Finite Group Databasebeta

← #46845889

Subgroups — #46845889

Order 34688.

458 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576192C211yesmaximal, characteristic
2C2×Q16×D271C211yesmaximal, characteristic, complemented
3#31774472C211yesmaximal, characteristic
4#31774373C211yesmaximal, characteristic, complemented
5#31774122C211yesmaximal, characteristic, complemented
6#31774121C211yesmaximal, characteristic, complemented
7#31774120C211yesmaximal, characteristic, complemented
8#31774119C211yesmaximal, characteristic, complemented
9#31774106C211yesmaximal, characteristic, complemented
10#31774105C211yesmaximal, characteristic, complemented
11#31774104C211yesmaximal, characteristic, complemented
12#31773922C211yesmaximal, characteristic, complemented
13#31773904C211yesmaximal, characteristic, complemented
14#31773892C211yesmaximal, characteristic
15#462×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×Q8×D271C2211yescharacteristic
18C2×Q8×D271C2211yescharacteristic
19#22576192—24no—
20C2×Dic1084C2211yescharacteristic
21C2×#16864644C2211yescharacteristic
22C2×#16864644C2211yescharacteristic
23C2×#16864644C2211yescharacteristic
24Q16×D271—24no—
25C2×#16864643C2211yescharacteristic
26C2×#16864641C2211yescharacteristic
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×Dic542—21no—
38C4×Dic542C2211yescharacteristic
39#22576125—21no—
40#22576125C2211yescharacteristic
41#22576118—21no—
42#22576118C2211yescharacteristic
43#22576117—21no—
44#22576117C2211yescharacteristic
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—
63C2×Dic542—21no—
64C2×Dic542—21no—
65C2×Dic542C2311yescharacteristic
66C2×Dic542C2311yescharacteristic
67Q8×D271—22no—
68Q8×D271—22no—
69Q8×D271—22no—
70Q8×D271D414yes—
71C2×D1084—21no—
72C2×D1084C2311yescharacteristic
73C2×C4×D271—21no—
74C2×C4×D271—21no—
75C2×C4×D271—21no—
76C2×C4×D271C2311yescharacteristic
77Dic1084—22no—
78#16864644—22no—
79#16864644—22no—
80#16864644—22no—
81#16864643—22no—
82#16864641—22no—
83#16864637—22no—
84C2×C271⋊C8C2311yescharacteristic
85C2×C271⋊C8C2311yescharacteristic
86C8×D271—22no—
87C4⋊Dic271—21no—
88C4⋊Dic271C2311yescharacteristic
89Dic271⋊C4—21no—
90Dic271⋊C4—21no—
91Dic271⋊C4—21no—
92Dic271⋊C4—21no—
93Dic271⋊C4D411yescharacteristic
94Dic271⋊C4D411yescharacteristic
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4—21no—
97(C2×D271)⋊C4—21no—
98(C2×D271)⋊C4—21no—
99(C2×D271)⋊C4—21no—
100(C2×D271)⋊C4D411yescharacteristic
101(C2×D271)⋊C4D411yescharacteristic
102C4×Dic271—21no—
103C4×Dic271—21no—
104C4×Dic271—21no—
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×D271C2×D411yescharacteristic
122Dic542—21no—
123Dic542—24no—
124Dic542—21no—
125Dic542—21no—
126Dic542—42no—
127Dic542C2×D412yes—
128D1084—21no—
129D1084—21no—
130D1084—42no—
131C2×Dic271—21no—
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×D271C2×D412yes—
141C4×D271#4112yes—
142C271⋊C8—21no—
143C271⋊C8—21no—
144Q8×C271—21no—
145Q8×C271—21no—
146Q8×C271—21no—
147Q8×C271—21no—
148Q8×C271—21no—
149Q8×C271C2×D412yes—
150C2×C4×C271—21no—
151C2×C4×C271—21no—
152C2×C4×C271—21no—
153C2×C4×C271—21no—
154C2×C4×C271C2×D411yescharacteristic
155C2×C4×C271C2×D411yescharacteristic
156C2×C4×C271C2411yescharacteristic
157C8×C271—21no—
158C8×C271—21no—
159C22×C271C22×D411yescharacteristic
160C2×D271—21no—
161C2×D271—21no—
162C2×D271—41no—
163C2×D271—41no—
164C2×D271—41no—
165C2×D271#12112yes—
166C4×C271—22no—
167C4×C271—21no—
168C4×C271—21no—
169C4×C271—41no—
170C4×C271—41no—
171C4×C271—41no—
172C4×C271C22×D411yescharacteristic
173C4×C271C2×#4111yescharacteristic
174Dic271—22no—
175Dic271—21no—
176Dic271—21no—
177Dic271—41no—
178Dic271—41no—
179Dic271#12112yes—
180C2×C271C2×#12111yescharacteristic
181C2×C271#57811yescharacteristic
182C2×C271#57811yescharacteristic
183D271—22no—
184D271—81no—
185C271#304411yescharacteristic, complemented, Sylow
186#3044—2711nomaximal, Sylow
187C2×#137—2711no—
188C22×Q16—2711no—
189#536—2711no—
190#519—2711no—
191#464—2711no—
192#464—2711no—
193#464—2711no—
194#464—2711no—
