Finite Group Databasebeta

← #33240624

Subgroups — #33240624

Order 18688.

546 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556613C211yesmaximal, characteristic
2#22612092C212yesmaximal, complemented
3#22612085C212yesmaximal, complemented
4#22612080C211yesmaximal, characteristic, complemented
5#22612075C211yesmaximal, characteristic, complemented
6#22612070C211yesmaximal, characteristic, complemented
7#22612065C211yesmaximal, characteristic, complemented
8#22611713C212yesmaximal, complemented
9C2×C4⋊#11924003C211yesmaximal, characteristic, complemented
10#22611636C212yesmaximal
11C73×#3160C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13C4×#11924028—21no—
14C22×#11924003C2212yes—
15C4×Dic292C2211yescharacteristic
16C4×Dic292C2211yescharacteristic
17C4×#11924017—21no—
18C4×#11924017—21no—
19C4×D584C2211yescharacteristic
20C4×D584C2211yescharacteristic
21#17988625C2211yescharacteristic
22#17988584—21no—
23#17988414C2212yes—
24#17988414C2212yes—
25#17988412C2212yes—
26#17988412C2212yes—
27C4⋊#11924003C2211yescharacteristic
28C4⋊#11924003C2211yescharacteristic
29C4⋊#11924003C2212yes—
30#17988390C2212yes—
31#17988390C2212yes—
32#17988387C2212yes—
33#17988387C2212yes—
34#533×C73—21no—
35#500×C73C2211yescharacteristic
36#500×C73C2211yescharacteristic
37#499×C73—21no—
38#499×C73C2212yes—
39#498×C73—21no—
40#427×C73—21no—
41#427×C73C2212yes—
42C2×C4×C8×C73C2211yescharacteristic
43C2×#11924028—21no—
44#14556613—22no—
45#14556613Q814yescomplemented
46C2×Dic292C2311yescharacteristic
47C2×#11924017—21no—
48C2×D584C2311yescharacteristic
49C4×Dic146—21no—
50C4×Dic146—21no—
51C4×D292—21no—
52C4×D292—21no—
53C4×C22⋊D73—41no—
54C4×C22⋊D73—41no—
55#14556550—21no—
56#14556548—41no—
57C42×D73—41no—
58C2×#11924003—24no—
59C2×#11924003C2312yes—
60C2×#11924003C2312yes—
61#14556525C2311yescharacteristic
62#14556525C2311yescharacteristic
63#14556523—21no—
64#14556516—21no—
65#14556516—21no—
66#14556513—21no—
67#14556513—21no—
68C22×C8×C73D411yescharacteristic
69C22×C8×C73C2311yescharacteristic
70C4⋊Q8×C73—21no—
71C4⋊Q8×C73—21no—
72#125×C73—22no—
73C22⋊Q8×C73—41no—
74C22⋊Q8×C73—41no—
75C4×Q8×C73—41no—
76C4×Q8×C73—41no—
77#117×C73—22no—
78C2×C42×C73D411yescharacteristic
79#107×C73—21no—
80#107×C73—21no—
81#107×C73C2312yes—
82#107×C73C2312yes—
83#106×C73—22no—
84#106×C73—22no—
85C4×C8×C73D411yescharacteristic
86C4×C8×C73D411yescharacteristic
87C4×C8×C73C2311yescharacteristic
88C4×C8×C73C2311yescharacteristic
89#11924028—24no—
90#11924028—42no—
91C2×Dic146—21no—
92C2×D292—21no—
93C2×C22⋊D73—41no—
94C2×C4×D73—41no—
95C2×C4×D73—41no—
96C2×C4×D73—41no—
97Dic292—21no—
98Dic292C2×Q812yes—
99#11924017—24no—
100D584—21no—
101D584C2×Q812yes—
102C4⋊Dic73—21no—
103C4⋊Dic73—21no—
104#11924007—41no—
105#11924007—41no—
106C22⋊Dic73—41no—
107#11924005—41no—
108#11924005—41no—
109C4×Dic73—41no—
110C4×Dic73—41no—
111C4×Dic73—41no—
112#11924003—22no—
113#11924003—22no—
114C2×Q8×C73—41no—
115C2×Q8×C73—41no—
116C22×C4×C73D811yescharacteristic
117C22×C4×C73D811yescharacteristic
118C22×C4×C73C2×D411yescharacteristic
119C2×C8×C73—22no—
120C2×C8×C73C2×D411yescharacteristic
121C2×C8×C73C2×D411yescharacteristic
122C2×C8×C73C2×D411yescharacteristic
123C2×C8×C73C2×Q812yes—
124C2×C8×C73#4112yes—
125C2×C8×C73C2411yescharacteristic
126C4⋊C4×C73—22no—
127C4⋊C4×C73—22no—
128C4⋊C4×C73—42no—
129C4⋊C4×C73—42no—
130C4⋊C4×C73—41no—
