Finite Group Databasebeta

← #46839882

Subgroups — #46839882

Order 34688.

500 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×#22576137C211yesmaximal, characteristic
2C2×#22576125C211yesmaximal, characteristic, complemented
3#31773833C211yesmaximal, characteristic
4#31773830C211yesmaximal, characteristic, complemented
5#31773828C211yesmaximal, characteristic, complemented
6#31773801C211yesmaximal, characteristic, complemented
7#31773792C211yesmaximal, characteristic, complemented
8#31773791C211yesmaximal, characteristic
9#31773785C211yesmaximal, characteristic
10#31773783C211yesmaximal, characteristic, complemented
11#31773782C211yesmaximal, characteristic, complemented
12#31773766C211yesmaximal, characteristic, complemented
13#31773755C211yesmaximal, characteristic
14C22⋊C4×Dic271C211yesmaximal, characteristic
15#399×C271C211yesmaximal, characteristic
16C22×Dic542C2211yescharacteristic
17C23×Dic271C2211yescharacteristic
18#22576137—24no—
19#22576125—24no—
20C2×C4⋊Dic271C2211yescharacteristic
21C2×C4⋊Dic271C2211yescharacteristic
22C2×Dic271⋊C4C2211yescharacteristic
23C2×Dic271⋊C4C2211yescharacteristic
24C2×Dic271⋊C4C2211yescharacteristic
25C2×Dic271⋊C4C2211yescharacteristic
26C2×Dic271⋊C4C2211yescharacteristic
27C2×Dic271⋊C4C2211yescharacteristic
28C2×Dic271⋊C4C2211yescharacteristic
29C2×Dic271⋊C4C2211yescharacteristic
30C2×C22⋊Dic271C2211yescharacteristic
31C2×C22⋊Dic271C2211yescharacteristic
32C2×C22⋊Dic271C2211yescharacteristic
33C2×C4×Dic271C2211yescharacteristic
34C2×C4×Dic271C2211yescharacteristic
35C2×C4×Dic271C2211yescharacteristic
36C2×C4×Dic271C2211yescharacteristic
37C271⋊#95C2211yescharacteristic
38C271⋊#95C2211yescharacteristic
39C271⋊#95C2211yescharacteristic
40C271⋊#95C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C2×C4⋊C4×C271C2211yescharacteristic
47C2×C4⋊C4×C271C2211yescharacteristic
48C2×C4⋊C4×C271C2211yescharacteristic
49C2×C22⋊C4×C271C2211yescharacteristic
50C2×C22⋊C4×C271C2211yescharacteristic
51C2×C22⋊C4×C271C2211yescharacteristic
52#95×C271C2211yescharacteristic
53C2×Dic542—22no—
54C2×Dic542—22no—
55C2×Dic542D414yes—
56C22×Dic271—22no—
57C22×Dic271—22no—
58C22×Dic271—22no—
59C22×Dic271C2311yescharacteristic
60C22×Dic271C2311yescharacteristic
61C22×Dic271C2311yescharacteristic
62C22×Dic271C2311yescharacteristic
63C22×Dic271C2311yescharacteristic
64C22×Dic271C2311yescharacteristic
65C22×Dic271C2311yescharacteristic
66C22×Dic271C2311yescharacteristic
67C4⋊Dic271—22no—
68C4⋊Dic271—22no—
69Dic271⋊C4—22no—
70Dic271⋊C4—22no—
71Dic271⋊C4—22no—
72Dic271⋊C4—22no—
73Dic271⋊C4—22no—
74Dic271⋊C4—41no—
75Dic271⋊C4—41no—
76Dic271⋊C4—41no—
77C22⋊Dic271—22no—
78C22⋊Dic271—41no—
79C22⋊Dic271Q814yescomplemented
80C4×Dic271—22no—
81C4×Dic271—22no—
82C4×Dic271—22no—
83C4×Dic271—22no—
84C24×C271C2311yescharacteristic
85C22×C4×C271C2311yescharacteristic
86C22×C4×C271C2311yescharacteristic
87C22×C4×C271C2311yescharacteristic
88C22×C4×C271C2311yescharacteristic
89C22×C4×C271C2311yescharacteristic
90C22×C4×C271C2311yescharacteristic
91C4⋊C4×C271—22no—
92C4⋊C4×C271—22no—
93C4⋊C4×C271—41no—
