Finite Group Databasebeta

← #46850014

Subgroups — #46850014

Order 34688.

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×#22576208C211yesmaximal, characteristic, complemented
2C22×Q8×D271C211yesmaximal, characteristic, complemented
3#31774562C213yesmaximal, complemented
4Q8×C22⋊D271C211yesmaximal, characteristic, complemented
5#31774560C213yesmaximal, complemented
6#31774559C211yesmaximal, characteristic, complemented
7#31774558C211yesmaximal, characteristic, complemented
8#31774557C213yesmaximal, complemented
9#31774556C213yesmaximal, complemented
10#31774472C213yesmaximal, complemented
11#31774470C213yesmaximal, complemented
12#31774410C213yesmaximal, complemented
13#31774406C213yesmaximal, complemented
14#31774373C211yesmaximal, characteristic, complemented
15D4×Q8×C271C211yesmaximal, characteristic, complemented
16#22576208—28no—
17C2×#16864655C2213yes—
18C2×#16864655C2211yescharacteristic, complemented
19C2×#16864654C2213yes—
20C2×#16864654C2213yescomplemented
21C22×Dic542C2213yes—
22C2×Q8×D271—28no—
23C2×Q8×D271C2213yes—
24C2×Q8×D271C2213yes—
25C2×Q8×D271C2211yescharacteristic
26C2×Q8×D271C2211yescharacteristic
27C2×Q8×D271C2213yescomplemented
28C22×C4×D271C2213yes—
29#22576159C2216yescomplemented
30#22576158C2213yescomplemented
31#22576149C2213yescomplemented
32#22576148C2213yescomplemented
33#22576146C2216yescomplemented
34#22576145C2216yes—
35#22576145C2213yes—
36#22576142C2213yes—
37#22576140C2213yescomplemented
38#22576139C2213yescomplemented
39#22576138C2213yes—
40#22576138C2216yescomplemented
41#22576138C2216yescomplemented
42#22576137C2213yes—
43#22576136C2213yes—
44#22576136C2213yescomplemented
45#22576135C2213yes—
46#22576134C2213yes—
47#22576133C2213yes—
48#22576132C2213yes—
49#22576130C2213yes—
50#22576128C2211yescharacteristic
51Q8×Dic271C2211yescharacteristic, complemented
52C4×Dic542C2213yescomplemented
53#22576125C2213yescomplemented
54C4×D1084C2213yes—
55#22576122C2216yes—
56C4×C22⋊D271C2213yes—
57C4×C22⋊D271C2213yes—
58(C22⋊D271)⋊C4C2216yes—
59#22576118C2213yes—
60#22576117C2213yes—
61C22×Q8×C271C2211yescharacteristic
62C22×Q8×C271C2211yescharacteristic, complemented
63C4⋊Q8×C271C2213yescomplemented
64C22⋊Q8×C271C2213yes—
65C22⋊Q8×C271C2213yescomplemented
66C4×Q8×C271C2211yescharacteristic, complemented
67C4×D4×C271C2213yes—
68C23×D271C2311yescharacteristic
69#16864655—212no—
70#16864655—24no—
71#16864654—212no—
72#16864654—212no—
73C2×Dic542—212no—
74C2×Dic542C2313yes—
75C2×Dic542C2316yes—
76C2×Dic542C2313yes—
77C2×Dic542C2313yes—
78Q8×D271—212no—
79Q8×D271—212no—
80Q8×D271—212no—
81Q8×D271—24no—
82Q8×D271D418yescomplemented
83C2×D1084C2313yes—
84C2×D1084C2313yes—
85C2×D1084C2313yes—
86C2×C22⋊D271C2311yescharacteristic
87C2×C22⋊D271C2311yescharacteristic
