Finite Group Databasebeta

← #46839835

Subgroups — #46839835

Order 34688.

560 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576141C211yesmaximal, characteristic
2#31773835C211yesmaximal, characteristic, complemented
3#31773828C211yesmaximal, characteristic, complemented
4#31773825C211yesmaximal, characteristic, complemented
5#31773825C211yesmaximal, characteristic, complemented
6#31773781C211yesmaximal, characteristic, complemented
7#31773781C211yesmaximal, characteristic, complemented
8#31773777C211yesmaximal, characteristic
9#31773770C211yesmaximal, characteristic, complemented
10#31773770C211yesmaximal, characteristic, complemented
11#31773760C211yesmaximal, characteristic, complemented
12#31773758C211yesmaximal, characteristic, complemented
13#31773755C211yesmaximal, characteristic, complemented
14C2×C271⋊#95C211yesmaximal, characteristic, complemented
15C2×C4×D4×C271C211yesmaximal, characteristic
16C23×Dic271C2211yescharacteristic
17C23×Dic271C2211yescharacteristic
18#22576141—24no—
19C2×Dic271⋊C4C2211yescharacteristic
20C2×Dic271⋊C4C2211yescharacteristic
21C2×Dic271⋊C4C2211yescharacteristic
22C2×Dic271⋊C4C2211yescharacteristic
23C2×Dic271⋊C4C2211yescharacteristic
24C2×Dic271⋊C4C2211yescharacteristic
25C2×C22⋊Dic271C2211yescharacteristic
26C2×C22⋊Dic271C2211yescharacteristic
27C2×C22⋊Dic271C2211yescharacteristic
28C2×C22⋊Dic271C2211yescharacteristic
29C2×C22⋊Dic271C2211yescharacteristic
30C2×C22⋊Dic271C2211yescharacteristic
31C271⋊#95C2211yescharacteristic
32C271⋊#95C2211yescharacteristic
33C271⋊#95C2211yescharacteristic
34C271⋊#95C2211yescharacteristic
35C271⋊#95C2211yescharacteristic
36C271⋊#95C2211yescharacteristic
37C271⋊#95C2211yescharacteristic
38C271⋊#95C2211yescharacteristic
39C271⋊#95C2211yescharacteristic
40C271⋊#95C2211yescharacteristic
41C271⋊#95C2211yescharacteristic
42C271⋊#95C2211yescharacteristic
43C271⋊#95C2211yescharacteristic
44C271⋊#95C2211yescharacteristic
45C22×D4×C271C2211yescharacteristic
46C23×C4×C271C2211yescharacteristic
47C23×C4×C271C2211yescharacteristic
48C4×D4×C271—24no—
49C2×C4⋊C4×C271C2211yescharacteristic
50C2×C22⋊C4×C271C2211yescharacteristic
51C2×C22⋊C4×C271C2211yescharacteristic
52C2×C42×C271C2211yescharacteristic
53C22×Dic271—22no—
54C22×Dic271—22no—
55C22×Dic271—22no—
56C22×Dic271—22no—
57C22×Dic271—22no—
58C22×Dic271—41no—
59C22×Dic271C2311yescharacteristic
60C22×Dic271C2311yescharacteristic
61C22×Dic271C2311yescharacteristic
62C22×Dic271C2311yescharacteristic
63C22×Dic271C2311yescharacteristic
64C22×Dic271C2311yescharacteristic
65C22×Dic271C2311yescharacteristic
66C22×Dic271C2311yescharacteristic
67Dic271⋊C4—22no—
68Dic271⋊C4—22no—
69Dic271⋊C4—22no—
70Dic271⋊C4—22no—
71Dic271⋊C4—41no—
72Dic271⋊C4—41no—
73C22⋊Dic271—22no—
74C22⋊Dic271—22no—
75C22⋊Dic271—22no—
76C22⋊Dic271—22no—
77C22⋊Dic271—22no—
78C22⋊Dic271—41no—
79C24×C271C2311yescharacteristic
80C24×C271C2311yescharacteristic
81C2×D4×C271—22no—
82C2×D4×C271—22no—
83C2×D4×C271D414yes—
84C22×C4×C271—22no—
