Finite Group Databasebeta

← #13248607

Subgroups — #13248607

Order 1792.

592 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89618C211yesmaximal, characteristic
2#11731769C211yesmaximal, characteristic
3#11728210C211yesmaximal, characteristic, complemented
4C2×#89278C211yesmaximal, characteristic
5#11727336C211yesmaximal, characteristic, complemented
6C7×#1296C211yesmaximal, characteristic, complemented
7#11725510C211yesmaximal, characteristic, complemented
8C2×#6455—21no—
9C2×C4×D28C2211yescharacteristic
10C2×#6432C2211yescharacteristic
11C2×C7×C4⋊C8C2211yescharacteristic
12C2×C7×C4⋊C8C2211yescharacteristic
13C7×#384C2211yescharacteristic
14C2×#6395—21no—
15C2×#6392—21no—
16C2×#6390—21no—
17C2×#6390—21no—
18#89762—22no—
19#89760—21no—
20#89759—21no—
21C4⋊C4×Dic7—21no—
22C4×C4⋊Dic7C2211yescharacteristic
23C2×C4×C7⋊C8C2211yescharacteristic
24#89618—22no—
25#89618C414yescomplemented
26#89324—22no—
27#89278—22no—
28#8984—71nomaximal, Sylow
29C22×D28D411yescharacteristic
30C22×C4×D7—21no—
31C22×Dic14—21no—
32C22×C7×C8—21no—
33C22×C7×C8—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4D412yes—
37C2×C42×C7C2311yescharacteristic
38C2×(C2×D7)⋊C4—21no—
39C2×(C2×D7)⋊C4—22no—
40C2×(C2×D7)⋊C4—22no—
41C2×C4⋊Dic7—21no—
42C2×C4⋊Dic7—21no—
43C2×C4⋊Dic7D411yescharacteristic
44C2×C4⋊Dic7D411yescharacteristic
45C2×C4⋊Dic7D411yescharacteristic
46C2×C4×Dic7—22no—
47C2×C4×Dic7—21no—
48C22×C7⋊C8—21no—
49#6455—42no—
50#6455—42no—
51C4×D28—21no—
52C4×D28—21no—
53C4×D28C2×C412yes—
54C4×D28D412yes—
55#6432—21no—
56#6432—21no—
57#6432—21no—
58#6432C2×C412yes—
59C7×C4⋊C8—21no—
60C7×C4⋊C8—21no—
61C7×C4⋊C8—21no—
62C7×C4⋊C8C2×C412yes—
63C7⋊#95—22no—
64C7⋊#95—21no—
65#6395—24no—
66#6392—24no—
67#6390—42no—
68#6390—42no—
69C4×C7⋊C8—21no—
70C4×C7⋊C8—21no—
71C2×#474—71no—
72#2354—71no—
73#1990—71no—
74#1981—71no—
75C2×D4⋊C8—71no—
76#1296D711yescharacteristic, complemented
77#1279—71no—
78C23×D7—21no—
79C22×C4×C7—22no—
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×Dic7—21no—
84C22×Dic7—21no—
85C22×Dic7—21no—
86C22×Dic7—41no—
87C2×D28—21no—
88C2×D28—21no—
89C2×D28—21no—
90C2×D28—21no—
91C2×D28—21no—
92C2×D28C22⋊C412yes—
93C2×C4×D7—22no—
94C2×C4×D7—22no—
95C2×C4×D7—41no—
96C2×C4×D7—41no—
97C2×C4×D7—41no—
98C2×C4×D7—41no—
99C2×Dic14—22no—
100C2×Dic14—41no—
101C2×Dic14—41no—
102C2×Dic14—41no—
103C2×Dic14—41no—
104C2×Dic14—41no—
105C2×C7×C8—22no—
106C2×C7×C8—41no—
107C2×C7×C8—41no—
108C2×C7×C8—41no—
109C2×C7×C8—41no—
110C2×C7×C8—41no—
111C7×C4⋊C4—42no—
112C7×C4⋊C4—41no—
113C7×C4⋊C4—41no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C42×C7C22×C411yescharacteristic
117C42×C7C2×D411yescharacteristic
118C42×C7#4111yescharacteristic
119C42×C7#4111yescharacteristic
120(C2×D7)⋊C4—22no—
121(C2×D7)⋊C4—22no—
122(C2×D7)⋊C4—44no—
123(C2×D7)⋊C4—82no—
124C4⋊Dic7—21no—
125C4⋊Dic7—21no—
126C4⋊Dic7—22no—
127C4⋊Dic7—22no—
128C4⋊Dic7—21no—
129C4⋊Dic7—21no—
130C4⋊Dic7—41no—
131C4⋊Dic7—41no—
132C4⋊Dic7C22⋊C412yes—
133C4⋊Dic7D812yes—
134C4×Dic7—44no—
135C4×Dic7—41no—
136C4×Dic7—41no—
137C2×C7⋊C8—22no—
138C2×C7⋊C8—41no—
139C2×C7⋊C8—41no—
140C2×C4⋊Q8—71no—
141C2×C4×D4—71no—
142C2×#117—141no—
143#474—74no—
144#474—142no—
145C2×#107—141no—
146C2×C4⋊C8C2×D711yescharacteristic
147C2×C4⋊C8C2×D711yescharacteristic
148C2×#103—141no—
149C2×#103—141no—
150C2×#102—141no—
