Finite Group Databasebeta

← #46847913

Subgroups — #46847913

Order 34688.

478 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×#22576181C211yesmaximal, characteristic, complemented
2#31774558C211yesmaximal, characteristic, complemented
3#31774546C211yesmaximal, characteristic, complemented
4#31774290C211yesmaximal, characteristic, complemented
5#31774143C211yesmaximal, characteristic, complemented
6#31774135C211yesmaximal, characteristic, complemented
7#31774114C211yesmaximal, characteristic, complemented
8#31774105C211yesmaximal, characteristic, complemented
9#31774058C211yesmaximal, characteristic, complemented
10#31774056C211yesmaximal, characteristic, complemented
11#31774053C211yesmaximal, characteristic, complemented
12#31774042C211yesmaximal, characteristic, complemented
13#31773998C211yesmaximal, characteristic, complemented
14#31773966C211yesmaximal, characteristic, complemented
15C2×#137×C271C211yesmaximal, characteristic, complemented
16C2×#16864655C2211yescharacteristic
17C2×#16864653C2211yescharacteristic
18#22576181—24no—
19C2×#16864644C2211yescharacteristic
20C2×#16864644C2211yescharacteristic
21C2×#16864643C2211yescharacteristic
22C2×#16864641C2211yescharacteristic
23C2×#16864639C2211yescharacteristic
24C2×C8×D271C2211yescharacteristic
25#22576158C2211yescharacteristic
26#22576149—21no—
27#22576149C2211yescharacteristic
28#22576143—21no—
29#22576141—21no—
30#22576139C2211yescharacteristic
31#22576136—21no—
32#22576136C2211yescharacteristic
33#22576136C2211yescharacteristic
34#22576134—21no—
35#22576134C2211yescharacteristic
36#22576132—21no—
37Q8×Dic271C2211yescharacteristic
38D4×Dic271C2211yescharacteristic
39C4×C22⋊D271—21no—
40C4×C22⋊D271C2211yescharacteristic
41C2×Dic271⋊C4—21no—
42#22576094C2211yescharacteristic
43Dic271⋊C8C2211yescharacteristic
44#22576085C2211yescharacteristic
45#22576084C2211yescharacteristic
46#22576084C2211yescharacteristic
47#22576084C2211yescharacteristic
48#22576082C2211yescharacteristic
49#22576081C2211yescharacteristic
50C22⋊(C271⋊C8)C2211yescharacteristic
51(C2×D271)⋊C8C2211yescharacteristic
52#22576070C2211yescharacteristic
53C2×#41×C271C2211yescharacteristic
54C22×Q8×C271C2211yescharacteristic
55#137×C271—24no—
56C2×Q16×C271C2211yescharacteristic
57C2×Q16×C271C2211yescharacteristic
58C2×SD16×C271C2211yescharacteristic
59C2×SD16×C271C2211yescharacteristic
60C2×M16×C271C2211yescharacteristic
61#16864655—22no—
62#16864655—41no—
63#16864653—22no—
64#16864653—41no—
65C2×Dic542C2311yescharacteristic
66C2×D1084—21no—
67C2×D1084C2311yescharacteristic
68C2×C22⋊D271—21no—
69C2×C22⋊D271—21no—
70C2×C22⋊D271D411yescharacteristic
71C22×Dic271—21no—
72C22×Dic271—21no—
73C2×C4×D271—21no—
74C2×C4×D271C2311yescharacteristic
75#16864644—22no—
76#16864644—41no—
77#16864643—22no—
78#16864641—41no—
79#16864639—22no—
80C2×C271⋊C8C2311yescharacteristic
81C2×C271⋊C8C2311yescharacteristic
82C8×D271—22no—
83C4⋊Dic271—21no—
84C4⋊Dic271C2311yescharacteristic
85C4⋊Dic271C2311yescharacteristic
86Dic271⋊C4—21no—
87Dic271⋊C4—21no—
88Dic271⋊C4—21no—
89Dic271⋊C4—21no—
90Dic271⋊C4—21no—
91Dic271⋊C4D411yescharacteristic
92C22⋊Dic271—21no—
93C22⋊Dic271—21no—
94C22⋊Dic271D411yescharacteristic
95(C2×D271)⋊C4—21no—
96(C2×D271)⋊C4—21no—
97(C2×D271)⋊C4—21no—
98(C2×D271)⋊C4—21no—
99(C2×D271)⋊C4—21no—
100(C2×D271)⋊C4D411yescharacteristic
101C4×Dic271—21no—