85C22×C4×C271—22no—
86C22×C4×C271—22no—
87C22×C4×C271—22no—
88C22×C4×C271—41no—
89C22×C4×C271—41no—
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—
98C42×C271—22no—
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—21no—
117C2×Dic271—21no—
118C2×Dic271—21no—
119C2×Dic271—21no—
120C2×Dic271—21no—
121C2×Dic271—21no—
122C2×Dic271—44no—
123C2×Dic271—44no—
124C2×Dic271#4112yes—
125C23×C271—21no—
126C23×C271—21no—
127C23×C271—21no—
128C23×C271—21no—
129C23×C271—21no—
130C23×C271—21no—
131C23×C271—21no—
132C23×C271—21no—
133C23×C271—21no—
134C23×C271—21no—
135C23×C271C2×D412yes—
136C23×C271C2×D412yes—
137C23×C271#4112yes—
138C23×C271#4112yes—
139C23×C271C2411yescharacteristic
140D4×C271—28no—
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—44no—
155C2×C4×C271—44no—
156C2×C4×C271C2×D412yes—
157C2×C4×C271#4112yes—
158C22×C271—22no—
159C22×C271—22no—
160C22×C271—22no—
161C22×C271—22no—
162C22×C271—22no—
163C22×C271—22no—
164C22×C271—22no—
165C22×C271—22no—
166C22×C271—22no—
167C22×C271—41no—
168C22×C271—41no—
169C22×C271—41no—
170C22×C271—41no—
171C22×C271#12314yes—
172C22×C271C22×D411yescharacteristic
173C22×C271C2×#4111yescharacteristic
174C22×C271C2×#4111yescharacteristic
175C22×C271C2×#4111yescharacteristic
176C22×C271C2×#4111yescharacteristic
177C22×C271H(5,𝔽2)11yescharacteristic
178C22×C2712−1+411yescharacteristic
179C4×C271—22no—
180C4×C271—22no—
181C4×C271—41no—
182C4×C271—41no—
183C4×C271—41no—
184Dic271—22no—
185Dic271—41no—
186Dic271—41no—
187Dic271—41no—
188Dic271—41no—
189Dic271—41no—
190Dic271—41no—
191Dic271—41no—
192C2×C271—24no—
193C2×C271—42no—
194C2×C271C2×#12311yescharacteristic
195C2×C271#52911yescharacteristic
196C2×C271#53911yescharacteristic
197C2×C271#54611yescharacteristic
198C2×C271#54711yescharacteristic
199C2×C271#55111yescharacteristic
200C2×C271#55211yescharacteristic
201C271#241911yescharacteristic, complemented, Sylow
202#2419—2711nomaximal, Sylow
203C2×#123—2711no—
204C2×C4×D4D27111yescharacteristic
205#400—2711no—
206#398—2711no—
207#397—2711no—
208#397—2711no—
209#387—2711no—
210#387—2711no—
211#386—2711no—
212#385—2711no—
213#385—2711no—
214#381—2711no—
215#381—2711no—
216#380—2711no—
217C2×#95—2711no—
218C22×D4C2×D27111yescharacteristic
219C23×C4—2711no—
220C23×C4—2711no—
221C23×C4C2×D27111yescharacteristic
222C23×C4C2×D27111yescharacteristic
223#123—5424no—
224C4×D4—24no—
225C2×C4⋊C4—2711no—
226C2×C4⋊C4—2711no—
227C2×C4⋊C4—2711no—
228C2×C4⋊C4—2711no—
229C2×C4⋊C4—2711no—
230C2×C4⋊C4—2711no—
231C2×C4⋊C4C2×D27111yescharacteristic
232C2×C22⋊C4—2711no—
233C2×C22⋊C4—2711no—
234C2×C22⋊C4—2711no—
235C2×C22⋊C4—2711no—
236C2×C22⋊C4—2711no—
237C2×C22⋊C4—2711no—
238C2×C22⋊C4C2×D27111yescharacteristic
239C2×C22⋊C4C2×D27111yescharacteristic
240C2×C42C2×D27111yescharacteristic
241#95—2711no—
242#95—2711no—
243#95—2711no—
244#95—2711no—
245#95—2711no—
246#95—2711no—
247#95—2711no—
248#95—2711no—
249#95—2711no—
250#95—2711no—
251#95—2711no—
252#95—2711no—
253#95—2711no—
