Finite Group Databasebeta

← #46839622

Subgroups — #46839622

Order 34688.

560 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C22⋊D271)⋊C4C211yesmaximal, characteristic
2C2×#22576116C211yesmaximal, characteristic, complemented
3#31773838C211yesmaximal, characteristic, complemented
4#31773832C211yesmaximal, characteristic, complemented
5#31773831C211yesmaximal, characteristic, complemented
6#31773826C211yesmaximal, characteristic, complemented
7#31773825C211yesmaximal, characteristic, complemented
8#31773813C211yesmaximal, characteristic
9#31773812C211yesmaximal, characteristic
10C4⋊C4×Dic271C211yesmaximal, characteristic, complemented
11#31773748C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic271C211yesmaximal, characteristic, complemented
13#31773744C211yesmaximal, characteristic, complemented
14#31773744C211yesmaximal, characteristic, complemented
15#397×C271C211yesmaximal, characteristic, complemented
16C22×C22⋊D271C2211yescharacteristic
17C23×Dic271C2211yescharacteristic
18C22×C4×D271C2211yescharacteristic
19(C22⋊D271)⋊C4—24no—
20#22576116—24no—
21C2×C4⋊Dic271C2211yescharacteristic
22C2×Dic271⋊C4C2211yescharacteristic
23C2×Dic271⋊C4C2211yescharacteristic
24C2×Dic271⋊C4C2211yescharacteristic
25C2×Dic271⋊C4C2211yescharacteristic
26C2×Dic271⋊C4C2211yescharacteristic
27C2×C22⋊Dic271C2211yescharacteristic
28C2×C22⋊Dic271C2211yescharacteristic
29C2×(C2×D271)⋊C4C2211yescharacteristic
30C2×(C2×D271)⋊C4C2211yescharacteristic
31C2×(C2×D271)⋊C4C2211yescharacteristic
32C2×(C2×D271)⋊C4C2211yescharacteristic
33C2×(C2×D271)⋊C4C2211yescharacteristic
34C2×C4×Dic271C2211yescharacteristic
35C2×C4×Dic271C2211yescharacteristic
36C2×C4×Dic271C2211yescharacteristic
37C2×C4×Dic271C2211yescharacteristic
38C2×C4×Dic271C2211yescharacteristic
39C271⋊#95C2211yescharacteristic
40C271⋊#95C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C2×C4⋊C4×C271C2211yescharacteristic
47C2×C22⋊C4×C271C2211yescharacteristic
48C2×C22⋊C4×C271C2211yescharacteristic
49C2×C22⋊C4×C271C2211yescharacteristic
50#95×C271C2211yescharacteristic
51#95×C271C2211yescharacteristic
52#95×C271C2211yescharacteristic
53C23×D271C2311yescharacteristic
54C2×C22⋊D271—22no—
55C2×C22⋊D271—22no—
56C2×C22⋊D271—22no—
57C22×Dic271—22no—
58C22×Dic271—22no—
59C22×Dic271D414yes—
60C22×Dic271C2311yescharacteristic
61C22×Dic271C2311yescharacteristic
62C22×Dic271C2311yescharacteristic
63C22×Dic271C2311yescharacteristic
64C22×Dic271C2311yescharacteristic
65C22×Dic271C2311yescharacteristic
66C22×Dic271C2311yescharacteristic
67C2×C4×D271—22no—
68C2×C4×D271—22no—
69C2×C4×D271—22no—
70C4⋊Dic271—22no—
71Dic271⋊C4—22no—
72Dic271⋊C4—22no—
73Dic271⋊C4—22no—
74Dic271⋊C4—22no—
75Dic271⋊C4—22no—
76C22⋊Dic271—22no—
77C22⋊Dic271—22no—
78(C2×D271)⋊C4—22no—
79(C2×D271)⋊C4—22no—
80(C2×D271)⋊C4—22no—
81(C2×D271)⋊C4—22no—
82(C2×D271)⋊C4—22no—
83C4×Dic271—22no—
84C4×Dic271—22no—
85C4×Dic271—22no—
86C4×Dic271—22no—
87C4×Dic271—22no—
88C24×C271C2311yescharacteristic
89C22×C4×C271C2311yescharacteristic
90C22×C4×C271C2311yescharacteristic
91C22×C4×C271C2311yescharacteristic
92C22×C4×C271C2311yescharacteristic
93C22×C4×C271C2311yescharacteristic
94C22×C4×C271C2311yescharacteristic
95C4⋊C4×C271—22no—