88C2×C22⋊D271C2313yes—
89C2×C22⋊D271C2313yes—
90C22×Dic271C2313yes—
91C2×C4×D271—212no—
92C2×C4×D271C2313yes—
93C2×C4×D271C2313yes—
94C2×C4×D271C2313yes—
95C2×C4×D271C2316yes—
96C2×C4×D271C2313yes—
97C2×C4×D271C2313yes—
98C4⋊Dic271C2313yes—
99C4⋊Dic271C2313yes—
100Dic271⋊C4C2313yes—
101Dic271⋊C4C2313yes—
102Dic271⋊C4C2316yes—
103Dic271⋊C4C2313yes—
104Dic271⋊C4C2313yes—
105C22⋊Dic271C2311yescharacteristic
106C22⋊Dic271C2313yes—
107(C2×D271)⋊C4C2311yescharacteristic
108(C2×D271)⋊C4C2311yescharacteristic
109(C2×D271)⋊C4C2313yes—
110(C2×D271)⋊C4C2313yes—
111(C2×D271)⋊C4C2313yes—
112(C2×D271)⋊C4C2313yes—
113(C2×D271)⋊C4C2316yes—
114(C2×D271)⋊C4C2316yes—
115C4×Dic271C2316yes—
116C4×Dic271C2313yes—
117C2×Q8×C271—24no—
118C2×Q8×C271—24no—
119C2×Q8×C271C2313yes—
120C2×Q8×C271C2313yes—
121C2×Q8×C271C2311yescharacteristic
122C2×D4×C271C2311yescharacteristic
123C22×C4×C271C2313yes—
124C22×C4×C271C2313yes—
125C4⋊C4×C271C2313yes—
126C4⋊C4×C271C2313yes—
127C4⋊C4×C271C2316yes—
128C22⋊C4×C271C2313yes—
129C22⋊C4×C271C2313yes—
130C42×C271C2313yes—
131C22×D271—22no—
132C22×D271—22no—
133C22×D271—22no—
134C22×D271C2413yes—
135C22×D271C2411yescharacteristic
136C22×D271C2411yescharacteristic
137Dic542—26no—
138Dic542—212no—
139Dic542—26no—
140Dic542C2×D4112yes—
141D1084—26no—
142D1084—26no—
143D1084—26no—
144C22⋊D271—26no—
145C22⋊D271—22no—
146C22⋊D271—22no—
147C22⋊D271—26no—
148C2×Dic271—26no—
149C2×Dic271C2411yescharacteristic
150C2×Dic271C2413yes—
151C2×Dic271C2413yes—
152C2×Dic271C2411yescharacteristic
153C2×Dic271C2413yes—
154C4×D271—26no—
155C4×D271—26no—
156C4×D271—26no—
157C4×D271—26no—
158C4×D271—212no—
159C4×D271C2×D4112yes—
160C23×C271C2411yescharacteristic
161C23×C271C2411yescharacteristic
162Q8×C271—26no—
163Q8×C271—26no—
164Q8×C271C2×D414yes—
165D4×C271—22no—
166C2×C4×C271—26no—
167C2×C4×C271—26no—
168C2×C4×C271C2411yescharacteristic
169C2×C4×C271C2413yes—
170C2×C4×C271C2413yes—
171C2×C4×C271C2413yes—
172C2×C4×C271C2413yes—
173C22×C271—21no—
174C22×C271—21no—
175C22×C271—21no—
176C22×C271—21no—
177C22×C271—21no—
178C22×C2712−1+412yescomplemented
179C22×C271C2511yescharacteristic
180C2×D271—28no—
181C2×D271—21no—
182C2×D271—21no—
183C2×D271—21no—
184C2×D271—21no—
185C2×D271—23no—
186C2×D271—23no—
187C2×D271—23no—
188C2×D271C22×D412yes—
189C2×D2712−1+412yescomplemented
190C4×C271—23no—
191C4×C271—23no—
192C4×C271—21no—
193C4×C271—23no—
194C4×C271C22×D416yes—
195Dic271—21no—
196Dic271—23no—
197Dic271—21no—
198Dic271—23no—
199Dic271C22×D416yes—
200C2×C271—22no—
201C2×C271—41no—
202C2×C271C23×D411yescharacteristic
203C2×C271C2×2−1+411yescharacteristic
204C2×C271C2×2−1+411yescharacteristic
205D271—22no—
206D271—22no—
207D271—43no—
208C271#348611yescharacteristic, complemented, Sylow
209#3486—2711nomaximal, Sylow