151C2×C4×C8—71no—
152#390—141no—
153#388—142no—
154#384C2×D711yescharacteristic
155#380—141no—
156C4×C4⋊C4—71no—
157C4×C4⋊C4—141no—
158#340—142no—
159D4⋊C8—142no—
160C23×C7#12011yescharacteristic
161C22×D7—22no—
162C22×D7—41no—
163C22×D7—41no—
164C22×D7—41no—
165C22×D7—41no—
166C22×D7—41no—
167C22×D7—41no—
168C2×C4×C7—21no—
169C2×C4×C7—21no—
170C2×C4×C7—42no—
171C2×C4×C7—42no—
172C2×C4×C7—42no—
173C2×C4×C7C2×C22⋊C411yescharacteristic
174C2×C4×C7C4×D411yescharacteristic
175C2×C4×C7C4×D411yescharacteristic
176C2×C4×C7#12111yescharacteristic
177C2×C4×C7#12111yescharacteristic
178C2×C4×C7#12311yescharacteristic
179C2×C4×C7C2×D811yescharacteristic
180C2×C4×C7#13511yescharacteristic
181C2×C4×C7#13511yescharacteristic
182C2×C4×C7#13511yescharacteristic
183C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C7#13711yescharacteristic
186C2×C4×C7#13711yescharacteristic
187C2×Dic7—22no—
188C2×Dic7—22no—
189C2×Dic7—41no—
190C2×Dic7—41no—
191C2×Dic7—41no—
192C2×Dic7—41no—
193C2×Dic7—42no—
194C2×Dic7—41no—
195C2×Dic7—41no—
196C2×Dic7—41no—
197C2×Dic7—81no—
198C2×Dic7—81no—
199D28—22no—
200D28—22no—
201D28—41no—
202D28—41no—
203C4×D7—42no—
204C4×D7—42no—
205C4×D7—42no—
206C4×D7—42no—
207Dic14—42no—
208Dic14—42no—
209Dic14—81no—
210Dic14—81no—
211C7×C8—42no—
212C7×C8—81no—
213C7⋊C8—42no—
214C22×Q8—141no—
215C22×D4—71no—
216C23×C4—141no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—141no—
220C4⋊Q8—72no—
221C4⋊Q8—141no—
222C4⋊Q8—141no—
223C4⋊Q8—141no—
224C4×D4—72no—
225C4×D4—72no—
226C4×D4—141no—
227C4×D4—141no—
228#117—282no—
229#117—282no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—71no—
233C2×C4⋊C4—71no—
234C2×C4⋊C4—71no—
235C2×C4⋊C4—141no—
236C2×C4⋊C4—141no—
237C2×C4⋊C4D2812yes—
238C2×C22⋊C4—141no—
239C2×C22⋊C4—142no—
240C2×C22⋊C4—142no—
241C2×C42—142no—
242C2×C42—141no—
243C2×C42C22×D711yescharacteristic
244#107—144no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8C4×D712yes—
249#103—282no—
250#103—282no—
251#102—144no—
252C4×C8—141no—
253C4×C8—141no—
254#95—142no—
255#95—141no—
256C22×C7#38611yescharacteristic
257C22×C7#44711yescharacteristic
258C22×C7#44911yescharacteristic
259C22×C7#45211yescharacteristic
260C22×C7#45211yescharacteristic
261C22×C7#45511yescharacteristic
262C22×C7#45611yescharacteristic
263C2×D7—42no—
264C2×D7—42no—
265C2×D7—42no—
266C2×D7—81no—
267C2×D7—81no—
268C2×D7—81no—
269C2×D7—81no—
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—82no—
273C4×C7#41611yescharacteristic
274C4×C7#41911yescharacteristic
275C4×C7C4×D811yescharacteristic
276C4×C7#44211yescharacteristic
277C4×C7#45911yescharacteristic
278C4×C7#46311yescharacteristic
279C4×C7#48511yescharacteristic
280C4×C7#48511yescharacteristic
281Dic7—42no—
282Dic7—42no—
283Dic7—82no—
284Dic7—81no—
285C24—141no—
286C2×Q8—142no—
287C2×Q8—281no—
288C2×Q8—281no—
289C2×Q8—281no—
290C2×Q8—281no—
291C2×Q8—281no—
292C2×D4—72no—
293C2×D4—141no—
294C2×D4—141no—
295C2×D4—141no—
296C2×D4—141no—
297C2×D4—141no—
298C22×C4—22no—
299C22×C4—141no—
300C22×C4—141no—
301C22×C4—142no—
302C22×C4—142no—
303C22×C4—141no—
304C22×C4—281no—
305C22×C4—281no—
306C22×C4—281no—
307C22×C4—281no—
308C22×C4—281no—
309C22×C4C2×D2811yescharacteristic
310C22×C4D4×D711yescharacteristic
311C22×C4D4×D711yescharacteristic
312C2×C8—22no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—142no—