96C22⋊C4×C271—22no—
97C22⋊C4×C271—22no—
98C22⋊C4×C271—22no—
99C22×D271—21no—
100C22×D271—21no—
101C22×D271—21no—
102C22×D271—21no—
103C22×D271—21no—
104C22×D271—21no—
105C22×D271#4112yes—
106C22⋊D271—28no—
107C2×Dic271—21no—
108C2×Dic271—21no—
109C2×Dic271—21no—
110C2×Dic271—21no—
111C2×Dic271—21no—
112C2×Dic271—21no—
113C2×Dic271—28no—
114C2×Dic271—21no—
115C2×Dic271—21no—
116C2×Dic271—21no—
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271—21no—
123C2×Dic271C2×D412yes—
124C2×Dic271C2×D412yes—
125C2×Dic271#4112yes—
126C2×Dic271#4112yes—
127C2×Dic271#4112yes—
128C2×Dic271#4112yes—
129C4×D271—44no—
130C23×C271—21no—
131C23×C271—21no—
132C23×C271—21no—
133C23×C271—21no—
134C23×C271—21no—
135C23×C271—21no—
136C23×C271C2×D412yes—
137C23×C271C2411yescharacteristic
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×C271—21no—
148C2×C4×C271—21no—
149C2×C4×C271—21no—
150C2×C4×C271—21no—
151C2×C4×C271—21no—
152C2×C4×C271—21no—
153C2×C4×C271—21no—
154C2×C4×C271—21no—
155C2×C4×C271#4112yes—
156C22×C271—22no—
157C22×C271—22no—
158C22×C271—22no—
159C22×C271—41no—
160C22×C271—41no—
161C22×C271—41no—
162C22×C271—41no—
163C22×C271C22×D411yescharacteristic
164C22×C271C2×#4111yescharacteristic
165C22×C271C2×#4111yescharacteristic
166C22×C271C2×#4111yescharacteristic
167C22×C271C2×#4111yescharacteristic
168C22×C271C2×#4111yescharacteristic
169C22×C271C2×#4111yescharacteristic
170C2×D271—22no—
171C2×D271—22no—
172C2×D271—22no—
173C2×D271—41no—
174C2×D271—41no—
175C2×D271—41no—
176C2×D271—41no—
177C4×C271—22no—
178C4×C271—41no—
179C4×C271—41no—
180C4×C271—41no—
181C4×C271—41no—
182C4×C271—41no—
183Dic271—22no—
184Dic271—22no—
185Dic271—22no—
186Dic271—41no—
187Dic271—41no—
188Dic271—41no—
189Dic271#12314yes—
190C2×C271—42no—
191C2×C271C2×#12311yescharacteristic
192C2×C271#52511yescharacteristic
193C2×C271#52911yescharacteristic
194C2×C271#52911yescharacteristic
195C2×C271#52911yescharacteristic
196C2×C271#52911yescharacteristic
197C2×C271#53211yescharacteristic
198D271—42no—
199C271#239511yescharacteristic, complemented, Sylow
200#2395—2711nomaximal, Sylow
201C2×C4×D4—2711no—
202C2×#117—2711no—
203#400—2711no—
204#399—2711no—
205#398—2711no—
206#397—2711no—
207#397—2711no—
208#397D27111yescharacteristic, complemented
209#394—2711no—
210#394—2711no—
211C4×C4⋊C4—2711no—
212C4×C4⋊C4—2711no—
213C4×C22⋊C4—2711no—
214C4×C22⋊C4—2711no—
215C4×C22⋊C4—2711no—
216C22×D4—2711no—
217C23×C4—2711no—
218C23×C4—2711no—
219C4×D4—5424no—
220#117—5424no—
221C2×C4⋊C4—2711no—
222C2×C4⋊C4—2711no—
223C2×C4⋊C4—2711no—
224C2×C4⋊C4—2711no—
225C2×C4⋊C4—2711no—
226C2×C4⋊C4—2711no—
227C2×C4⋊C4C2×D27111yescharacteristic
228C2×C22⋊C4—2711no—
229C2×C22⋊C4—2711no—
230C2×C22⋊C4—2711no—
231C2×C22⋊C4—2711no—
232C2×C22⋊C4—2711no—
233C2×C22⋊C4—2711no—
234C2×C22⋊C4—2711no—
235C2×C22⋊C4C2×D27111yescharacteristic
236C2×C22⋊C4C2×D27111yescharacteristic
237C2×C22⋊C4C2×D27111yescharacteristic
238C2×C42—2711no—
239C2×C42—2711no—
240C2×C42—2711no—
241C2×C42—2711no—
242C2×C42—2711no—
243#95—2711no—
244#95—2711no—
245#95—2711no—
246#95—2711no—
247#95—2711no—
248#95—2711no—
249#95—2711no—
250#95C2×D27111yescharacteristic
251#95C2×D27111yescharacteristic
252#95C2×D27111yescharacteristic
