Finite Group Databasebeta

← #46839786

Subgroups — #46839786

Order 34688.

604 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576137C211yesmaximal, characteristic, complemented
2C2×#22576137C211yesmaximal, characteristic, complemented
3C2×D4×Dic271C211yesmaximal, characteristic
4C2×#22576116C211yesmaximal, characteristic, complemented
5#31773825C211yesmaximal, characteristic, complemented
6#31773822C211yesmaximal, characteristic, complemented
7#31773811C211yesmaximal, characteristic
8#31773801C211yesmaximal, characteristic, complemented
9#31773796C211yesmaximal, characteristic, complemented
10#31773794C211yesmaximal, characteristic, complemented
11#31773792C211yesmaximal, characteristic, complemented
12#31773790C211yesmaximal, characteristic, complemented
13#31773785C211yesmaximal, characteristic, complemented
14#31773755C211yesmaximal, characteristic, complemented
15#394×C271C211yesmaximal, characteristic
16C22×Dic542C2211yescharacteristic
17C23×Dic271C2211yescharacteristic
18C23×Dic271C2211yescharacteristic
19#22576137—24no—
20#22576137—24no—
21D4×Dic271—24no—
22#22576116—24no—
23C2×C4⋊Dic271C2211yescharacteristic
24C2×C4⋊Dic271C2211yescharacteristic
25C2×C4⋊Dic271C2211yescharacteristic
26C2×Dic271⋊C4C2211yescharacteristic
27C2×Dic271⋊C4C2211yescharacteristic
28C2×Dic271⋊C4C2211yescharacteristic
29C2×Dic271⋊C4C2211yescharacteristic
30C2×Dic271⋊C4C2211yescharacteristic
31C2×Dic271⋊C4C2211yescharacteristic
32C2×C22⋊Dic271C2211yescharacteristic
33C2×C22⋊Dic271C2211yescharacteristic
34C2×C22⋊Dic271C2211yescharacteristic
35C2×C22⋊Dic271C2211yescharacteristic
36C2×C22⋊Dic271C2211yescharacteristic
37C2×C22⋊Dic271C2211yescharacteristic
38C2×C4×Dic271C2211yescharacteristic
39C2×C4×Dic271C2211yescharacteristic
40C2×C4×Dic271C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C271⋊#95C2211yescharacteristic
46C271⋊#95C2211yescharacteristic
47C271⋊#95C2211yescharacteristic
48C22×D4×C271C2211yescharacteristic
49C2×C4⋊C4×C271C2211yescharacteristic
50C2×C22⋊C4×C271C2211yescharacteristic
51C2×C22⋊C4×C271C2211yescharacteristic
52C2×C22⋊C4×C271C2211yescharacteristic
53C2×C22⋊C4×C271C2211yescharacteristic
54#95×C271C2211yescharacteristic
55C2×Dic542—22no—
56C2×Dic542—22no—
57C2×Dic542—22no—
58C22×Dic271—22no—
59C22×Dic271—22no—
60C22×Dic271—22no—
61C22×Dic271—22no—
62C22×Dic271—22no—
63C22×Dic271D414yes—
64C22×Dic271C2311yescharacteristic
65C22×Dic271C2311yescharacteristic
66C22×Dic271C2311yescharacteristic
67C22×Dic271C2311yescharacteristic
68C22×Dic271C2311yescharacteristic
69C22×Dic271C2311yescharacteristic
70C22×Dic271C2311yescharacteristic
71C22×Dic271C2311yescharacteristic
72C4⋊Dic271—22no—
73C4⋊Dic271—22no—
74C4⋊Dic271—22no—
75Dic271⋊C4—22no—
76Dic271⋊C4—22no—
77Dic271⋊C4—22no—
78Dic271⋊C4—22no—
79Dic271⋊C4—22no—
80Dic271⋊C4D414yescomplemented
81C22⋊Dic271—22no—
82C22⋊Dic271—22no—
83C22⋊Dic271—22no—
84C22⋊Dic271—22no—
85C22⋊Dic271—22no—
86C22⋊Dic271—22no—
87C4×Dic271—22no—
88C4×Dic271—22no—
89C4×Dic271—22no—
90C24×C271C2311yescharacteristic
91C24×C271C2311yescharacteristic
92C2×D4×C271—22no—
93C2×D4×C271—22no—
94C2×D4×C271—22no—
95C22×C4×C271C2311yescharacteristic
96C22×C4×C271C2311yescharacteristic
97C22×C4×C271C2311yescharacteristic
98C22×C4×C271C2311yescharacteristic