102C4×Dic271C2311yescharacteristic
103#41×C271—22no—
104#41×C271—41no—
105C2×Q8×C271—21no—
106C2×Q8×C271—22no—
107C2×Q8×C271—41no—
108C2×Q8×C271C2311yescharacteristic
109C2×Q8×C271C2311yescharacteristic
110C2×Q8×C271C2311yescharacteristic
111C2×D4×C271—21no—
112C2×D4×C271C2311yescharacteristic
113C22×C4×C271—21no—
114C22×C4×C271—21no—
115C22×C4×C271C2311yescharacteristic
116Q16×C271—22no—
117Q16×C271—22no—
118SD16×C271—22no—
119SD16×C271D414yescomplemented
120M16×C271—22no—
121C2×C8×C271C2311yescharacteristic
122C2×C8×C271C2311yescharacteristic
123C22×D271—21no—
124C22×D271C2×D411yescharacteristic
125Dic542—21no—
126Dic542—21no—
127D1084—21no—
128D1084—21no—
129D1084—42no—
130C22⋊D271—41no—
131C22⋊D271—42no—
132C22⋊D271—42no—
133C2×Dic271—21no—
134C2×Dic271—21no—
135C2×Dic271—21no—
136C2×Dic271—42no—
137C2×Dic271—42no—
138C2×Dic271C2×D411yescharacteristic
139C2×Dic271C2×D411yescharacteristic
140C2×Dic271C2×D411yescharacteristic
141C4×D271—21no—
142C4×D271—21no—
143C4×D271—42no—
144C271⋊C8—21no—
145C271⋊C8—21no—
146C23×C271—21no—
147C23×C271C2×D411yescharacteristic
148Q8×C271—21no—
149Q8×C271—21no—
150Q8×C271—21no—
151Q8×C271—21no—
152Q8×C271—21no—
153Q8×C271—42no—
154Q8×C271C2×D412yes—
155D4×C271—21no—
156D4×C271—42no—
157D4×C271C2×D412yes—
158C2×C4×C271—21no—
159C2×C4×C271—21no—
160C2×C4×C271—21no—
161C2×C4×C271—21no—
162C2×C4×C271—21no—
163C2×C4×C271—42no—
164C2×C4×C271—42no—
165C2×C4×C271C2×D411yescharacteristic
166C2×C4×C271C2411yescharacteristic
167C8×C271—21no—
168C8×C271C2×D412yes—
169C22×C271—21no—
170C22×C271—22no—
171C22×C271—21no—
172C22×C271—21no—
173C22×C271—41no—
174C22×C271—41no—
175C22×C271C22×D411yescharacteristic
176C2×D271—21no—
177C2×D271—21no—
178C2×D271—21no—
179C2×D271—41no—
180C2×D271—41no—
181C2×D271—41no—
182C4×C271—21no—
183C4×C271—22no—
184C4×C271—41no—
185C4×C271—41no—
186C4×C271C22×D411yescharacteristic
187C4×C271H(5,𝔽2)11yescharacteristic
188Dic271—21no—
189Dic271—21no—
190Dic271—21no—
191Dic271—41no—
192Dic271—41no—
193Dic271—41no—
194C2×C271—41no—
195C2×C271—42no—
196C2×C271D4211yescharacteristic
197C2×C271#57511yescharacteristic
198C2×C271#57811yescharacteristic
199D271—41no—
200D271—81no—
201C271#326911yescharacteristic, complemented, Sylow
202#3269—2711nomaximal, Sylow
203C2×#137D27111yescharacteristic, complemented
204C2×#135—2711no—
205#548—2711no—
206#547—2711no—
207#497—2711no—
208#468—2711no—
209#467—2711no—
210#463—2711no—
211#462—2711no—
212#452—2711no—
213#452—2711no—
214#451—2711no—
215#449—2711no—
216#440—2711no—
217#435—2711no—
218C2×#41—2711no—
219C2×#41—2711no—
220C2×#41C2×D27111yescharacteristic
221C22×Q8C2×D27111yescharacteristic
222#137—24no—
223#135—5424no—
224C2×Q16—2711no—
225C2×Q16—2711no—
226C2×Q16C2×D27111yescharacteristic
227C2×Q16C2×D27111yescharacteristic
228C2×SD16—2711no—
229C2×SD16—2711no—
230C2×SD16C2×D27111yescharacteristic
231C2×SD16C2×D27111yescharacteristic
232C2×D8—2711no—
233C2×M16C2×D27111yescharacteristic
234C22×C8—2711no—
235C4⋊Q8—2711no—
236#124—2711no—
237#124—5421no—
238#123—5421no—
239#123—5421no—
240C22⋊Q8—2711no—
241C22⋊Q8—2711no—
242C22⋊Q8—2711no—
243C22⋊Q8—5421no—
244#121—2711no—
245#121—5421no—
246#121—5421no—
247C4×Q8—2711no—
248C4×D4—2711no—
249C4×D4—2711no—
250C4×D4—5421no—
251C2×C4⋊C4—5421no—
252#107—2711no—
253C4⋊C8—2711no—
254#103—2711no—
255#103—2711no—
256#103—2711no—
257#103—2711no—
