Finite Group Databasebeta

← #46845915

Subgroups — #46845915

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×#22576189C211yesmaximal, characteristic, complemented
2C2×#22576179C211yesmaximal, characteristic
3#31774467C211yesmaximal, characteristic, complemented
4#31774376C211yesmaximal, characteristic, complemented
5#31774118C211yesmaximal, characteristic, complemented
6#31774116C211yesmaximal, characteristic, complemented
7#31774110C211yesmaximal, characteristic, complemented
8#31774108C211yesmaximal, characteristic, complemented
9#31774106C211yesmaximal, characteristic, complemented
10#31774102C211yesmaximal, characteristic, complemented
11#31774100C211yesmaximal, characteristic, complemented
12#31773922C211yesmaximal, characteristic, complemented
13#31773895C211yesmaximal, characteristic, complemented
14#103×D271C211yesmaximal, characteristic, complemented
15#463×C271C211yesmaximal, characteristic, complemented
16C2×#16864653C2211yescharacteristic
17C2×#16864653C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic
19#22576189—24no—
20#22576179—24no—
21C2×Dic1084C2211yescharacteristic
22C2×Dic1084C2211yescharacteristic
23C2×#16864644C2211yescharacteristic
24C2×#16864642C2211yescharacteristic
25C2×#16864642C2211yescharacteristic
26C2×#16864642C2211yescharacteristic
27C2×#16864637C2211yescharacteristic
28C2×C8×D271C2211yescharacteristic
29#22576160C2211yescharacteristic
30#22576158C2211yescharacteristic
31#22576147C2211yescharacteristic
32#22576142—21no—
33#22576142—21no—
34#22576137—21no—
35#22576137—21no—
36C4×Dic542C2211yescharacteristic
37#22576125C2211yescharacteristic
38D4×Dic271C2211yescharacteristic
39C4×C22⋊D271—21no—
40(C22⋊D271)⋊C4—21no—
41#22576117C2211yescharacteristic
42#22576116—21no—
43C2×Dic271⋊C4—21no—
44C4⋊C4×D271C2211yescharacteristic
45C4⋊(C271⋊C8)C2211yescharacteristic
46Dic271⋊C8C2211yescharacteristic
47Dic271⋊C8C2211yescharacteristic
48#22576085C2211yescharacteristic
49#22576085C2211yescharacteristic
50#22576084C2211yescharacteristic
51#22576083C2211yescharacteristic
52#22576083C2211yescharacteristic
53#22576081C2211yescharacteristic
54C2×SD16×C271C2211yescharacteristic
55C2×D8×C271C2211yescharacteristic
56#124×C271C2211yescharacteristic
57C4×D4×C271C2211yescharacteristic
58C4⋊C8×C271C2211yescharacteristic
59#103×C271C2211yescharacteristic
60#102×C271C2211yescharacteristic
61#16864653—22no—
62#16864653—22no—
63#16864653—22no—
64#16864653D414yes—
65C2×Dic542—21no—
66C2×Dic542C2311yescharacteristic
67C2×Dic542C2311yescharacteristic
68C2×Dic542C2311yescharacteristic
69Q8×D271—22no—
70Q8×D271—22no—
71C2×C22⋊D271—21no—
72C2×C22⋊D271—21no—
73C22×Dic271—21no—
74C22×Dic271—21no—
75C22×Dic271—21no—
76C2×C4×D271—21no—
77C2×C4×D271—21no—
78C2×C4×D271C2311yescharacteristic
79Dic1084—22no—
80Dic1084—22no—
81#16864644—22no—
82#16864642—22no—
83#16864642—22no—
84#16864642—22no—
85#16864637—22no—
86C2×C271⋊C8C2311yescharacteristic
87C2×C271⋊C8C2311yescharacteristic
88C8×D271—22no—
89C4⋊Dic271—21no—
90C4⋊Dic271—21no—
91C4⋊Dic271C2311yescharacteristic
92Dic271⋊C4—21no—
