Finite Group Databasebeta

← #46839807

Subgroups — #46839807

Order 34688.

788 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×#22576133C212yesmaximal, complemented
2C2×#22576131C211yesmaximal, characteristic, complemented
3C2×#22576131C211yesmaximal, characteristic, complemented
4C2×C4×D1084C211yesmaximal, characteristic, complemented
5C2×C4⋊C4×D271C211yesmaximal, characteristic, complemented
6#31773820C212yesmaximal, complemented
7#31773812C212yesmaximal, complemented
8#31773795C211yesmaximal, characteristic, complemented
9#31773773C212yesmaximal, complemented
10#31773754C211yesmaximal, characteristic, complemented
11#390×C271C211yesmaximal, characteristic, complemented
12C22×D1084C2211yescharacteristic
13C22×C22⋊D271C2212yes—
14C22×C22⋊D271C2212yes—
15C22×C4×D271C2211yescharacteristic
16C22×C4×D271C2212yes—
17C22×C4×D271C2211yescharacteristic
18#22576133—28no—
19#22576131—24no—
20#22576131—24no—
21C4×D1084—24no—
22C2×C4⋊Dic271C2211yescharacteristic
23C2×C4⋊Dic271C2211yescharacteristic
24C2×C4⋊Dic271C2211yescharacteristic
25C2×Dic271⋊C4C2212yes—
26C2×Dic271⋊C4C2212yes—
27C4⋊C4×D271—24no—
28C2×C22⋊Dic271C2212yes—
29C2×C22⋊Dic271C2212yes—
30C2×(C2×D271)⋊C4C2212yes—
31C2×(C2×D271)⋊C4C2211yescharacteristic
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C271⋊#95C2212yes—
34C271⋊#95C2212yes—
35C22×D4×C271C2211yescharacteristic
36C2×C4⋊C4×C271C2211yescharacteristic
37C2×C22⋊C4×C271C2212yes—
38C2×C22⋊C4×C271C2212yes—
39C2×C42×C271C2211yescharacteristic
40C23×D271C2311yescharacteristic
41C23×D271C2311yescharacteristic
42C2×D1084—22no—
43C2×D1084—22no—
44C2×D1084D414yescomplemented
45C2×C22⋊D271—24no—
46C2×C22⋊D271—24no—
47C2×C22⋊D271—24no—
48C2×C22⋊D271—24no—
49C2×C22⋊D271—24no—
50C2×C22⋊D271—42no—
51C22×Dic271C2311yescharacteristic
52C22×Dic271C2312yes—
53C22×Dic271C2312yes—
54C22×Dic271C2311yescharacteristic
55C2×C4×D271—24no—
56C2×C4×D271—24no—
57C2×C4×D271—22no—
58C2×C4×D271—22no—
59C2×C4×D271—22no—
60C2×C4×D271—22no—
61C2×C4×D271—22no—
62C2×C4×D271—42no—
63C2×C4×D271D414yescomplemented
64C4⋊Dic271—22no—
65C4⋊Dic271—22no—
66C4⋊Dic271D414yescomplemented
67Dic271⋊C4—24no—
68Dic271⋊C4—24no—
69C22⋊Dic271—24no—
70C22⋊Dic271—24no—
71(C2×D271)⋊C4—24no—
72(C2×D271)⋊C4—22no—
73(C2×D271)⋊C4—22no—
74C24×C271C2312yes—
75C2×D4×C271—22no—
76C2×D4×C271—22no—
77C2×D4×C271—22no—
78C22×C4×C271C2312yes—
79C22×C4×C271C2311yescharacteristic
80C22×C4×C271C2311yescharacteristic
81C22×C4×C271C2311yescharacteristic
82C4⋊C4×C271—22no—
83C22⋊C4×C271—24no—
84C22⋊C4×C271—42no—
85C42×C271—22no—
86C22×D271—21no—
87C22×D271—21no—
88C22×D271—21no—
89C22×D271—21no—
90C22×D271—21no—
91C22×D271—21no—
92C22×D271—21no—
93C22×D271—21no—
94C22×D271—21no—
95C22×D271—21no—
96C22×D271C2×D412yes—
97C22×D271C2×D412yes—
98C22×D271C2×D412yes—
99C22×D271#4112yes—
100D1084—28no—
101C22⋊D271—48no—
102C22⋊D271—48no—
103C2×Dic271—21no—
104C2×Dic271—22no—
105C2×Dic271—21no—
106C2×Dic271—22no—
107C2×Dic271—22no—
108C2×Dic271—21no—
109C2×Dic271—21no—