99C22×C4×C271C2311yescharacteristic
100C4⋊C4×C271—22no—
101C22⋊C4×C271—22no—
102C22⋊C4×C271—22no—
103C22⋊C4×C271—22no—
104C22⋊C4×C271—22no—
105Dic542—44no—
106C2×Dic271—21no—
107C2×Dic271—21no—
108C2×Dic271—21no—
109C2×Dic271—21no—
110C2×Dic271—28no—
111C2×Dic271—21no—
112C2×Dic271—21no—
113C2×Dic271—21no—
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×Dic271—21no—
124C2×Dic271—21no—
125C2×Dic271—21no—
126C2×Dic271—44no—
127C2×Dic271C2×D412yes—
128C2×Dic271C2×D412yes—
129C2×Dic271C2×D412yes—
130C2×Dic271C2×D412yes—
131C2×Dic271#4112yes—
132C23×C271—21no—
133C23×C271—21no—
134C23×C271—21no—
135C23×C271—21no—
136C23×C271—21no—
137C23×C271—21no—
138C23×C271—21no—
139C23×C271—21no—
140C23×C271—21no—
141C23×C271—21no—
142C23×C271—21no—
143C23×C271—21no—
144C23×C271C2×D412yes—
145C23×C271#4112yes—
146C23×C271C2411yescharacteristic
147D4×C271—28no—
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—21no—
156C2×C4×C271—21no—
157C2×C4×C271—21no—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271C2×D412yes—
162C2×C4×C271#4112yes—
163C22×C271—22no—
164C22×C271—22no—
165C22×C271—22no—
166C22×C271—22no—
167C22×C271—22no—
168C22×C271—22no—
169C22×C271—41no—
170C22×C271—41no—
171C22×C271—41no—
172C22×C271—41no—
173C22×C271—41no—
174C22×C271—41no—
175C22×C271—41no—
176C22×C271—41no—
177C22×C271C22×D411yescharacteristic
178C22×C271C22×D411yescharacteristic
179C22×C271C2×#4111yescharacteristic
180C22×C271C2×#4111yescharacteristic
181C22×C271C2×#4111yescharacteristic
182C22×C2712−1+411yescharacteristic
183C22×C2712−1+411yescharacteristic
184C4×C271—22no—
185C4×C271—22no—
186C4×C271—41no—
187C4×C271—41no—
188C4×C271—41no—
189Dic271—22no—
190Dic271—22no—
191Dic271—41no—
192Dic271—41no—
193Dic271—41no—
194Dic271—41no—
195Dic271—41no—
196Dic271#12114yes—
197C2×C271—42no—
198C2×C271—42no—
199C2×C271C2×#12111yescharacteristic
200C2×C271#52511yescharacteristic
201C2×C271#53611yescharacteristic
202C2×C271#54711yescharacteristic
203C2×C271#54811yescharacteristic
204C2×C271#55211yescharacteristic
205C2×C271#55611yescharacteristic
206C271#241511yescharacteristic, complemented, Sylow
207#2415—2711nomaximal, Sylow
208C2×C22⋊Q8—2711no—
209C2×C22⋊Q8—2711no—
210C2×C4×D4—2711no—
211C2×#117—2711no—
212#397—2711no—
213#397—2711no—
214#394—2711no—
215#394D27111yescharacteristic
216#391—2711no—
217#389—2711no—
218#389—2711no—
219#389—2711no—
220#388—2711no—
221#388—2711no—
222#380—2711no—
223C22×Q8—2711no—
224C22×D4C2×D27111yescharacteristic
225C23×C4—2711no—
226C23×C4—2711no—
227C22⋊Q8—5424no—
228C22⋊Q8—5424no—
229C4×D4—5424no—
230#117—5424no—
231C2×C4⋊C4—2711no—
232C2×C4⋊C4—2711no—
233C2×C4⋊C4—2711no—
234C2×C4⋊C4—2711no—
235C2×C4⋊C4—2711no—
236C2×C4⋊C4—2711no—
237C2×C4⋊C4—2711no—
238C2×C4⋊C4—2711no—
239C2×C4⋊C4—2711no—
240C2×C4⋊C4C2×D27111yescharacteristic
241C2×C22⋊C4—2711no—
242C2×C22⋊C4—2711no—
243C2×C22⋊C4—2711no—
244C2×C22⋊C4—2711no—
245C2×C22⋊C4—2711no—
246C2×C22⋊C4—2711no—
247C2×C22⋊C4C2×D27111yescharacteristic
248C2×C22⋊C4C2×D27111yescharacteristic
249C2×C22⋊C4C2×D27111yescharacteristic
250C2×C22⋊C4C2×D27111yescharacteristic
251C2×C42—2711no—
252C2×C42—2711no—
253C2×C42—2711no—
254#95—2711no—
255#95—2711no—
256#95—2711no—
257#95—2711no—
258#95—2711no—
259#95—2711no—