93Dic271⋊C4—21no—
94Dic271⋊C4—21no—
95Dic271⋊C4—21no—
96Dic271⋊C4—21no—
97Dic271⋊C4D411yescharacteristic
98Dic271⋊C4D411yescharacteristic
99C22⋊Dic271—21no—
100C22⋊Dic271—21no—
101C22⋊Dic271—21no—
102(C2×D271)⋊C4D411yescharacteristic
103(C2×D271)⋊C4D411yescharacteristic
104C4×Dic271—21no—
105C4×Dic271—21no—
106C4×Dic271C2311yescharacteristic
107C2×Q8×C271C2311yescharacteristic
108C2×D4×C271C2311yescharacteristic
109C2×D4×C271C2311yescharacteristic
110C22×C4×C271—21no—
111SD16×C271—22no—
112D8×C271—22no—
113C2×C8×C271C2311yescharacteristic
114C2×C8×C271C2311yescharacteristic
115C4⋊C4×C271C2311yescharacteristic
116C22⋊C4×C271—21no—
117C22⋊C4×C271—21no—
118C22⋊C4×C271—21no—
119C42×C271C2311yescharacteristic
120C22×D271C2×D411yescharacteristic
121Dic542—21no—
122Dic542—21no—
123Dic542—21no—
124Dic542—21no—
125Dic542—21no—
126Dic542—42no—
127Dic542C2×D412yes—
128C22⋊D271—24no—
129C22⋊D271—42no—
130C2×Dic271—21no—
131C2×Dic271—24no—
132C2×Dic271—21no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—42no—
136C2×Dic271—81no—
137C2×Dic271C2×D411yescharacteristic
138C2×Dic271C2×D411yescharacteristic
139C2×Dic271C2×D411yescharacteristic
140C4×D271—42no—
141C4×D271—81no—
142C4×D271C2×D412yes—
143C4×D271#4112yes—
144C271⋊C8—21no—
145C271⋊C8—21no—
146C23×C271—21no—
147C23×C271—21no—
148Q8×C271—21no—
149Q8×C271—21no—
150D4×C271—21no—
151D4×C271—21no—
152D4×C271—21no—
153D4×C271C2×D412yes—
154C2×C4×C271—21no—
155C2×C4×C271—21no—
156C2×C4×C271—81no—
157C2×C4×C271C2×D411yescharacteristic
158C2×C4×C271C2×D411yescharacteristic
159C2×C4×C271C2411yescharacteristic
160C8×C271—21no—
161C8×C271—21no—
162C22×C271—22no—
163C22×C271—41no—
164C22×C271—41no—
165C22×C271—41no—
166C22×C271—41no—
167C22×C271—41no—
168C22×C271C22×D411yescharacteristic
169C2×D271—21no—
170C2×D271—21no—
171C2×D271#12112yes—
172C4×C271—21no—
173C4×C271—21no—
174C4×C271—41no—
175C4×C271—41no—
176C4×C271C22×D411yescharacteristic
177C4×C271C2×#4111yescharacteristic
178Dic271—22no—
179Dic271—21no—
180Dic271—21no—
181Dic271—41no—
182Dic271—41no—
183Dic271—41no—
184Dic271#12112yes—
185C2×C271—42no—
186C2×C271—81no—
187C2×C271C2×#12111yescharacteristic
188C2×C271#57711yescharacteristic
189C2×C271#57811yescharacteristic
190D271—22no—
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#463D27111yescharacteristic, complemented
201#462—2711no—
202#462—2711no—
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×Q16—2711no—
215C2×Q16—2711no—
216C2×SD16—2711no—
217C2×SD16—2711no—
218C2×SD16—2711no—
219C2×SD16C2×D27111yescharacteristic
220C2×D8C2×D27111yescharacteristic
221C2×M16—2711no—
222C22×C8—2711no—
223C4⋊Q8—2711no—
224C4⋊Q8—2711no—
225#124—2711no—
226#124C2×D27111yescharacteristic
227#123—5421no—
228#123—5421no—
229C22⋊Q8—5421no—
230C22⋊Q8—5421no—
231C4×Q8—2711no—
232C4×Q8—2711no—
233C4×D4—2711no—
234C4×D4—5421no—
235C4×D4—5421no—
236C4×D4C2×D27111yescharacteristic
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#103—2711no—
250#103C2×D27111yescharacteristic
251#102—2711no—
252#102C2×D27111yescharacteristic
253#41—2714no—
