Finite Group Databasebeta

← #46839712

Subgroups — #46839712

Order 34688.

908 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#16864655C211yesmaximal, characteristic, complemented
2C2×#22576136C211yesmaximal, characteristic, complemented
3C2×#22576135C211yesmaximal, characteristic, complemented
4C2×C4×D1084C211yesmaximal, characteristic, complemented
5#31773815C212yesmaximal, complemented
6#31773807C212yesmaximal, complemented
7#31773799C211yesmaximal, characteristic, complemented
8#31773775C212yesmaximal, complemented
9#31773771C212yesmaximal, complemented
10#31773754C211yesmaximal, characteristic, complemented
11#391×C271C211yesmaximal, characteristic, complemented
12C2×#16864655—24no—
13C2×#16864655—24no—
14C2×#16864655—24no—
15C22×D1084C2212yes—
16C22×D1084C2211yescharacteristic
17C22×D1084C2212yes—
18C22×D1084C2211yescharacteristic
19C22×C4×D271C2212yes—
20C22×C4×D271C2211yescharacteristic
21C22×C4×D271C2211yescharacteristic
22#22576136—24no—
23#22576135—24no—
24C4×D1084—24no—
25C2×C4⋊Dic271C2211yescharacteristic
26C2×C4⋊Dic271C2211yescharacteristic
27C2×Dic271⋊C4C2212yes—
28C2×(C2×D271)⋊C4C2212yes—
29C2×(C2×D271)⋊C4C2212yes—
30C2×(C2×D271)⋊C4C2211yescharacteristic
31C2×(C2×D271)⋊C4C2212yes—
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C2×(C2×D271)⋊C4C2212yes—
34C2×(C2×D271)⋊C4C2212yes—
35C271⋊#95C2212yes—
36C22×Q8×C271C2211yescharacteristic
37C2×C4⋊C4×C271C2211yescharacteristic
38C2×C42×C271C2211yescharacteristic
39#95×C271C2212yes—
40#95×C271C2212yes—
41C23×D271C2312yes—
42C23×D271C2311yescharacteristic
43C23×D271C2311yescharacteristic
44#16864655—232no—
45C2×D1084—24no—
46C2×D1084—24no—
47C2×D1084—22no—
48C2×D1084—22no—
49C2×D1084—24no—
50C2×D1084—24no—
51C2×D1084—22no—
52C2×D1084—24no—
53C2×D1084—22no—
54C2×D1084—22no—
55C2×D1084—42no—
56C2×D1084D414yescomplemented
57C22×Dic271C2311yescharacteristic
58C22×Dic271C2312yes—
59C22×Dic271C2311yescharacteristic
60C2×C4×D271—24no—
61C2×C4×D271—24no—
62C2×C4×D271—22no—
63C2×C4×D271—24no—
64C2×C4×D271—22no—
65C2×C4×D271—22no—
66C2×C4×D271—22no—
67C2×C4×D271—22no—
68C2×C4×D271D414yescomplemented
69C4⋊Dic271—22no—
70C4⋊Dic271—22no—
71Dic271⋊C4—24no—
72(C2×D271)⋊C4—24no—
73(C2×D271)⋊C4—22no—
74(C2×D271)⋊C4—22no—
75(C2×D271)⋊C4—24no—
76(C2×D271)⋊C4—24no—
77(C2×D271)⋊C4—42no—
78(C2×D271)⋊C4—42no—
79C2×Q8×C271—22no—
80C2×Q8×C271—22no—
81C2×Q8×C271D414yescomplemented
82C22×C4×C271C2312yes—
83C22×C4×C271C2311yescharacteristic
84C22×C4×C271C2312yes—
85C22×C4×C271C2311yescharacteristic
86C22×C4×C271C2311yescharacteristic
87C4⋊C4×C271—22no—
88C42×C271—22no—
89C22×D271—21no—
90C22×D271—21no—
91C22×D271—22no—
92C22×D271—21no—
93C22×D271—21no—
94C22×D271—22no—
95C22×D271—21no—
96C22×D271—22no—
97C22×D271—21no—
98C22×D271—22no—
99C22×D271—21no—
100C22×D271—21no—
101C22×D271—21no—
102C22×D271—22no—
103C22×D271—21no—
104C22×D271—21no—
105C22×D271—22no—
106C22×D271—21no—
107C22×D271C2×D414yes—
108C22×D271C2×D412yes—
109C22×D271#4112yescomplemented
110D1084—28no—
111D1084—216no—
112D1084—28no—