195#462—2711no—
196#462—2711no—
197#462—2711no—
198#462D27111yescharacteristic, complemented
199#424—2711no—
200#419—2711no—
201#416—2711no—
202C2×#41—2711no—
203C22×Q8—2711no—
204C22×Q8—2711no—
205#137—5424no—
206C2×Q16—2711no—
207C2×Q16—2711no—
208C2×Q16—2711no—
209C2×Q16—2711no—
210C2×Q16—5424no—
211C2×Q16C2×D27111yescharacteristic
212C2×Q16C2×D27111yescharacteristic
213C2×SD16—2711no—
214C2×SD16—2711no—
215C2×M16—2711no—
216C22×C8—2711no—
217C4⋊Q8—2711no—
218C4⋊Q8C2×D27111yescharacteristic
219#124—2711no—
220#124—2711no—
221#123—5421no—
222#123—5421no—
223C22⋊Q8—5421no—
224C22⋊Q8—5421no—
225C4×Q8—2711no—
226C4×Q8—2711no—
227C4×Q8—2711no—
228C4×Q8—5421no—
229C4×Q8—5421no—
230C4×Q8C2×D27111yescharacteristic
231#117—2711no—
232#117—2711no—
233#117—5421no—
234#117—5421no—
235C4⋊C8—2711no—
236C4⋊C8—2711no—
237C4⋊C8—2711no—
238C4⋊C8C2×D27111yescharacteristic
239#103—2711no—
240#103—2711no—
241#103—2711no—
242#103—2711no—
243#103C2×D27111yescharacteristic
244#103C2×D27111yescharacteristic
245#102—2711no—
246#102—2711no—
247#41—5422no—
248#41—5422no—
249C2×Q8—2714no—
250C2×Q8—2711no—
251C2×Q8—2711no—
252C2×Q8—5421no—
253C2×Q8—5421no—
254C2×Q8—5422no—
255C2×Q8—5422no—
256C2×Q8—5422no—
257C2×Q8C22×D27111yescharacteristic
258C2×Q8C22×D27111yescharacteristic
259C2×Q8C22×D27111yescharacteristic
260C2×D4—2711no—
261C2×D4—5421no—
262C22×C4—2711no—
263C22×C4—5421no—
264C22×C4—5421no—
265C22×C4—5421no—
266Q16—22no—
267Q16—22no—
268Q16—5422no—
269Q16—5422no—
270Q16—5422no—
271Q16—5422no—
272SD16—5422no—
273SD16—5422no—
274M16—5422no—
275C2×C8—2711no—
276C2×C8—2711no—
277C2×C8—5422no—
278C2×C8C22×D27111yescharacteristic
279C2×C8C22×D27111yescharacteristic
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—2711no—
284C4⋊C4—2711no—
285C4⋊C4—2711no—
286C4⋊C4—5421no—
287C4⋊C4—5421no—
288C4⋊C4—5421no—
289C4⋊C4—5421no—
290C4⋊C4—5421no—
291C4⋊C4C22×D27111yescharacteristic
292C22⋊C4—2711no—
293C22⋊C4—2711no—
294C22⋊C4—5421no—
295C22⋊C4—5421no—
296C22⋊C4—5421no—
297C22⋊C4—5421no—
298C22⋊C4—5421no—
299C42—21no—
300C42—2711no—
301C42—5421no—
302C42—5421no—
303C42—5421no—
304C42—5421no—
305C42C22×D27111yescharacteristic
306C23—2711no—
307C23—5421no—
308Q8—21no—
309Q8—21no—
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8—2712no—
314Q8—5424no—
315Q8—5421no—
316Q8—5421no—
317Q8—5421no—
318Q8—10842no—
319Q8D4×D27112yes—
320D4—5421no—
321D4—5421no—
322D4—10842no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—2711no—
328C2×C4—2712no—
329C2×C4—2712no—
330C2×C4—2711no—
331C2×C4—2711no—
332C2×C4—5424no—
333C2×C4—5421no—
334C2×C4—5421no—
335C2×C4—5421no—
336C2×C4—10842no—
337C2×C4—10842no—
338C2×C4D4×D27111yescharacteristic
339C2×C4D4×D27111yescharacteristic
340C2×C4C23×D27111yescharacteristic
341C8—21no—
342C8—21no—
343C8—5421no—
344C8—5421no—
345C22—2712no—
346C22—5421no—
347C22—5421no—
348C22—10841no—
349C22—10841no—
350C22—10841no—
351C22C2×D4×D27111yescharacteristic
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—2712no—
359C4—5422no—
360C4—5421no—
361C4—5421no—
362C4—10841no—
363C4—10841no—
364C4C2×D4×D27111yescharacteristic
365C4#41×D27111yescharacteristic
366C2—5422no—
367C2—21681no—
368C2#121×D27111yescharacteristic
369C2#3177471011yescharacteristic
370C2#3177471311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#462×C271C2
socleC22×C271C22×D4
prime radicalC2×#16864655C22
hypercenter#462D271
hypocenterC271#3044
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
2. C2×Q16×D271C2semidirect
4. #31774373C2semidirect
5. #31774122C2semidirect
6. #31774121C2semidirect
7. #31774120C2semidirect
8. #31774119C2semidirect
9. #31774106C2semidirect
10. #31774105C2semidirect
11. #31774104C2semidirect
12. #31773922C2semidirect
13. #31773904C2semidirect
15. #462×C271C2semidirect
185. C271#3044semidirect
198. #462D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761922
C2×Q16×D2712
#317744722
#317743732
#317741222
#317741212
#317741202
#317741192
#317741062
#317741052
#317741042
#317739222
#317739042
#317738922
#462×C2712
#3044271