Finite Group Databasebeta

← #33240640

Subgroups — #33240640

Order 18688.

598 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620442C211yesmaximal, characteristic, complemented
2C4×D8×D73C211yesmaximal, characteristic
3#22612087C212yesmaximal, complemented
4#22612086C212yesmaximal, complemented
5#22612075C211yesmaximal, characteristic, complemented
6#22612069C211yesmaximal, characteristic, complemented
7#22612068C211yesmaximal, characteristic, complemented
8#22611718C212yesmaximal, complemented
9#22611685C211yesmaximal, characteristic, complemented
10#22611645C212yesmaximal
11C73×#2226C211yesmaximal, characteristic, complemented
12C2×D8×D73C2211yescharacteristic
13#17989142—21no—
14C4×D4×D73—21no—
15C2×#14556530C2212yes—
16#17988992C2211yescharacteristic, complemented
17#17988988—21no—
18#17988988C2212yescomplemented
19#17988983—21no—
20C4×D584C2211yescharacteristic
21D8×Dic73C2211yescharacteristic
22#17988669C2211yescharacteristic
23C4×#11924013—21no—
24#17988667—21no—
25#17988624—21no—
26#17988624C2212yes—
27#107×D73C2211yescharacteristic
28#102×D73—21no—
29C4×C8×D73C2211yescharacteristic
30#17988414C2212yescomplemented
31#17988413C2212yescomplemented
32#17988412C2212yescomplemented
33C4⋊#11924003C2211yescharacteristic, complemented
34#17988405C2212yescomplemented
35#17988389C2212yes—
36#17988388C2212yes—
37#17988387C2212yes—
38#500×C73C2211yescharacteristic, complemented
39C4×D8×C73C2211yescharacteristic
40#367×C73C2212yescomplemented
41C4⋊C16×C73C2211yescharacteristic, complemented
42#357×C73C2212yes—
43C2×D4×D73—21no—
44C22×C4×D73—41no—
45C2×D584C2311yescharacteristic
46C2×#11924013—21no—
47D8×D73—22no—
48D8×D73Q814yescomplemented
49C2×C8×D73D411yescharacteristic
50C2×C8×D73C2311yescharacteristic
51#14556591—21no—
52#14556582—22no—
53#14556571—41no—
54#14556567—41no—
55Q8×Dic73—41no—
56#14556556—41no—
57C4×D292—21no—
58D4×Dic73—21no—
59#14556553—21no—
60C4×C22⋊D73—41no—
61#14556551—41no—
62#14556549—22no—
63C4⋊C4×D73—21no—
64C22⋊C4×D73—41no—
65C42×D73D411yescharacteristic
66#14556530—24no—
67C2×#11924003C2312yes—
68#14556525—21no—
69#14556525C2311yescharacteristic
70#14556525C2312yes—
71#14556524—21no—
72#14556522—22no—
73#14556522—22no—
74#14556513—21no—
75#14556512—21no—
76#14556511—21no—
77C4×#10596831D411yescharacteristic
78C8×Dic73D411yescharacteristic
79C8×Dic73C2311yescharacteristic
80C2×D8×C73C2311yescharacteristic
81C4⋊Q8×C73—21no—
82C4×D4×C73—21no—
83C2×C16×C73C2312yes—
84#107×C73—21no—
85#107×C73C2311yescharacteristic
86#107×C73C2312yes—
87#102×C73—21no—
88C4×C8×C73C2311yescharacteristic
89C23×D73—41no—
90C2×Dic146—41no—
91C2×D292—21no—
92C2×C22⋊D73—41no—
93D4×D73—24no—
94D4×D73—42no—
95C22×Dic73—41no—
96C2×C4×D73—41no—
97C2×C4×D73—41no—
98C2×C4×D73—41no—
99C2×C4×D73—84no—
100C2×C4×D73D811yescharacteristic
101C2×C4×D73D811yescharacteristic
102C2×C4×D73C2×D411yescharacteristic
103D584—21no—
104D584C2×Q812yes—
105#11924013—24no—
106C2×#10596831C2×D411yescharacteristic
107C2×#10596831C2×D411yescharacteristic
108C8×D73—22no—
109C8×D73C2×Q812yes—
110C8×D73#4112yes—
111C4⋊Dic73—21no—
112C4⋊Dic73—22no—
113C4⋊Dic73—41no—
114C4⋊Dic73—42no—
115#11924007—41no—
116#11924007—42no—
117C22⋊Dic73—41no—
118#11924005—42no—
119#11924005—41no—
120#11924005—41no—
121C4×Dic73—42no—
122C4×Dic73D811yescharacteristic
123C4×Dic73D811yescharacteristic