113D1084—48no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—22no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—22no—
121C2×Dic271—22no—
122C2×Dic271C2×D412yes—
123C4×D271—216no—
124C4×D271—28no—
125C4×D271—44no—
126C23×C271C2411yescharacteristic
127Q8×C271—28no—
128C2×C4×C271—21no—
129C2×C4×C271—22no—
130C2×C4×C271—22no—
131C2×C4×C271—21no—
132C2×C4×C271—21no—
133C2×C4×C271—22no—
134C2×C4×C271—22no—
135C2×C4×C271—21no—
136C2×C4×C271—22no—
137C2×C4×C271—21no—
138C2×C4×C271C2×D412yes—
139C2×C4×C271C2×D412yes—
140C2×C4×C271C2×D414yes—
141C2×C4×C271C2×D412yes—
142C2×C4×C271#4112yescomplemented
143C22×C271C22×D411yescharacteristic
144C22×C271C22×D411yescharacteristic
145C22×C271C22×D411yescharacteristic
146C22×C271C2×#4111yescharacteristic
147C22×C271C2×#4111yescharacteristic
148C22×C271H(5,𝔽2)11yescharacteristic
149C22×C2712−1+411yescharacteristic
150C2×D271—24no—
151C2×D271—22no—
152C2×D271—22no—
153C2×D271—22no—
154C2×D271—22no—
155C2×D271—22no—
156C2×D271—24no—
157C2×D271—24no—
158C2×D271—22no—
159C2×D271—41no—
160C2×D271—42no—
161C2×D271—41no—
162C2×D271—42no—
163C2×D271—41no—
164C2×D271—41no—
165C2×D271—41no—
166C2×D271—41no—
167C2×D271—42no—
168C2×D271—41no—
169C2×D271—42no—
170C2×D271—41no—
171C4×C271—24no—
172C4×C271—22no—
173C4×C271—41no—
174C4×C271—42no—
175C4×C271#12014yes—
176Dic271—22no—
177Dic271—41no—
178Dic271—42no—
179C2×C271C2×#12011yescharacteristic
180C2×C271#52511yescharacteristic
181C2×C271#53711yescharacteristic
182C2×C271#54611yescharacteristic
183C2×C271#54711yescharacteristic
184C2×C271#54811yescharacteristic
185C2×C271#55011yescharacteristic
186D271—42no—
187D271—44no—
188D271—42no—
189C271#240711yescharacteristic, complemented, Sylow
190#2407—2711nomaximal, Sylow
191C22×#41—2711no—
192C2×C22⋊Q8—2711no—
193C2×#121—2711no—
194C2×C4×D4—2711no—
195#394—2712no—
196#392—2712no—
197#391D27111yescharacteristic, complemented
198#390—2711no—
199#386—2712no—
200#385—2712no—
201#380—2711no—
202C2×#41—5424no—
203C2×#41—5424no—
204C2×#41—5424no—
205C22×Q8C2×D27111yescharacteristic
206C22×D4—2711no—
207C22×D4—2711no—
208C22×D4—2712no—
209C22×D4—2712no—
210C23×C4—2711no—
211C23×C4—2711no—
212C23×C4—2712no—
213C22⋊Q8—5424no—
214#121—5424no—
215C4×D4—5424no—
216C2×C4⋊C4—2711no—
217C2×C4⋊C4—2711no—
218C2×C4⋊C4—2712no—
219C2×C4⋊C4C2×D27111yescharacteristic
220C2×C22⋊C4—2711no—
221C2×C22⋊C4—2711no—
222C2×C22⋊C4—2712no—
223C2×C22⋊C4—2712no—
224C2×C22⋊C4—2712no—
225C2×C22⋊C4—2712no—
226C2×C22⋊C4—2712no—
227C2×C42C2×D27111yescharacteristic
228#95—2712no—
229#95C2×D27112yes—
230#95C2×D27112yes—
231C24—2711no—
232C24—2712no—
233C24—2711no—
234#41—54232no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8D108414yescomplemented
238C2×D4—2714no—
239C2×D4—5422no—
240C2×D4—5422no—
241C2×D4—5424no—
242C2×D4—5424no—
243C2×D4—5424no—
244C2×D4—5424no—
245C2×D4—5422no—
246C2×D4—5424no—
247C2×D4—5422no—
248C2×D4—5422no—
249C2×D4—10842no—
250C22×C4—2711no—
251C22×C4—2714no—
252C22×C4—2712no—
253C22×C4—2711no—
254C22×C4—5424no—
255C22×C4—5424no—
256C22×C4—5422no—
257C22×C4—5424no—
258C22×C4—5422no—
259C22×C4—5422no—
260C22×C4—5422no—
261C22×C4—5422no—