94C22⋊C4×C271—22no—
95C22⋊C4×C271—22no—
96C22⋊C4×C271—41no—
97Dic542—28no—
98C2×Dic271—21no—
99C2×Dic271—21no—
100C2×Dic271—21no—
101C2×Dic271—21no—
102C2×Dic271—21no—
103C2×Dic271—21no—
104C2×Dic271—21no—
105C2×Dic271—21no—
106C2×Dic271—21no—
107C2×Dic271—21no—
108C2×Dic271—21no—
109C2×Dic271—21no—
110C2×Dic271—21no—
111C2×Dic271—21no—
112C2×Dic271—21no—
113C2×Dic271—21no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271—44no—
117C2×Dic271C2×D412yes—
118C2×Dic271C2×D412yes—
119C2×Dic271C2×Q812yes—
120C2×Dic271C2×Q812yes—
121C2×Dic271#4112yes—
122C2×Dic271#4112yes—
123C23×C271—21no—
124C23×C271—21no—
125C23×C271—21no—
126C23×C271—21no—
127C23×C271—21no—
128C23×C271—21no—
129C23×C271C2×Q812yes—
130C23×C271C2411yescharacteristic
131C2×C4×C271—21no—
132C2×C4×C271—21no—
133C2×C4×C271—21no—
134C2×C4×C271—21no—
135C2×C4×C271—21no—
136C2×C4×C271—21no—
137C2×C4×C271—21no—
138C2×C4×C271—21no—
139C2×C4×C271—21no—
140C2×C4×C271—21no—
141C2×C4×C271—21no—
142C2×C4×C271—21no—
143C2×C4×C271—21no—
144C2×C4×C271—21no—
145C2×C4×C271—21no—
146C2×C4×C271—21no—
147C2×C4×C271C2×D412yes—
148C2×C4×C271#4112yes—
149C22×C271—22no—
150C22×C271—22no—
151C22×C271—22no—
152C22×C271—41no—
153C22×C271—41no—
154C22×C271—41no—
155C22×C271—41no—
156C22×C271C22×D411yescharacteristic
157C22×C271C22×Q811yescharacteristic
158C22×C271C2×#4111yescharacteristic
159C22×C271C2×#4111yescharacteristic
160C22×C271C2×#4111yescharacteristic
161C22×C271H(5,𝔽2)11yescharacteristic
162C22×C2712−1+411yescharacteristic
163C4×C271—22no—
164C4×C271—22no—
165C4×C271—41no—
166C4×C271—41no—
167C4×C271—41no—
168C4×C271—41no—
169Dic271—22no—
170Dic271—22no—
171Dic271—22no—
172Dic271—41no—
173Dic271—41no—
174Dic271—41no—
175Dic271—41no—
176Dic271#12314yes—
177C2×C271—42no—
178C2×C271C2×#12311yescharacteristic
179C2×C271#53311yescharacteristic
180C2×C271#54211yescharacteristic
181C2×C271D4×Q811yescharacteristic
182C2×C271#55011yescharacteristic
183C2×C271#55211yescharacteristic
184C2×C271#55411yescharacteristic
185C271#242311yescharacteristic, complemented, Sylow
186#2423—2711nomaximal, Sylow
187C2×C22⋊Q8—2711no—
188C2×C4×Q8—2711no—
189#399—2711no—
190#399D27111yescharacteristic
191#398—2711no—
192#398—2711no—
193#391—2711no—
194#389—2711no—
195#388—2711no—
196#388—2711no—
197#387—2711no—
198#387—2711no—
199#384—2711no—
200#380—2711no—
201C4×C22⋊C4—2711no—
202C22×Q8—2711no—
203C23×C4—2711no—
204C22⋊Q8—5424no—
205C4×Q8—5424no—
206C2×C4⋊C4—2711no—
207C2×C4⋊C4—2711no—
208C2×C4⋊C4—2711no—
209C2×C4⋊C4—2711no—
210C2×C4⋊C4—2711no—
211C2×C4⋊C4—2711no—
212C2×C4⋊C4—2711no—
213C2×C4⋊C4—2711no—
214C2×C4⋊C4—2711no—
215C2×C4⋊C4—2711no—
216C2×C4⋊C4C2×D27111yescharacteristic
217C2×C4⋊C4C2×D27111yescharacteristic
218C2×C4⋊C4C2×D27111yescharacteristic
219C2×C22⋊C4—2711no—
220C2×C22⋊C4—2711no—
221C2×C22⋊C4—2711no—
222C2×C22⋊C4C2×D27111yescharacteristic
223C2×C22⋊C4C2×D27111yescharacteristic
224C2×C22⋊C4C2×D27111yescharacteristic
225C2×C42—2711no—
226C2×C42—2711no—
227C2×C42—2711no—
228C2×C42—2711no—
229#95—2711no—
230#95—2711no—
231#95—2711no—
232#95—2711no—
233#95—2711no—