210C2×2−1+4—2711no—
211C23×Q8—2711no—
212D4×Q8—2711no—
213D4×Q8—2713no—
214D4×Q8—2713no—
215D4×Q8D27111yescharacteristic, complemented
216#548—2711no—
217#548—2713no—
218#548—2711no—
219#548—2713no—
220#536—2713no—
221#536—2713no—
222#525—2713no—
223#525—2713no—
224#519—2711no—
2252−1+4—5428no—
226C2×#41—2713no—
227C2×#41—2711no—
228C2×#41—2713no—
229C2×#41—2713no—
230C22×Q8—2711no—
231C22×Q8—2713no—
232C22×Q8—2713no—
233C22×Q8—2711no—
234C22×Q8—2713no—
235C22×Q8—2713no—
236C22×Q8—5428no—
237C22×Q8C2×D27111yescharacteristic
238C22×Q8C2×D27111yescharacteristic, complemented
239C23×C4—2713no—
240C4⋊Q8—2713no—
241C4⋊Q8—2716no—
242C4⋊Q8C2×D27113yescomplemented
243#124—2713no—
244#124—2716no—
245#124—2713no—
246#123—2716no—
247#123—2713no—
248#123—2713no—
249C22⋊Q8—2713no—
250C22⋊Q8—2713no—
251C22⋊Q8—2713no—
252C22⋊Q8—2713no—
253C22⋊Q8—2716no—
254C22⋊Q8—2716no—
255C22⋊Q8—2713no—
256C22⋊Q8—2713no—
257C22⋊Q8C2×D27113yes—
258C22⋊Q8C2×D27113yescomplemented
259#121—2713no—
260#121—2713no—
261#121—2713no—
262#121—2713no—
263#120—2711no—
264#120—2713no—
265C4×Q8—2711no—
266C4×Q8—2713no—
267C4×Q8—2713no—
268C4×Q8C2×D27111yescharacteristic, complemented
269C4×D4—2713no—
270C4×D4—2713no—
271C4×D4—2716no—
272C4×D4—2716no—
273C4×D4—2713no—
274C4×D4C2×D27113yes—
275#117—2713no—
276#117—2713no—
277C24—2711no—
278#41—54212no—
279#41—54212no—
280#41—54212no—
281#41—5424no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—2718no—
285C2×Q8—2713no—
286C2×Q8—2716no—
287C2×Q8—2713no—
288C2×Q8—2713no—
289C2×Q8—54212no—
290C2×Q8—54212no—
291C2×Q8—54212no—
292C2×Q8—54212no—
293C2×Q8—5424no—
294C2×Q8C22×D27113yes—
295C2×Q8C22×D27113yes—
296C2×Q8C22×D27111yescharacteristic
297C2×D4—2711no—
298C2×D4—2713no—
299C2×D4—2711no—
300C2×D4—2713no—
301C2×D4—2713no—
302C2×D4—2713no—
303C2×D4—2713no—
304C2×D4C22×D27111yescharacteristic
305C22×C4—2713no—
306C22×C4—2713no—
307C22×C4—2713no—
308C22×C4—2713no—
309C22×C4—2716no—
310C22×C4—2713no—
311C22×C4—2713no—
312C22×C4—54212no—
313C22×C4C22×D27113yes—
314C22×C4C22×D27113yes—
315C4⋊C4—2713no—
316C4⋊C4—2713no—
317C4⋊C4—2713no—
318C4⋊C4—2713no—
319C4⋊C4—2716no—
320C4⋊C4—2713no—
321C4⋊C4—2713no—
322C4⋊C4C22×D27113yes—
323C4⋊C4C22×D27116yes—
324C4⋊C4C22×D27113yes—
325C22⋊C4—2711no—
326C22⋊C4—2713no—
327C22⋊C4—2713no—
328C22⋊C4—2711no—
329C22⋊C4—2713no—
330C22⋊C4—2711no—
331C22⋊C4—2713no—
332C22⋊C4—2716no—
333C22⋊C4—2716no—
334C22⋊C4—2713no—
335C22⋊C4C22×D27113yes—
336C22⋊C4C22×D27113yes—
337C42—2716no—
338C42—2713no—
339C42C22×D27113yes—
340C23—2711no—
341C23—2711no—
342C23—2713no—
343C23—5422no—
344C23—5422no—
345C23—5422no—
346C23C23×D27111yescharacteristic
347C23C23×D27111yescharacteristic
348Q8—26no—
349Q8—26no—
350Q8—27112no—
351Q8—54212no—
352Q8—5426no—
353Q8—5426no—
354Q8D4×D27114yes—