110C2×Dic271—22no—
111C2×Dic271—21no—
112C2×Dic271—22no—
113C2×Dic271C2×D412yes—
114C2×Dic271C2×D414yes—
115C4×D271—28no—
116C4×D271—48no—
117C4×D271—44no—
118C23×C271—22no—
119C23×C271—22no—
120C23×C271—22no—
121C23×C271—22no—
122C23×C271—22no—
123C23×C271—22no—
124C23×C271—22no—
125C23×C271C2411yescharacteristic
126D4×C271—44no—
127C2×C4×C271—21no—
128C2×C4×C271—21no—
129C2×C4×C271—21no—
130C2×C4×C271—22no—
131C2×C4×C271—22no—
132C2×C4×C271—21no—
133C2×C4×C271—22no—
134C2×C4×C271—21no—
135C2×C4×C271C2×D412yes—
136C2×C4×C271C2×D412yes—
137C2×C4×C271C2×D412yes—
138C2×C4×C271#4112yes—
139C22×C271—42no—
140C22×C271—42no—
141C22×C271—42no—
142C22×C271—42no—
143C22×C271—42no—
144C22×C271—42no—
145C22×C271—42no—
146C22×C271C22×D411yescharacteristic
147C22×C271C22×D411yescharacteristic
148C22×C271C22×D411yescharacteristic
149C22×C271C2×#4111yescharacteristic
150C22×C271C2×#4111yescharacteristic
151C22×C271H(5,𝔽2)11yescharacteristic
152C22×C2712−1+411yescharacteristic
153C2×D271—22no—
154C2×D271—22no—
155C2×D271—22no—
156C2×D271—22no—
157C2×D271—22no—
158C2×D271—22no—
159C2×D271—22no—
160C2×D271—22no—
161C2×D271—22no—
162C2×D271—41no—
163C2×D271—41no—
164C2×D271—41no—
165C2×D271—41no—
166C2×D271#12114yescomplemented
167C4×C271—22no—
168C4×C271—41no—
169C4×C271—42no—
170C4×C271#12114yescomplemented
171Dic271—22no—
172Dic271—24no—
173Dic271—41no—
174Dic271—42no—
175C2×C271—82no—
176C2×C271C2×#12111yescharacteristic
177C2×C271C2×#12111yescharacteristic
178C2×C271#53711yescharacteristic
179C2×C271D4211yescharacteristic
180C2×C271#54711yescharacteristic
181C2×C271#54811yescharacteristic
182C2×C271#55311yescharacteristic
183D271—24no—
184D271—42no—
185C271#241611yescharacteristic, complemented, Sylow
186#2416—2711nomaximal, Sylow
187C2×#121—2711no—
188C2×#121—2711no—
189C2×#121—2712no—
190C2×C4×D4—2711no—
191C22×C4⋊C4—2711no—
192#396—2712no—
193#394—2712no—
194#390D27111yescharacteristic, complemented
195#389—2711no—
196#386—2712no—
197#380—2711no—
198C22×D4—2712no—
199C22×D4—2712no—
200C22×D4—2711no—
201C22×D4C2×D27111yescharacteristic
202C23×C4—2711no—
203C23×C4—2711no—
204C23×C4—2712no—
205#121—5428no—
206#121—5424no—
207#121—5424no—
208C4×D4—5424no—
209C2×C4⋊C4—2711no—
210C2×C4⋊C4—2712no—
211C2×C4⋊C4—2711no—
212C2×C4⋊C4—2711no—
213C2×C4⋊C4—2712no—
214C2×C4⋊C4—5424no—
215C2×C4⋊C4C2×D27111yescharacteristic
216C2×C22⋊C4—2712no—
217C2×C22⋊C4—2712no—
218C2×C22⋊C4—2711no—
219C2×C22⋊C4—2711no—
220C2×C22⋊C4—2712no—
221C2×C22⋊C4C2×D27112yes—
222C2×C22⋊C4C2×D27112yes—
223C2×C42C2×D27111yescharacteristic
224#95—2712no—
225#95—2712no—
226C24—2711no—
227C24—2711no—
228C24C22×D27112yes—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—2714no—
233C2×D4—5422no—
234C2×D4—5422no—
235C2×D4—5424no—
236C2×D4—5424no—
237C2×D4—5424no—
238C2×D4—5424no—
239C2×D4—5424no—
240C2×D4—10842no—
241C22×C4—2712no—
242C22×C4—2711no—
243C22×C4—2714no—
244C22×C4—2711no—
245C22×C4—2712no—
246C22×C4—5424no—
247C22×C4—5424no—
248C22×C4—5422no—
