8C2×D26911yescharacteristic, complemented
269C4×D4—2693no—
270C4×D4—2693no—
271C4×D4—2696no—
272C4×D4—2696no—
273C4×D4—2693no—
274C4×D4C2×D26913yes—
275#117—2693no—
276#117—2693no—
277C24—2691no—
278#41—53812no—
279#41—53812no—
280#41—53812no—
281#41—5384no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—2698no—
285C2×Q8—2693no—
286C2×Q8—2696no—
287C2×Q8—2693no—
288C2×Q8—2693no—
289C2×Q8—53812no—
290C2×Q8—53812no—
291C2×Q8—53812no—
292C2×Q8—53812no—
293C2×Q8—5384no—
294C2×Q8C22×D26913yes—
295C2×Q8C22×D26913yes—
296C2×Q8C22×D26911yescharacteristic
297C2×D4—2691no—
298C2×D4—2693no—
299C2×D4—2691no—
300C2×D4—2693no—
301C2×D4—2693no—
302C2×D4—2693no—
303C2×D4—2693no—
304C2×D4C22×D26911yescharacteristic
305C22×C4—2693no—
306C22×C4—2693no—
307C22×C4—2693no—
308C22×C4—2693no—
309C22×C4—2696no—
310C22×C4—2693no—
311C22×C4—2693no—
312C22×C4—53812no—
313C22×C4C22×D26913yes—
314C22×C4C22×D26913yes—
315C4⋊C4—2693no—
316C4⋊C4—2693no—
317C4⋊C4—2693no—
318C4⋊C4—2693no—
319C4⋊C4—2696no—
320C4⋊C4—2693no—
321C4⋊C4—2693no—
322C4⋊C4C22×D26913yes—
323C4⋊C4C22×D26916yes—
324C4⋊C4C22×D26913yes—
325C22⋊C4—2691no—
326C22⋊C4—2693no—
327C22⋊C4—2693no—
328C22⋊C4—2691no—
329C22⋊C4—2693no—
330C22⋊C4—2691no—
331C22⋊C4—2693no—
332C22⋊C4—2696no—
333C22⋊C4—2696no—
334C22⋊C4—2693no—
335C22⋊C4C22×D26913yes—
336C22⋊C4C22×D26913yes—
337C42—2696no—
338C42—2693no—
339C42C22×D26913yes—
340C23—2691no—
341C23—2691no—
342C23—2693no—
343C23—5382no—
344C23—5382no—
345C23—5382no—
346C23C23×D26911yescharacteristic
347C23C23×D26911yescharacteristic
348Q8—26no—
349Q8—26no—
350Q8—26912no—
351Q8—53812no—
352Q8—5386no—
353Q8—5386no—
354Q8D4×D26914yes—
355D4—22no—
356D4—5386no—
357D4—5386no—
358D4—5386no—
359D4—5382no—
360D4—5382no—
361D4—5386no—
362D4—5386no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—2693no—
366C2×C4—26912no—
367C2×C4—2691no—
368C2×C4—2693no—
369C2×C4—2693no—
370C2×C4—2691no—
371C2×C4—5386no—
372C2×C4—53812no—
373C2×C4—5386no—
374C2×C4—5386no—
375C2×C4—5386no—
376C2×C4—5386no—
377C2×C4C23×D26913yes—
378C2×C4C23×D26911yescharacteristic
379C2×C4C23×D26913yes—
380C2×C4C23×D26913yes—
381C2×C4C23×D26913yes—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—2692no—
388C22—2692no—
389C22—5388no—
390C22—5381no—
391C22—5381no—
392C22—5381no—
393C22—5381no—
394C22—5383no—
395C22—5383no—
396C22—5383no—
397C22#2257550112yescomplemented
398C22C24×D26911yescharacteristic
399C4—23no—
400C4—23no—
401C4—23no—
402C4—21no—
403C4—2696no—
404C4—5381no—
405C4—5383no—
406C4—5381no—
407C4—5383no—
408C4C2×D4×D26916yes—
409C2—22no—
410C2—41no—
411C2—5382no—
412C2—5382no—
413C2—10763no—
414C2C22×D4×D26911yescharacteristic
415C22−1+4×D26911yescharacteristic
416C2C2×#2257550111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D269
commutator, socle, cosocleC22×C269C25
FittingD4×Q8×C269C2
hypercenterD4×Q8D269
hypocenterC269#3486
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#22575501C2semidirect
2. C22×Q8×D269C2semidirect
3. #31771270C2semidirect
4. Q8×C22⋊D269C2semidirect
5. #31771268C2semidirect
6. #31771267C2semidirect
7. #31771266C2semidirect
8. #31771265C2semidirect
9. #31771264C2semidirect
10. #31771180C2semidirect
11. #31771178C2semidirect
12. #31771118C2semidirect
13. #31771114C2semidirect
14. #31771081C2semidirect
15. D4×Q8×C269C2semidirect
18. C2×#16864403C22semidirect
20. C2×#16864402C22semidirect
27. C2×Q8×D269C22semidirect
29. #22575452C22semidirect
30. #22575451C22semidirect
31. #22575442C22semidirect
32. #22575441C22semidirect
33. #22575439C22semidirect
37. #22575433C22semidirect
38. #22575432C22semidirect
40. #22575431C22semidirect
41. #22575431C22semidirect
44. #22575429C22semidirect
51. Q8×Dic269C22semidirect
52. C4×Dic538C22semidirect
53. #22575418C22semidirect
62. C22×Q8×C269C22semidirect
63. C4⋊Q8×C269C22semidirect
65. C22⋊Q8×C269C22semidirect
66. C4×Q8×C269C22semidirect
82. Q8×D269D4semidirect
178. C22×C2692−1+4semidirect
189. C2×D2692−1+4semidirect
208. C269#3486semidirect
215. D4×Q8D269semidirect
238. C22×Q8C2×D269semidirect
242. C4⋊Q8C2×D269semidirect
258. C22⋊Q8C2×D269semidirect
268. C4×Q8C2×D269semidirect
397. C22#22575501semidirect

Maximal subgroups

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

subgroupindex
C2×#225755012
C22×Q8×D2692
#317712702
Q8×C22⋊D2692
#317712682
#317712672
#317712662
#317712652
#317712642
#317711802
#317711782
#317711182
#317711142
#317710812
D4×Q8×C2692
#3486269