Finite Group Databasebeta

← #46845911

Subgroups — #46845911

Order 34688.

472 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576192C211yesmaximal, characteristic, complemented
2C2×#22576183C211yesmaximal, characteristic
3#31774472C211yesmaximal, characteristic, complemented
4#31774379C211yesmaximal, characteristic, complemented
5#31774121C211yesmaximal, characteristic, complemented
6#31774119C211yesmaximal, characteristic, complemented
7#31774115C211yesmaximal, characteristic, complemented
8#31774113C211yesmaximal, characteristic, complemented
9#31774105C211yesmaximal, characteristic, complemented
10#31774101C211yesmaximal, characteristic, complemented
11#31774099C211yesmaximal, characteristic, complemented
12#31773922C211yesmaximal, characteristic, complemented
13#31773897C211yesmaximal, characteristic, complemented
14#103×D271C211yesmaximal, characteristic, complemented
15#462×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864655C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic
19#22576192—24no—
20#22576183—24no—
21C2×#16864644C2211yescharacteristic
22C2×#16864643C2211yescharacteristic
23C2×#16864641C2211yescharacteristic
24C2×#16864641C2211yescharacteristic
25C2×#16864641C2211yescharacteristic
26C2×D2168C2211yescharacteristic
27C2×#16864637C2211yescharacteristic
28C2×C8×D271C2211yescharacteristic
29#22576158C2211yescharacteristic
30#22576149C2211yescharacteristic
31#22576148C2211yescharacteristic
32#22576145—21no—
33#22576145—21no—
34#22576138—21no—
35#22576138—21no—
36Q8×Dic271C2211yescharacteristic
37C4×D1084—21no—
38C4×D1084C2211yescharacteristic
39#22576122—21no—
40#22576122C2211yescharacteristic
41#22576117—21no—
42#22576117C2211yescharacteristic
43C4⋊C4×D271—21no—
44C4⋊C4×D271C2211yescharacteristic
45C4⋊(C271⋊C8)C2211yescharacteristic
46Dic271⋊C8C2211yescharacteristic
47Dic271⋊C8C2211yescharacteristic
48#22576085C2211yescharacteristic
49#22576084C2211yescharacteristic
50#22576084C2211yescharacteristic
51#22576083C2211yescharacteristic
52#22576082C2211yescharacteristic
53#22576080C2211yescharacteristic
54C2×Q16×C271C2211yescharacteristic
55C2×Q16×C271C2211yescharacteristic
56C4⋊Q8×C271C2211yescharacteristic
57C4×Q8×C271C2211yescharacteristic
58C4⋊C8×C271C2211yescharacteristic
59#103×C271C2211yescharacteristic
60#103×C271C2211yescharacteristic
61#16864655—22no—
62#16864655—22no—
63#16864655—22no—
64#16864655D414yes—
65C2×Dic542—21no—
66C2×Dic542C2311yescharacteristic
67Q8×D271—22no—
68Q8×D271—22no—
69C2×D1084—21no—
70C2×D1084—21no—
71C2×D1084C2311yescharacteristic
72C2×D1084C2311yescharacteristic
73C2×C4×D271—21no—
74C2×C4×D271—21no—
75C2×C4×D271—21no—
76C2×C4×D271—21no—
77C2×C4×D271—21no—
78C2×C4×D271—21no—
79C2×C4×D271C2311yescharacteristic
80#16864644—22no—
81#16864643—22no—
82#16864641—22no—
83#16864641—22no—
84#16864641—22no—
85D2168—22no—
86#16864637—22no—
87C2×C271⋊C8C2311yescharacteristic
88C2×C271⋊C8C2311yescharacteristic
89C8×D271—22no—
90C4⋊Dic271—21no—
91C4⋊Dic271C2311yescharacteristic
92Dic271⋊C4—21no—
93Dic271⋊C4—21no—
94Dic271⋊C4—21no—
95Dic271⋊C4D411yescharacteristic
96Dic271⋊C4D411yescharacteristic
97(C2×D271)⋊C4—21no—
98(C2×D271)⋊C4—21no—
99(C2×D271)⋊C4—21no—
100(C2×D271)⋊C4—21no—