258#102—2711no—
259#102—2711no—
260C22⋊C8—2711no—
261C22⋊C8—2711no—
262#97—2711no—
263#41—22no—
264#41—41no—
265#41—5422no—
266#41—5422no—
267#41—10841no—
268#41—10841no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—2711no—
273C2×Q8C22×D27111yescharacteristic
274C2×Q8C22×D27111yescharacteristic
275C2×Q8C22×D27111yescharacteristic
276C2×D4—21no—
277C2×D4—2711no—
278C2×D4—2711no—
279C2×D4—5421no—
280C2×D4—5421no—
281C2×D4—5421no—
282C2×D4C22×D27111yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—2711no—
286C22×C4—5421no—
287C22×C4—5421no—
288C22×C4—5421no—
289C22×C4C22×D27111yescharacteristic
290Q16—22no—
291Q16—22no—
292Q16—5422no—
293Q16—10841no—
294SD16—22no—
295SD16—5422no—
296SD16—10841no—
297SD16C22⋊D27114yescomplemented
298D8—5422no—
299M16—22no—
300C2×C8—2711no—
301C2×C8—2711no—
302C2×C8—5422no—
303C2×C8C22×D27111yescharacteristic
304C2×C8C22×D27111yescharacteristic
305C4⋊C4—2711no—
306C4⋊C4—2711no—
307C4⋊C4—2711no—
308C4⋊C4—5421no—
309C4⋊C4—5421no—
310C4⋊C4—5421no—
311C4⋊C4—5421no—
312C4⋊C4—5421no—
313C4⋊C4—5421no—
314C22⋊C4—2711no—
315C22⋊C4—2711no—
316C22⋊C4—5421no—
317C22⋊C4—5421no—
318C22⋊C4—5421no—
319C22⋊C4—5421no—
320C22⋊C4—5421no—
321C22⋊C4—5421no—
322C22⋊C4—5421no—
323C42—2711no—
324C42—5421no—
325C23—21no—
326C23—2711no—
327C23—5421no—
328C23D4×D27111yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—42no—
335Q8—5421no—
336Q8—5421no—
337Q8C2×C22⋊D27112yes—
338D4—21no—
339D4—42no—
340D4—5421no—
341D4—5421no—
342D4—10842no—
343D4—10841no—
344D4—10842no—
345D4—10842no—
346D4C2×C22⋊D27112yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—2711no—
355C2×C4—2711no—
356C2×C4—2711no—
357C2×C4—5421no—
358C2×C4—5421no—
359C2×C4—5421no—
360C2×C4—5421no—
361C2×C4—5421no—
362C2×C4—10842no—
363C2×C4—10842no—
364C2×C4—10842no—
365C2×C4D4×D27111yescharacteristic
366C2×C4C23×D27111yescharacteristic
367C8—21no—
368C8—5421no—
369C8—5421no—
370C8C2×C22⋊D27112yes—
371C22—21no—
372C22—22no—
373C22—21no—
374C22—21no—
375C22—41no—
376C22—41no—
377C22—5421no—
378C22—5421no—
379C22—5421no—
380C22—10841no—
381C22—10841no—
382C22—10841no—
383C22C2×D4×D27111yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—5421no—
389C4—5421no—
390C4—5421no—
391C4—10841no—
392C4—10841no—
393C4—10841no—
394C4C22×C22⋊D27111yescharacteristic
395C4#2257620511yescharacteristic
396C2—41no—
397C2—42no—
398C2—10841no—
399C2—21681no—
400C2D4×C22⋊D27111yescharacteristic
401C2#3177469611yescharacteristic
402C2#3177471311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×#137×C271C2
socleC22×C271C22×D4
hypercenterC2×#137D271
hypocenterC271#3269
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#22576181C2semidirect
2. #31774558C2semidirect
3. #31774546C2semidirect
4. #31774290C2semidirect
5. #31774143C2semidirect
6. #31774135C2semidirect
7. #31774114C2semidirect
8. #31774105C2semidirect
9. #31774058C2semidirect
10. #31774056C2semidirect
11. #31774053C2semidirect
12. #31774042C2semidirect
13. #31773998C2semidirect
14. #31773966C2semidirect
15. C2×#137×C271C2semidirect
119. SD16×C271D4semidirect
201. C271#3269semidirect
203. C2×#137D271semidirect
297. SD16C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761812
#317745582
#317745462
#317742902
#317741432
#317741352
#317741142
#317741052
#317740582
#317740562
#317740532
#317740422
#317739982
#317739662
C2×#137×C2712
#3269271