249C22×C4—5422no—
250C22×C4—5422no—
251C22×C4—5422no—
252C22×C4—5422no—
253C22×C4—10842no—
254C22×C4C22×D27111yescharacteristic
255C22×C4C22×D27112yes—
256C22×C4C22×D27111yescharacteristic
257C22×C4C22×D27111yescharacteristic
258C4⋊C4—22no—
259C4⋊C4—2714no—
260C4⋊C4—5424no—
261C4⋊C4—5422no—
262C4⋊C4—5424no—
263C4⋊C4—5422no—
264C22⋊C4—24no—
265C22⋊C4—42no—
266C22⋊C4—5422no—
267C22⋊C4—5422no—
268C22⋊C4—5424no—
269C22⋊C4—5424no—
270C22⋊C4—5424no—
271C42—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—2712no—
280C23—2712no—
281C23—2712no—
282C23—2712no—
283C23—5421no—
284C23—5421no—
285C23—5421no—
286C23—5421no—
287C23—5421no—
288C23—5421no—
289C23—5421no—
290C23—5421no—
291C23—5421no—
292C23—5421no—
293C23C23×D27111yescharacteristic
294D4—44no—
295D4—5428no—
296D4—10848no—
297D4—10848no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—2712no—
307C2×C4—2714no—
308C2×C4—5421no—
309C2×C4—5422no—
310C2×C4—5421no—
311C2×C4—5422no—
312C2×C4—5428no—
313C2×C4—5421no—
314C2×C4—5421no—
315C2×C4—5422no—
316C2×C4—5422no—
317C2×C4—5422no—
318C2×C4—5421no—
319C2×C4—10848no—
320C2×C4—10844no—
321C2×C4D4×D27112yes—
322C2×C4D4×D27112yes—
323C2×C4D4×D27112yes—
324C2×C4#1686465412yes—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—2714no—
333C22—5422no—
334C22—5422no—
335C22—5422no—
336C22—5422no—
337C22—5422no—
338C22—5422no—
339C22—5422no—
340C22—5422no—
341C22—5422no—
342C22—10841no—
343C22—10841no—
344C22—10841no—
345C22—10841no—
346C22C2×D4×D27111yescharacteristic
347C22C2×D4×D27111yescharacteristic
348C22C2×D4×D27111yescharacteristic
349C22#41×D27111yescharacteristic
350C22C2×#1686465411yescharacteristic
351C22#2257620511yescharacteristic
352C22#2257620711yescharacteristic
353C4—22no—
354C4—41no—
355C4—42no—
356C4—5424no—
357C4—5422no—
358C4—10842no—
359C4—10841no—
360C4#2257613314yescomplemented
361C2—82no—
362C2—5424no—
363C2—10842no—
364C2#121×D27111yescharacteristic
365C2C2×#2257613311yescharacteristic
366C2#3177447611yescharacteristic
367C2#3177453211yescharacteristic
368C2#3177455211yescharacteristic
369C2#3177455511yescharacteristic
370C2#3177459711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#390×C271C2
hypercenter#390D271
hypocenterC271#2416
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#22576133C2semidirect
2. C2×#22576131C2semidirect
3. C2×#22576131C2semidirect
4. C2×C4×D1084C2semidirect
5. C2×C4⋊C4×D271C2semidirect
6. #31773820C2semidirect
7. #31773812C2semidirect
8. #31773795C2semidirect
9. #31773773C2semidirect
10. #31773754C2semidirect
11. #390×C271C2semidirect
44. C2×D1084D4semidirect
63. C2×C4×D271D4semidirect
66. C4⋊Dic271D4semidirect
166. C2×D271#121semidirect
170. C4×C271#121semidirect
185. C271#2416semidirect
194. #390D271semidirect
360. C4#22576133semidirect

Maximal subgroups

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

subgroupindex
C2×#225761332
C2×#225761312
C2×C4×D10842
C2×C4⋊C4×D2712
#317738202
#317738122
#317737952
#317737732
#317737542
#390×C2712
#2416271