262C22×C4C22×D27112yes—
263C22×C4C22×D27111yescharacteristic
264C22×C4C22×D27112yes—
265C22×C4C22×D27111yescharacteristic
266C22×C4C22×D27111yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—5422no—
269C4⋊C4—5424no—
270C4⋊C4—5422no—
271C22⋊C4—5422no—
272C22⋊C4—5422no—
273C22⋊C4—5424no—
274C22⋊C4—5424no—
275C22⋊C4—5424no—
276C22⋊C4—10842no—
277C22⋊C4—10842no—
278C42—22no—
279C23—2712no—
280C23—2714no—
281C23—2712no—
282C23—5421no—
283C23—5421no—
284C23—5422no—
285C23—5421no—
286C23—5421no—
287C23—5422no—
288C23—5421no—
289C23—5422no—
290C23—5421no—
291C23—5422no—
292C23—5421no—
293C23—5421no—
294C23—5421no—
295C23—5422no—
296C23—5421no—
297C23—5421no—
298C23—5422no—
299C23—5421no—
300C23C23×D27111yescharacteristic
301Q8—28no—
302D4—54216no—
303D4—5428no—
304D4—5428no—
305D4—10848no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—2712no—
317C2×C4—54216no—
318C2×C4—5421no—
319C2×C4—5421no—
320C2×C4—5428no—
321C2×C4—5421no—
322C2×C4—5421no—
323C2×C4—5422no—
324C2×C4—5421no—
325C2×C4—5422no—
326C2×C4—5422no—
327C2×C4—10844no—
328C2×C4D4×D27112yes—
329C2×C4D4×D27112yes—
330C2×C4C2×D108414yes—
331C2×C4C2×D108412yes—
332C2×C4#1686465512yescomplemented
333C22—5422no—
334C22—5422no—
335C22—5424no—
336C22—5424no—
337C22—5422no—
338C22—5424no—
339C22—5422no—
340C22—5422no—
341C22—5422no—
342C22—10841no—
343C22—10842no—
344C22—10841no—
345C22—10841no—
346C22—10841no—
347C22—10841no—
348C22—10842no—
349C22—10841no—
350C22—10842no—
351C22—10841no—
352C22—10841no—
353C22—10842no—
354C22C2×D4×D27111yescharacteristic
355C22C2×D4×D27111yescharacteristic
356C22C22×D108411yescharacteristic
357C22#41×D27111yescharacteristic
358C22C2×#1686465511yescharacteristic
359C22#2257620611yescharacteristic
360C22#2257620811yescharacteristic
361C4—24no—
362C4—22no—
363C4—41no—
364C4—42no—
365C4—5422no—
366C4—10842no—
367C4—10841no—
368C4#2257613014yes—
369C2—10842no—
370C2—10842no—
371C2—10844no—
372C2C2×#2257613011yescharacteristic
373C2#3177441011yescharacteristic
374C2#3177447711yescharacteristic
375C2#3177454011yescharacteristic
376C2#3177455011yescharacteristic
377C2#3177455911yescharacteristic
378C2#3177456911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#391×C271C2
hypercenter#391D271
hypocenterC271#2407
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C22×#16864655C2semidirect
2. C2×#22576136C2semidirect
3. C2×#22576135C2semidirect
4. C2×C4×D1084C2semidirect
5. #31773815C2semidirect
6. #31773807C2semidirect
7. #31773799C2semidirect
8. #31773775C2semidirect
9. #31773771C2semidirect
10. #31773754C2semidirect
11. #391×C271C2semidirect
56. C2×D1084D4semidirect
68. C2×C4×D271D4semidirect
81. C2×Q8×C271D4semidirect
109. C22×D271#41semidirect
142. C2×C4×C271#41semidirect
189. C271#2407semidirect
197. #391D271semidirect
237. C2×Q8D1084semidirect
332. C2×C4#16864655semidirect

Maximal subgroups

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

subgroupindex
C22×#168646552
C2×#225761362
C2×#225761352
C2×C4×D10842
#317738152
#317738072
#317737992
#317737752
#317737712
#317737542
#391×C2712
#2407271