124C4×Dic73C2×D411yescharacteristic
125#11924003—22no—
126C2×Q8×C73—41no—
127C2×D4×C73—21no—
128C22×C4×C73—41no—
129D8×C73—21no—
130D8×C73C2×Q812yes—
131C2×C8×C73C2×D411yescharacteristic
132C2×C8×C73C2411yescharacteristic
133C4⋊C4×C73—21no—
134C4⋊C4×C73—22no—
135C4⋊C4×C73—41no—
136C22⋊C4×C73—41no—
137C42×C73C2×D411yescharacteristic
138C16×C73—22no—
139C23×C73—41no—
140C22×D73—41no—
141C22×D73—44no—
142C22×D73—82no—
143C22×D73—82no—
144C22×D73C2×D811yescharacteristic
145Q8×C73—82no—
146D4×C73—22no—
147D4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—42no—
150C2×C4×C73—82no—
151C2×C4×C73C2×D811yescharacteristic
152C2×C4×C73C2×D811yescharacteristic
153C2×C4×C73C22×D411yescharacteristic
154C22⋊D73—44no—
155C2×Dic73—42no—
156C2×Dic73—41no—
157C2×Dic73—82no—
158C2×Dic73C2×D811yescharacteristic
159C2×Dic73C2×D811yescharacteristic
160C2×Dic73C2×D811yescharacteristic
161D292—22no—
162D292—41no—
163C4×D73—22no—
164C4×D73—41no—
165C4×D73—82no—
166C4×D73—82no—
167C4×D73—82no—
168C4×D73C22⋊Q812yes—
169C4×D73#13712yes—
170Dic146—82no—
171C8×C73—21no—
172C8×C73C22×Q811yescharacteristic
173C8×C73C2×#4111yescharacteristic
174#10596831—21no—
175#10596831C22⋊Q812yes—
176C22×C73—42no—
177C22×C73—81no—
178C22×C73—81no—
179C22×C73C22×D811yescharacteristic
180C2×D73—21no—
181C2×D73—21no—
182C2×D73—42no—
183C2×D73—88no—
184C2×D73—81no—
185C2×D73—81no—
186C2×D73#47412yes—
187C4×C73—21no—
188C4×C73—81no—
189C4×C73—82no—
190C4×C73#50912yes—
191C4×C73C2×C22⋊Q811yescharacteristic
192C4×C73C2×#13711yescharacteristic
193Dic73—21no—
194Dic73—21no—
195Dic73—82no—
196Dic73—81no—
197Dic73#47412yes—
198#33627—731nomaximal, Sylow
199C2×C73—82no—
200C2×C73C2×#47411yescharacteristic
201C2×C73C2×#50911yescharacteristic
202C2×C73#341411yescharacteristic
203D73—22no—
204D73—82no—
205#3160—731no—
206C2×C4×D8—731no—
207#2228—732no—
208#2228—732no—
209#2226—731no—
210#2226D7311yescharacteristic, complemented
211#2224—731no—
212#2224—731no—
213#2159—732no—
214#2150—731no—
215#2140—732no—
216C73#3362711yescharacteristic, complemented, Sylow
217C22×D8—731no—
218#533—1461no—
219C2×C4×D4—1461no—
220C2×M32—732no—
221#500—731no—
222#500C2×D7311yescharacteristic, complemented
223#499—732no—
224#499—1461no—
225#498—1461no—
226C4×D8—731no—
227C4×D8—731no—
228C4×D8—731no—
229C4×D8—1461no—
230C4×D8—1461no—
231C4×D8C2×D7311yescharacteristic
232#427—732no—
233#427—1461no—
234C2×#107—731no—
235C2×#102—1461no—
236C2×C4×C8—731no—
237#367—732no—
238#367C2×D7312yescomplemented
239#366—732no—
240#366—732no—
241C4⋊C16—732no—
242C4⋊C16—731no—
243C4⋊C16C2×D7311yescharacteristic, complemented
244#357—732no—
245#357—732no—
246#357—732no—
247#357C2×D7312yes—
248C22×D4—1461no—
249C23×C4—2921no—
250C2×D8—734no—
251C2×D8—731no—
252C2×D8—1462no—
253C2×D8—1461no—
254C2×D8C22×D7311yescharacteristic
255C22×C8—731no—
256C22×C8—731no—
257C4⋊Q8—21no—
258C4⋊Q8—1461no—
259#125—1462no—
260C22⋊Q8—2921no—
261C22⋊Q8—2921no—
262C4×Q8—2921no—
263C4×Q8—2921no—
264C4×D4—21no—
265C4×D4—2921no—
266C4×D4—2921no—
267C4×D4—1461no—
268C4×D4—1461no—
269C4×D4—1461no—
270#117—1462no—
271C2×C4⋊C4—1461no—
272C2×C22⋊C4—2921no—
273C2×C42—731no—
274M32—1464no—
275C2×C16—732no—
276C2×C16C22×D7312yes—
277#107—21no—
278#107—732no—
279#107—731no—
280#107—1461no—
281#107—1461no—
282#107C22×D7312yes—
283#107C22×D7311yescharacteristic