260#95—2711no—
261#95C2×D27111yescharacteristic
262C24C22×D27111yescharacteristic
263C24C22×D27111yescharacteristic
264C2×Q8—5422no—
265C2×Q8—5422no—
266C2×Q8—5422no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C22×C4—2711no—
271C22×C4—2714no—
272C22×C4—2711no—
273C22×C4—2711no—
274C22×C4—2711no—
275C22×C4—2711no—
276C22×C4—2711no—
277C22×C4—2711no—
278C22×C4—2711no—
279C22×C4—5422no—
280C22×C4—5422no—
281C22×C4—5422no—
282C22×C4—5422no—
283C22×C4—5422no—
284C22×C4C22×D27111yescharacteristic
285C22×C4C22×D27111yescharacteristic
286C22×C4C22×D27111yescharacteristic
287C22×C4C22×D27111yescharacteristic
288C22×C4C22×D27111yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—2714no—
291C4⋊C4—5422no—
292C4⋊C4—5422no—
293C4⋊C4—5422no—
294C4⋊C4—5422no—
295C4⋊C4—5422no—
296C4⋊C4—5422no—
297C4⋊C4—5422no—
298C4⋊C4—5422no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—5422no—
304C22⋊C4—5422no—
305C22⋊C4—5422no—
306C22⋊C4—5422no—
307C22⋊C4—5422no—
308C22⋊C4—5422no—
309C42—5422no—
310C42—5422no—
311C42—5422no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23D4×D27112yes—
325C23#1686465312yes—
326C23C23×D27111yescharacteristic
327Q8—10844no—
328D4—28no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—2712no—
343C2×C4—2712no—
344C2×C4—2712no—
345C2×C4—2712no—
346C2×C4—2712no—
347C2×C4—5428no—
348C2×C4—5421no—
349C2×C4—5421no—
350C2×C4—5421no—
351C2×C4—5421no—
352C2×C4—5421no—
353C2×C4—5421no—
354C2×C4—5421no—
355C2×C4—5421no—
356C2×C4—5421no—
357C2×C4—5421no—
358C2×C4—5421no—
359C2×C4—5421no—
360C2×C4—5421no—
361C2×C4—5421no—
362C2×C4—5421no—
363C2×C4—5421no—
364C2×C4—5421no—
365C2×C4—5421no—
366C2×C4—5421no—
367C2×C4—10844no—
368C2×C4D4×D27112yes—
369C2×C4#1686465312yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22C2×D4×D27111yescharacteristic
385C22C2×D4×D27111yescharacteristic
386C22#41×D27111yescharacteristic
387C22C2×#1686465311yescharacteristic
388C22C2×#1686465311yescharacteristic
389C22#2257620711yescharacteristic
390C22#2257620711yescharacteristic
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—2714no—
397C4—5422no—
398C4—5422no—
399C4—10841no—
400C4—10841no—
401C4—10841no—
402C4—10841no—
403C4—10841no—
404C2—42no—
405C2—42no—
406C2#121×D27111yescharacteristic
407C2#3177440311yescharacteristic
408C2#3177446911yescharacteristic
409C2#3177454711yescharacteristic
410C2#3177455311yescharacteristic
411C2#3177458411yescharacteristic
412C2#3177461611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#394×C271C2
prime radicalC22×D4×C271C22
hypercenter#394D271
hypocenterC271#2415
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#22576137C2semidirect
2. C2×#22576137C2semidirect
4. C2×#22576116C2semidirect
5. #31773825C2semidirect
6. #31773822C2semidirect
8. #31773801C2semidirect
9. #31773796C2semidirect
10. #31773794C2semidirect
11. #31773792C2semidirect
12. #31773790C2semidirect
13. #31773785C2semidirect
14. #31773755C2semidirect
80. Dic271⋊C4D4semidirect
206. C271#2415semidirect

Maximal subgroups

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

subgroupindex
C2×#225761372
C2×D4×Dic2712
C2×#225761162
#317738252
#317738222
#317738112
#317738012
#317737962
#317737942
#317737922
#317737902
#317737852
#317737552
#394×C2712
#2415271