101(C2×D271)⋊C4—21no—
102(C2×D271)⋊C4—21no—
103(C2×D271)⋊C4D411yescharacteristic
104(C2×D271)⋊C4D411yescharacteristic
105C4×Dic271—21no—
106C4×Dic271C2311yescharacteristic
107C2×Q8×C271C2311yescharacteristic
108C2×Q8×C271C2311yescharacteristic
109C2×Q8×C271C2311yescharacteristic
110Q16×C271—22no—
111Q16×C271—22no—
112C2×C8×C271C2311yescharacteristic
113C2×C8×C271C2311yescharacteristic
114C4⋊C4×C271—21no—
115C4⋊C4×C271—21no—
116C4⋊C4×C271—21no—
117C4⋊C4×C271C2311yescharacteristic
118C42×C271—21no—
119C42×C271C2311yescharacteristic
120C22×D271—21no—
121C22×D271—21no—
122C22×D271C2×D411yescharacteristic
123Dic542—21no—
124Dic542—21no—
125Dic542—42no—
126D1084—24no—
127D1084—21no—
128D1084—21no—
129D1084—21no—
130D1084—42no—
131D1084C2×D412yes—
132C2×Dic271—21no—
133C2×Dic271—21no—
134C2×Dic271C2×D411yescharacteristic
135C2×Dic271C2×D411yescharacteristic
136C2×Dic271C2×D411yescharacteristic
137C4×D271—24no—
138C4×D271—42no—
139C4×D271—42no—
140C4×D271—81no—
141C4×D271—81no—
142C4×D271—81no—
143C4×D271C2×D412yes—
144C4×D271#4112yes—
145C271⋊C8—21no—
146C271⋊C8—21no—
147Q8×C271—21no—
148Q8×C271—21no—
149Q8×C271—21no—
150Q8×C271—21no—
151Q8×C271—21no—
152Q8×C271C2×D412yes—
153C2×C4×C271—21no—
154C2×C4×C271—21no—
155C2×C4×C271—21no—
156C2×C4×C271—21no—
157C2×C4×C271C2×D411yescharacteristic
158C2×C4×C271C2×D411yescharacteristic
159C2×C4×C271C2411yescharacteristic
160C8×C271—21no—
161C8×C271—21no—
162C22×C271C22×D411yescharacteristic
163C2×D271—21no—
164C2×D271—21no—
165C2×D271—22no—
166C2×D271—41no—
167C2×D271—41no—
168C2×D271—41no—
169C2×D271—41no—
170C2×D271—41no—
171C2×D271#12112yes—
172C4×C271—22no—
173C4×C271—21no—
174C4×C271—21no—
175C4×C271—41no—
176C4×C271—41no—
177C4×C271—41no—
178C4×C271C22×D411yescharacteristic
179C4×C271C2×#4111yescharacteristic
180Dic271—21no—
181Dic271—21no—
182Dic271—41no—
183Dic271—41no—
184Dic271#12112yes—
185C2×C271C2×#12111yescharacteristic
186C2×C271#57711yescharacteristic
187C2×C271#57811yescharacteristic
188D271—22no—
189D271—42no—
190D271—81no—
191C271#304611yescharacteristic, complemented, Sylow
192#3046—2711nomaximal, Sylow
193C2×#137—2711no—
194C2×#135—2711no—
195#536—2711no—
196#520—2711no—
197#464—2711no—
198#464—2711no—
199#463—2711no—
200#463—2711no—
201#462—2711no—
202#462D27111yescharacteristic, complemented
203#461—2711no—
204#461—2711no—
205#424—2711no—
206#417—2711no—
207C2×#103—2711no—
208C2×#41—2711no—
209C2×#41—2711no—
210C22×Q8—2711no—
211#137—5424no—
212#135—5424no—
213C2×Q16—2711no—
214C2×Q16C2×D27111yescharacteristic
215C2×Q16C2×D27111yescharacteristic
216C2×SD16—2711no—
217C2×SD16—2711no—
218C2×SD16—2711no—
219C2×SD16—2711no—
220C2×D8—2711no—
221C2×M16—2711no—
222C22×C8—2711no—
223C4⋊Q8—2711no—
224C4⋊Q8C2×D27111yescharacteristic
225#124—2711no—
226#124—2711no—
227#123—5421no—
228#123—5421no—
229C22⋊Q8—5421no—
230C22⋊Q8—5421no—
231C4×Q8—2711no—
232C4×Q8C2×D27111yescharacteristic
233C4×D4—2711no—
234C4×D4—2711no—
235C4×D4—5421no—
236C4×D4—5421no—
237#117—2711no—
238#117—5421no—
239C2×C4⋊C4—2711no—
240C2×C4⋊C4—5421no—
241C4⋊C8—2711no—
242C4⋊C8—2711no—
243C4⋊C8—2711no—
244C4⋊C8C2×D27111yescharacteristic
245#103—2711no—
246#103—2711no—
247#103—2711no—
248#103—2711no—
249#103C2×D27111yescharacteristic