284#106—1462no—
285#106—1462no—
286#102—21no—
287#102—1461no—
288#102—1461no—
289#102—1461no—
290C4×C8—731no—
291C4×C8—731no—
292C4×C8—731no—
293C4×C8C22×D7311yescharacteristic
294C24—2921no—
295C2×Q8—41no—
296C2×Q8—2921no—
297C2×D4—21no—
298C2×D4—2922no—
299C2×D4—2921no—
300C2×D4—1464no—
301C2×D4—1461no—
302C22×C4—41no—
303C22×C4—2921no—
304C22×C4—2921no—
305C22×C4—2921no—
306C22×C4—2921no—
307C22×C4—5844no—
308C22×C4—731no—
309C22×C4—731no—
310C22×C4—731no—
311D8—21no—
312D8—732no—
313D8—1464no—
314D8—1461no—
315D8Q8×D7312yes—
316C2×C8—731no—
317C2×C8—732no—
318C2×C8—732no—
319C2×C8—731no—
320C2×C8—1462no—
321C2×C8D4×D7311yescharacteristic
322C2×C8C23×D7311yescharacteristic
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—2921no—
327C4⋊C4—2922no—
328C4⋊C4—2922no—
329C4⋊C4—2921no—
330C4⋊C4—1461no—
331C4⋊C4—1462no—
332C22⋊C4—41no—
333C22⋊C4—2922no—
334C22⋊C4—2921no—
335C22⋊C4—2921no—
336C22⋊C4—2921no—
337C42—2922no—
338C42—731no—
339C42—731no—
340C42—731no—
341C42D4×D7311yescharacteristic
342C16—22no—
343C16—1462no—
344C23—41no—
345C23—2924no—
346C23—2921no—
347C23—5842no—
348C23—5842no—
349C23—731no—
350Q8—82no—
351Q8—5842no—
352D4—22no—
353D4—41no—
354D4—2924no—
355D4—2921no—
356D4—1462no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—82no—
360C2×C4—2921no—
361C2×C4—2922no—
362C2×C4—2921no—
363C2×C4—5842no—
364C2×C4—5842no—
365C2×C4—5842no—
366C2×C4—5842no—
367C2×C4—731no—
368C2×C4—732no—
369C2×C4—732no—
370C2×C4—731no—
371C2×C4—731no—
372C2×C4—1462no—
373C2×C4D8×D7311yescharacteristic
374C2×C4D8×D7311yescharacteristic
375C2×C4C2×D4×D7311yescharacteristic
376C8—21no—
377C8—732no—
378C8—1461no—
379C8C2×Q8×D7311yescharacteristic
380C8#41×D7311yescharacteristic
381C22—42no—
382C22—81no—
383C22—81no—
384C22—2922no—
385C22—5848no—
386C22—5841no—
387C22—5841no—
388C22—732no—
389C22—1461no—
390C22—1461no—
391C22C2×D8×D7311yescharacteristic
392C4—21no—
393C4—81no—
394C4—82no—
395C4—5841no—
396C4—5842no—
397C4—732no—
398C4—1461no—
399C4—1461no—
400C4#1798902412yes—
401C4C22⋊Q8×D7311yescharacteristic
402C4#137×D7311yescharacteristic
403C2—82no—
404C2—5842no—
405C2—1462no—
406C2#474×D7311yescharacteristic
407C2C2×#1798902411yescharacteristic
408C2#2262306311yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2226C2
socleC22×C73C22×D8
prime radicalC2×D8×D73C22
hypercenter#2226D73
hypocenterC73#33627
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2226

Complements

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

NH ≅ G/Ntype
1. #22620442C2semidirect
3. #22612087C2semidirect
4. #22612086C2semidirect
5. #22612075C2semidirect
6. #22612069C2semidirect
7. #22612068C2semidirect
8. #22611718C2semidirect
9. #22611685C2semidirect
11. C73×#2226C2semidirect
16. #17988992C22semidirect
18. #17988988C22semidirect
30. #17988414C22semidirect
31. #17988413C22semidirect
32. #17988412C22semidirect
33. C4⋊#11924003C22semidirect
34. #17988405C22semidirect
38. #500×C73C22semidirect
40. #367×C73C22semidirect
41. C4⋊C16×C73C22semidirect
48. D8×D73Q8semidirect
210. #2226D73semidirect
216. C73#33627semidirect
222. #500C2×D73semidirect
238. #367C2×D73semidirect
243. C4⋊C16C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226204422
C4×D8×D732
#226120872
#226120862
#226120752
#226120692
#226120682
#226117182
#226116852
#226116452
C73×#22262
#3362773