254#95—2711no—
255C24C22×D27111yescharacteristic
256C24C22×D27111yescharacteristic
257C2×D4—22no—
258C2×D4—22no—
259C2×D4C22⋊D27114yes—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—2711no—
267C22×C4—2711no—
268C22×C4—2711no—
269C22×C4—2711no—
270C22×C4—2711no—
271C22×C4—2711no—
272C22×C4—2711no—
273C22×C4—2711no—
274C22×C4—5422no—
275C22×C4—5422no—
276C22×C4—5422no—
277C22×C4—5422no—
278C22×C4—5422no—
279C22×C4—10841no—
280C22×C4C22×D27111yescharacteristic
281C22×C4C22×D27111yescharacteristic
282C22×C4C22×D27111yescharacteristic
283C22×C4C22×D27111yescharacteristic
284C22×C4C22×D27111yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—5422no—
287C4⋊C4—5422no—
288C4⋊C4—5422no—
289C4⋊C4—5422no—
290C4⋊C4—10841no—
291C4⋊C4—10841no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—5422no—
295C22⋊C4—5422no—
296C22⋊C4—5422no—
297C22⋊C4—5422no—
298C22⋊C4—5422no—
299C22⋊C4—10841no—
300C42—22no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23C2×C22⋊D27112yes—
312C23C2×C22⋊D27112yes—
313C23#1686465412yes—
314C23#1686465412yes—
315C23C23×D27111yescharacteristic
316D4—28no—
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—44no—
331C2×C4—44no—
332C2×C4—2712no—
333C2×C4—5421no—
334C2×C4—5421no—
335C2×C4—5421no—
336C2×C4—5421no—
337C2×C4—5421no—
338C2×C4—5421no—
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—5421no—
355C2×C4—5421no—
356C2×C4—10844no—
357C2×C4—10844no—
358C2×C4C2×C22⋊D27112yes—
359C2×C4#1686465312yes—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22#2257614314yes—
374C22C22×C22⋊D27111yescharacteristic
375C22#41×D27111yescharacteristic
376C22C2×#1686465311yescharacteristic
377C22C2×#1686465411yescharacteristic
378C22C2×#1686465411yescharacteristic
379C22#2257620511yescharacteristic
380C22#2257620711yescharacteristic
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—5422no—
387C4—10841no—
388C4—10841no—
389C4—10841no—
390C4—10841no—
391C4—10841no—
392C4—10841no—
393C4—10841no—
394C2—24no—
395C2—42no—
396C2C2×#2257614311yescharacteristic
397C2#3177442711yescharacteristic
398C2#3177448911yescharacteristic
399C2#3177454111yescharacteristic
400C2#3177454611yescharacteristic
401C2#3177457611yescharacteristic
402C2#3177458711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC2×C4×D4×C271C2
prime radicalC22×D4×C271C22
hypercenterC2×C4×D4D271
hypocenterC271#2419
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #31773835C2semidirect
3. #31773828C2semidirect
4. #31773825C2semidirect
5. #31773825C2semidirect
6. #31773781C2semidirect
7. #31773781C2semidirect
9. #31773770C2semidirect
10. #31773770C2semidirect
11. #31773760C2semidirect
12. #31773758C2semidirect
13. #31773755C2semidirect
14. C2×C271⋊#95C2semidirect
201. C271#2419semidirect

Maximal subgroups

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

subgroupindex
C2×#225761412
#317738352
#317738282
#317738252
#317737812
#317737772
#317737702
#317737602
#317737582
#317737552
C2×C271⋊#952
C2×C4×D4×C2712
#2419271