254#41—5422no—
255#41—5422no—
256#41—5422no—
257C2×Q8—2711no—
258C2×Q8—2711no—
259C2×Q8—2711no—
260C2×Q8—5421no—
261C2×Q8—5422no—
262C2×Q8—5422no—
263C2×Q8C22×D27111yescharacteristic
264C2×D4—5421no—
265C2×D4—5421no—
266C2×D4C22×D27111yescharacteristic
267C2×D4C22×D27111yescharacteristic
268C22×C4—21no—
269C22×C4—2711no—
270C22×C4—5421no—
271C22×C4—5421no—
272C22×C4—5421no—
273C22×C4—5421no—
274C22×C4—5421no—
275Q16—5422no—
276Q16—5422no—
277Q16—5422no—
278SD16—22no—
279SD16—5422no—
280SD16—5422no—
281SD16—5422no—
282D8—22no—
283M16—5422no—
284C2×C8—2711no—
285C2×C8—2711no—
286C2×C8—5422no—
287C2×C8C22×D27111yescharacteristic
288C2×C8C22×D27111yescharacteristic
289C4⋊C4—2711no—
290C4⋊C4—2711no—
291C4⋊C4—2711no—
292C4⋊C4—5421no—
293C4⋊C4—5421no—
294C4⋊C4—5421no—
295C4⋊C4—5421no—
296C4⋊C4—5421no—
297C4⋊C4—5421no—
298C4⋊C4—5421no—
299C4⋊C4C22×D27111yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—2711no—
304C22⋊C4—2711no—
305C22⋊C4—5421no—
306C22⋊C4—5421no—
307C22⋊C4—5421no—
308C42—2711no—
309C42—5421no—
310C42—5421no—
311C42C22×D27111yescharacteristic
312C23—21no—
313C23—21no—
314C23—2711no—
315Q8—21no—
316Q8—21no—
317Q8—2712no—
318Q8—5421no—
319Q8—5421no—
320Q8—5421no—
321Q8—5421no—
322Q8—5421no—
323Q8—10842no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—5424no—
328D4—10842no—
329D4D4×D27112yes—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—2711no—
334C2×C4—2712no—
335C2×C4—2712no—
336C2×C4—2711no—
337C2×C4—2711no—
338C2×C4—5424no—
339C2×C4—5421no—
340C2×C4—5421no—
341C2×C4—5421no—
342C2×C4—5421no—
343C2×C4—10842no—
344C2×C4—10842no—
345C2×C4—21681no—
346C2×C4—21681no—
347C2×C4D4×D27111yescharacteristic
348C2×C4D4×D27111yescharacteristic
349C2×C4C23×D27111yescharacteristic
350C8—21no—
351C8—21no—
352C8—5421no—
353C8—5421no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—2712no—
361C22—5421no—
362C22—5421no—
363C22C2×D4×D27111yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—2712no—
369C4—5422no—
370C4—5421no—
371C4—5421no—
372C4—10841no—
373C4—10841no—
374C4—10841no—
375C4C2×D4×D27111yescharacteristic
376C4#41×D27111yescharacteristic
377C2—42no—
378C2—81no—
379C2—5422no—
380C2#121×D27111yescharacteristic
381C2#3177470711yescharacteristic
382C2#3177471111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#463×C271C2
socleC22×C271C22×D4
prime radicalC2×#22576179C2
hypercenter#463D271
hypocenterC271#3046
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#22576189C2semidirect
3. #31774467C2semidirect
4. #31774376C2semidirect
5. #31774118C2semidirect
6. #31774116C2semidirect
7. #31774110C2semidirect
8. #31774108C2semidirect
9. #31774106C2semidirect
10. #31774102C2semidirect
11. #31774100C2semidirect
12. #31773922C2semidirect
13. #31773895C2semidirect
14. #103×D271C2semidirect
15. #463×C271C2semidirect
191. C271#3046semidirect
200. #463D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761892
C2×#225761792
#317744672
#317743762
#317741182
#317741162
#317741102
#317741082
#317741062
#317741022
#317741002
#317739222
#317738952
#103×D2712
#463×C2712
#3046271