319C2×C8—281no—
320C2×C8—281no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—72no—
327C4⋊C4—72no—
328C4⋊C4—141no—
329C4⋊C4—141no—
330C4⋊C4—142no—
331C4⋊C4—142no—
332C4⋊C4—141no—
333C4⋊C4—141no—
334C4⋊C4—281no—
335C4⋊C4—281no—
336C22⋊C4—142no—
337C22⋊C4—142no—
338C22⋊C4—284no—
339C22⋊C4—562no—
340C42—284no—
341C42—281no—
342C42—281no—
343C42C2×C4×D711yescharacteristic
344C42#115711yescharacteristic
345C42D4×D711yescharacteristic
346C42#116111yescharacteristic
347C2×C7#164111yescharacteristic
348C2×C7#164611yescharacteristic
349C2×C7#164711yescharacteristic
350C2×C7#165211yescharacteristic
351C2×C7#187311yescharacteristic
352C2×C7#187911yescharacteristic
353C2×C7#188211yescharacteristic
354D7—82no—
355C23—142no—
356C23—281no—
357C23—281no—
358C23—281no—
359C23—281no—
360C23—281no—
361C23—281no—
362C23#644511yescharacteristic
363Q8—282no—
364Q8—282no—
365Q8—561no—
366Q8—561no—
367D4—142no—
368D4—142no—
369D4—281no—
370D4—281no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—142no—
377C2×C4—142no—
378C2×C4—281no—
379C2×C4—281no—
380C2×C4—281no—
381C2×C4—281no—
382C2×C4—282no—
383C2×C4—282no—
384C2×C4—282no—
385C2×C4—282no—
386C2×C4—282no—
387C2×C4—281no—
388C2×C4—281no—
389C2×C4—281no—
390C2×C4—561no—
391C2×C4—561no—
392C2×C4C4×D2811yescharacteristic
393C2×C4D7×C22⋊C411yescharacteristic
394C2×C4#644711yescharacteristic
395C2×C4#645611yescharacteristic
396C2×C4#645711yescharacteristic
397C2×C4#645811yescharacteristic
398C2×C4D7×D811yescharacteristic
399C2×C4#647511yescharacteristic
400C2×C4#647711yescharacteristic
401C2×C4#647711yescharacteristic
402C2×C4#647811yescharacteristic
403C2×C4#647911yescharacteristic
404C2×C4#648111yescharacteristic
405C2×C4#648211yescharacteristic
406C8—42no—
407C8—81no—
408C8—282no—
409C7#898411yescharacteristic, complemented, Sylow
410C22—282no—
411C22—282no—
412C22—282no—
413C22—561no—
414C22—561no—
415C22—561no—
416C22—561no—
417C22#8943711yescharacteristic
418C22#8951811yescharacteristic
419C22#8952411yescharacteristic
420C22#8954511yescharacteristic
421C22#8955511yescharacteristic
422C22#8958211yescharacteristic
423C22#8958311yescharacteristic
424C4—21no—
425C4—21no—
426C4—82no—
427C4—282no—
428C4—282no—
429C4—562no—
430C4—561no—
431C4#8954411yescharacteristic
432C4#8955811yescharacteristic
433C4#8957411yescharacteristic
434C4#8959111yescharacteristic
435C4#8963911yescharacteristic
436C4#8964111yescharacteristic
437C4#8964411yescharacteristic
438C4#8966011yescharacteristic
439C2—562no—
440C2#1172616111yescharacteristic
441C2#1172619011yescharacteristic
442C2#1172621611yescharacteristic
443C2#1172624911yescharacteristic
444C2#1172663311yescharacteristic
445C2#1172665111yescharacteristic
446C2#1172675811yescharacteristic
subgroupG/H
centerC23#6445
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1296C2
socleC23×C7#120
prime radicalC22×D28D4
cosocleC2×C42×C7C23
hypercenter#1296D7
hypocenterC7#8984
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. #11728210C2semidirect
5. #11727336C2semidirect
6. C7×#1296C2semidirect
7. #11725510C2semidirect
25. #89618C4semidirect
76. #1296D7semidirect
409. C7#8984semidirect

Maximal subgroups

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

subgroupindex
C2×#896182
#117317692
#117282102
C2×#892782
#117273362
C7×#12962
#117255102
#89847