253C24—2711no—
254C24C22×D27111yescharacteristic
255C2×D4—5422no—
256C2×D4—5422no—
257C2×D4—5422no—
258C22×C4—2711no—
259C22×C4—2714no—
260C22×C4—2711no—
261C22×C4—2711no—
262C22×C4—2711no—
263C22×C4—2711no—
264C22×C4—2711no—
265C22×C4—2711no—
266C22×C4—5422no—
267C22×C4—5422no—
268C22×C4—5422no—
269C22×C4—5422no—
270C22×C4—5422no—
271C22×C4C22×D27111yescharacteristic
272C22×C4C22×D27111yescharacteristic
273C22×C4C22×D27111yescharacteristic
274C22×C4C22×D27111yescharacteristic
275C22×C4C22×D27111yescharacteristic
276C22×C4C22×D27111yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—5422no—
279C4⋊C4—5422no—
280C4⋊C4—5422no—
281C4⋊C4—5422no—
282C4⋊C4—5422no—
283C4⋊C4—5422no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—5422no—
288C22⋊C4—5422no—
289C22⋊C4—5422no—
290C22⋊C4—5422no—
291C22⋊C4—5422no—
292C22⋊C4—5422no—
293C22⋊C4—5422no—
294C42—5422no—
295C42—5422no—
296C42—5422no—
297C42—5422no—
298C42—5422no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—2712no—
306C23—5421no—
307C23—5421no—
308C23—5421no—
309C23—5421no—
310C23—5421no—
311C23—5421no—
312C23D4×D27112yes—
313C23C23×D27111yescharacteristic
314D4—5428no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—2712no—
333C2×C4—2712no—
334C2×C4—2712no—
335C2×C4—2712no—
336C2×C4—2712no—
337C2×C4—2712no—
338C2×C4—5428no—
339C2×C4—5421no—
340C2×C4—5421no—
341C2×C4—5421no—
342C2×C4—5421no—
343C2×C4—5421no—
344C2×C4—5421no—
345C2×C4—5421no—
346C2×C4—5421no—
347C2×C4—5421no—
348C2×C4—5421no—
349C2×C4—5421no—
350C2×C4—5421no—
351C2×C4—5421no—
352C2×C4—5421no—
353C2×C4—5421no—
354C2×C4—10844no—
355C2×C4#1686465312yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—5422no—
364C22—5422no—
365C22—5422no—
366C22—10841no—
367C22—10841no—
368C22—10841no—
369C22—10841no—
370C22C2×D4×D27111yescharacteristic
371C22#41×D27111yescharacteristic
372C22#41×D27111yescharacteristic
373C22#41×D27111yescharacteristic
374C22#41×D27111yescharacteristic
375C22#41×D27111yescharacteristic
376C22C2×#1686465311yescharacteristic
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—2714no—
384C4—5422no—
385C4—5422no—
386C4—5422no—
387C4—10841no—
388C4—10841no—
389C4—10841no—
390C2—42no—
391C2—10842no—
392C2#123×D27111yescharacteristic
393C2#3177440511yescharacteristic
394C2#3177442411yescharacteristic
395C2#3177442611yescharacteristic
396C2#3177443011yescharacteristic
397C2#3177443011yescharacteristic
398C2#3177444211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#397×C271C2
prime radicalC22×C22⋊D271C22
hypercenter#397D271
hypocenterC271#2395
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#22576116C2semidirect
3. #31773838C2semidirect
4. #31773832C2semidirect
5. #31773831C2semidirect
6. #31773826C2semidirect
7. #31773825C2semidirect
10. C4⋊C4×Dic271C2semidirect
11. #31773748C2semidirect
12. C22⋊C4×Dic271C2semidirect
13. #31773744C2semidirect
14. #31773744C2semidirect
15. #397×C271C2semidirect
199. C271#2395semidirect
208. #397D271semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D271)⋊C42
C2×#225761162
#317738382
#317738322
#317738312
#317738262
#317738252
#317738132
#317738122
C4⋊C4×Dic2712
#317737482
C22⋊C4×Dic2712
#317737442
#397×C2712
#2395271