355D4—22no—
356D4—5426no—
357D4—5426no—
358D4—5426no—
359D4—5422no—
360D4—5422no—
361D4—5426no—
362D4—5426no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—2713no—
366C2×C4—27112no—
367C2×C4—2711no—
368C2×C4—2713no—
369C2×C4—2713no—
370C2×C4—2711no—
371C2×C4—5426no—
372C2×C4—54212no—
373C2×C4—5426no—
374C2×C4—5426no—
375C2×C4—5426no—
376C2×C4—5426no—
377C2×C4C23×D27113yes—
378C2×C4C23×D27111yescharacteristic
379C2×C4C23×D27113yes—
380C2×C4C23×D27113yes—
381C2×C4C23×D27113yes—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—2712no—
388C22—2712no—
389C22—5428no—
390C22—5421no—
391C22—5421no—
392C22—5421no—
393C22—5421no—
394C22—5423no—
395C22—5423no—
396C22—5423no—
397C22#2257620812yescomplemented
398C22C24×D27111yescharacteristic
399C4—23no—
400C4—21no—
401C4—23no—
402C4—23no—
403C4—2716no—
404C4—5421no—
405C4—5423no—
406C4—5421no—
407C4—5423no—
408C4C2×D4×D27116yes—
409C2—22no—
410C2—41no—
411C2—5422no—
412C2—5422no—
413C2—10843no—
414C2C22×D4×D27111yescharacteristic
415C22−1+4×D27111yescharacteristic
416C2C2×#2257620811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D271
commutator, socle, cosocleC22×C271C25
FittingD4×Q8×C271C2
hypercenterD4×Q8D271
hypocenterC271#3486
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#22576208C2semidirect
2. C22×Q8×D271C2semidirect
3. #31774562C2semidirect
4. Q8×C22⋊D271C2semidirect
5. #31774560C2semidirect
6. #31774559C2semidirect
7. #31774558C2semidirect
8. #31774557C2semidirect
9. #31774556C2semidirect
10. #31774472C2semidirect
11. #31774470C2semidirect
12. #31774410C2semidirect
13. #31774406C2semidirect
14. #31774373C2semidirect
15. D4×Q8×C271C2semidirect
18. C2×#16864655C22semidirect
20. C2×#16864654C22semidirect
27. C2×Q8×D271C22semidirect
29. #22576159C22semidirect
30. #22576158C22semidirect
31. #22576149C22semidirect
32. #22576148C22semidirect
33. #22576146C22semidirect
37. #22576140C22semidirect
38. #22576139C22semidirect
40. #22576138C22semidirect
41. #22576138C22semidirect
44. #22576136C22semidirect
51. Q8×Dic271C22semidirect
52. C4×Dic542C22semidirect
53. #22576125C22semidirect
62. C22×Q8×C271C22semidirect
63. C4⋊Q8×C271C22semidirect
65. C22⋊Q8×C271C22semidirect
66. C4×Q8×C271C22semidirect
82. Q8×D271D4semidirect
178. C22×C2712−1+4semidirect
189. C2×D2712−1+4semidirect
208. C271#3486semidirect
215. D4×Q8D271semidirect
238. C22×Q8C2×D271semidirect
242. C4⋊Q8C2×D271semidirect
258. C22⋊Q8C2×D271semidirect
268. C4×Q8C2×D271semidirect
397. C22#22576208semidirect

Maximal subgroups

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

subgroupindex
C2×#225762082
C22×Q8×D2712
#317745622
Q8×C22⋊D2712
#317745602
#317745592
#317745582
#317745572
#317745562
#317744722
#317744702
#317744102
#317744062
#317743732
D4×Q8×C2712
#3486271