234#95—2711no—
235#95—2711no—
236#95—2711no—
237#95—2711no—
238#95C2×D27111yescharacteristic
239C24C22×D27111yescharacteristic
240C2×Q8—2714no—
241C2×Q8—5422no—
242C2×Q8—5422no—
243C22×C4—2711no—
244C22×C4—2711no—
245C22×C4—2711no—
246C22×C4—2711no—
247C22×C4—2711no—
248C22×C4—2711no—
249C22×C4—2711no—
250C22×C4—2711no—
251C22×C4—5422no—
252C22×C4—5422no—
253C22×C4—5422no—
254C22×C4C22×D27111yescharacteristic
255C22×C4C22×D27111yescharacteristic
256C22×C4C22×D27111yescharacteristic
257C22×C4C22×D27111yescharacteristic
258C22×C4C22×D27111yescharacteristic
259C22×C4C22×D27111yescharacteristic
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—41no—
263C4⋊C4—5422no—
264C4⋊C4—5422no—
265C4⋊C4—5422no—
266C4⋊C4—5422no—
267C4⋊C4—5422no—
268C4⋊C4—5422no—
269C4⋊C4—5422no—
270C4⋊C4—10841no—
271C4⋊C4—10841no—
272C4⋊C4—10841no—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—41no—
276C22⋊C4—2714no—
277C22⋊C4—5422no—
278C22⋊C4—10841no—
279C42—5422no—
280C42—5422no—
281C42—5422no—
282C42—5422no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23Q8×D27112yes—
290C23C23×D27111yescharacteristic
291Q8—5428no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—2712no—
309C2×C4—2712no—
310C2×C4—2712no—
311C2×C4—2712no—
312C2×C4—2712no—
313C2×C4—2712no—
314C2×C4—5421no—
315C2×C4—5421no—
316C2×C4—5421no—
317C2×C4—5421no—
318C2×C4—5421no—
319C2×C4—5421no—
320C2×C4—5421no—
321C2×C4—5421no—
322C2×C4—5421no—
323C2×C4—5421no—
324C2×C4—5421no—
325C2×C4—5421no—
326C2×C4—5421no—
327C2×C4—5421no—
328C2×C4—5421no—
329C2×C4—5421no—
330C2×C4—5421no—
331C2×C4—5421no—
332C2×C4—10844no—
333C2×C4D4×D27112yes—
334C2×C4#1686465312yes—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22C2×D4×D27111yescharacteristic
343C22C2×Q8×D27111yescharacteristic
344C22#41×D27111yescharacteristic
345C22#41×D27111yescharacteristic
346C22C2×#1686465311yescharacteristic
347C22#2257620511yescharacteristic
348C22#2257620711yescharacteristic
349C4—22no—
350C4—22no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—2714no—
356C4—5422no—
357C4—5422no—
358C4—5422no—
359C4—10841no—
360C4—10841no—
361C4—10841no—
362C4—10841no—
363C2—42no—
364C2#123×D27111yescharacteristic
365C2#3177444911yescharacteristic
366C2#3177451511yescharacteristic
367C2#3177456011yescharacteristic
368C2#3177456711yescharacteristic
369C2#3177458411yescharacteristic
370C2#3177460111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#399×C271C2
prime radicalC24×C271C23
hypercenter#399D271
hypocenterC271#2423
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. C2×#22576125C2semidirect
4. #31773830C2semidirect
5. #31773828C2semidirect
6. #31773801C2semidirect
7. #31773792C2semidirect
10. #31773783C2semidirect
11. #31773782C2semidirect
12. #31773766C2semidirect
79. C22⋊Dic271Q8semidirect
185. C271#2423semidirect

Maximal subgroups

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

subgroupindex
C2×#225761372
C2×#225761252
#317738332
#317738302
#317738282
#317738012
#317737922
#317737912
#317737852
#317737832
#317737822
#317737662
#317737552
C22⋊C4×Dic2712
#399×C2712
#2423271