250#103C2×D27111yescharacteristic
251#102—2711no—
252#102—2711no—
253#41—2714no—
254#41—5422no—
255#41—5422no—
256#41—5422no—
257C2×Q8—2711no—
258C2×Q8—5421no—
259C2×Q8—5422no—
260C2×Q8—5422no—
261C2×Q8C22×D27111yescharacteristic
262C2×Q8C22×D27111yescharacteristic
263C2×Q8C22×D27111yescharacteristic
264C2×D4—2711no—
265C2×D4—2711no—
266C2×D4—5421no—
267C2×D4—5421no—
268C22×C4—2711no—
269C22×C4—5421no—
270C22×C4—5421no—
271C22×C4—5421no—
272C22×C4—5421no—
273C22×C4—5421no—
274C22×C4—5421no—
275Q16—22no—
276Q16—22no—
277Q16—5422no—
278SD16—5422no—
279SD16—5422no—
280SD16—5422no—
281SD16—5422no—
282D8—5422no—
283M16—5422no—
284C2×C8—2711no—
285C2×C8—2711no—
286C2×C8—5422no—
287C2×C8C22×D27111yescharacteristic
288C2×C8C22×D27111yescharacteristic
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—2711no—
293C4⋊C4—2711no—
294C4⋊C4—2711no—
295C4⋊C4—5421no—
296C4⋊C4—5421no—
297C4⋊C4—5421no—
298C4⋊C4—5421no—
299C4⋊C4C22×D27111yescharacteristic
300C22⋊C4—2711no—
301C22⋊C4—2711no—
302C22⋊C4—5421no—
303C22⋊C4—5421no—
304C22⋊C4—5421no—
305C22⋊C4—5421no—
306C22⋊C4—5421no—
307C22⋊C4—5421no—
308C42—21no—
309C42—2711no—
310C42—5421no—
311C42C22×D27111yescharacteristic
312C23—2711no—
313C23—5421no—
314C23—5421no—
315Q8—21no—
316Q8—21no—
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—5421no—
321Q8—5421no—
322Q8—10842no—
323Q8D4×D27112yes—
324D4—2712no—
325D4—5424no—
326D4—5421no—
327D4—5421no—
328D4—5421no—
329D4—10842no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—2711no—
335C2×C4—2712no—
336C2×C4—2712no—
337C2×C4—2711no—
338C2×C4—2711no—
339C2×C4—5424no—
340C2×C4—5421no—
341C2×C4—5421no—
342C2×C4—10842no—
343C2×C4—10842no—
344C2×C4—21681no—
345C2×C4—21681no—
346C2×C4—21681no—
347C2×C4D4×D27111yescharacteristic
348C2×C4D4×D27111yescharacteristic
349C2×C4C23×D27111yescharacteristic
350C8—21no—
351C8—21no—
352C8—5421no—
353C8—5421no—
354C22—2712no—
355C22—5421no—
356C22—5421no—
357C22—5422no—
358C22—10841no—
359C22—10841no—
360C22—10841no—
361C22—10841no—
362C22—10841no—
363C22C2×D4×D27111yescharacteristic
364C4—22no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—2712no—
371C4—5421no—
372C4—5421no—
373C4—10841no—
374C4—10841no—
375C4C2×D4×D27111yescharacteristic
376C4#41×D27111yescharacteristic
377C2—5422no—
378C2—10842no—
379C2—21681no—
380C2#121×D27111yescharacteristic
381C2#3177470611yescharacteristic
382C2#3177471011yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#462×C271C2
socleC22×C271C22×D4
prime radicalC2×#22576183C2
hypercenter#462D271
hypocenterC271#3046
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. C2×#22576192C2semidirect
3. #31774472C2semidirect
4. #31774379C2semidirect
5. #31774121C2semidirect
6. #31774119C2semidirect
7. #31774115C2semidirect
8. #31774113C2semidirect
9. #31774105C2semidirect
10. #31774101C2semidirect
11. #31774099C2semidirect
12. #31773922C2semidirect
13. #31773897C2semidirect
14. #103×D271C2semidirect
15. #462×C271C2semidirect
191. C271#3046semidirect
202. #462D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761922
C2×#225761832
#317744722
#317743792
#317741212
#317741192
#317741152
#317741132
#317741052
#317741012
#317740992
#317739222
#317738972
#103×D2712
#462×C2712
#3046271