131C4⋊C4×C73—41no—
132C22⋊C4×C73—42no—
133C42×C73—42no—
134C42×C73D811yescharacteristic
135C42×C73D811yescharacteristic
136C42×C73C2×D411yescharacteristic
137C42×C73C2×D411yescharacteristic
138C23×C73C2×D811yescharacteristic
139C22×D73—41no—
140Q8×C73—82no—
141Q8×C73—82no—
142C2×C4×C73—22no—
143C2×C4×C73—41no—
144C2×C4×C73—42no—
145C2×C4×C73—42no—
146C2×C4×C73C22⋊Q812yes—
147C2×C4×C73C2×D811yescharacteristic
148C2×C4×C73C2×D811yescharacteristic
149C2×C4×C73C2×D811yescharacteristic
150C2×C4×C73C2×D811yescharacteristic
151C2×C4×C73C2×D811yescharacteristic
152C2×C4×C73#13712yes—
153C2×C4×C73C22×D411yescharacteristic
154C22⋊D73—44no—
155C2×Dic73—41no—
156C2×Dic73—41no—
157C2×Dic73—41no—
158D292—22no—
159D292—41no—
160C4×D73—44no—
161C4×D73—82no—
162C4×D73—82no—
163Dic146—22no—
164Dic146—41no—
165C8×C73—21no—
166C8×C73—21no—
167C8×C73C22⋊Q812yes—
168C8×C73C22×Q811yescharacteristic
169C8×C73C2×#4111yescharacteristic
170C22×C73—21no—
171C22×C73—21no—
172C22×C73#47412yes—
173C22×C73C22×D811yescharacteristic
174C2×D73—42no—
175C2×D73—81no—
176C2×D73—81no—
177C4×C73—21no—
178C4×C73—21no—
179C4×C73—21no—
180C4×C73—82no—
181C4×C73—82no—
182C4×C73#47412yes—
183C4×C73#50812yes—
184C4×C73C2×C22⋊Q811yescharacteristic
185C4×C73C2×#13711yescharacteristic
186Dic73—42no—
187Dic73—81no—
188Dic73—81no—
189#33626—731nomaximal, Sylow
190C2×C73—22no—
191C2×C73C2×#47411yescharacteristic
192C2×C73C2×#50811yescharacteristic
193C2×C73#341411yescharacteristic
194D73—82no—
195#3160D7311yescharacteristic, complemented
196C4×#135—731no—
197#2229—732no—
198#2228—732no—
199#2227—731no—
200#2226—731no—
201#2225—731no—
202#2224—731no—
203#2158—732no—
204C2×C4⋊C16—731no—
205#2138—732no—
206C73#3362611yescharacteristic, complemented, Sylow
207C2×#135—731no—
208#533—21no—
209C4×#41—1461no—
210C22×C16—732no—
211#500C2×D7311yescharacteristic
212#500C2×D7311yescharacteristic
213#499—21no—
214#499C2×D7312yes—
215#498—21no—
216C4×Q16—731no—
217C4×Q16—731no—
218C4×SD16—1461no—
219C4×SD16—1461no—
220C4×D8—731no—
221C4×D8—731no—
222#427—21no—
223#427—731no—
224#427C2×D7312yes—
225#416—1461no—
226C2×C4×C8C2×D7311yescharacteristic
227#367—732no—
228#367—732no—
229#366—732no—
230#366—732no—
231C4⋊C16—731no—
232C4⋊C16—731no—
233C4⋊C16—732no—
234#358—732no—
235#358—732no—
236#357—732no—
237#357—732no—
238C2×#41—1461no—
239#135—734no—
240#135—1462no—
241C2×Q16—731no—
242C2×SD16—1461no—
243C2×D8—731no—
244C22×C8C22⋊D7311yescharacteristic
245C22×C8C22×D7311yescharacteristic
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248#125—22no—
249C22⋊Q8—41no—
250C22⋊Q8—41no—
251C4×Q8—41no—
252C4×Q8—41no—
253C4×Q8—1461no—
254C4×Q8—1461no—
255C4×D4—2921no—
256C4×D4—2921no—
257C4×D4—1461no—
258C4×D4—1461no—
259#117—22no—
260#117—2921no—
261#117—1461no—
262C2×C42—2921no—
263C2×C42C22⋊D7311yescharacteristic
264C2×C16—732no—
265C2×C16—732no—
266C2×C16—1464no—
267#107—21no—
268#107—21no—
269#107—731no—
270#107—731no—
271#107C22×D7312yes—
272#107C22×D7312yes—
273#106—22no—
274#106—22no—
275#106—1461no—
276#103—1461no—
277#103—1461no—
278#102—1461no—
279#102—1461no—
280C4×C8C22⋊D7311yescharacteristic
281C4×C8C22⋊D7311yescharacteristic
282C4×C8C22×D7311yescharacteristic
283C4×C8C22×D7311yescharacteristic
284#41—2922no—
285#41—1464no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—1461no—
289C2×D4—2921no—
290C2×D4—1461no—
291C22×C4—2921no—
292C22×C4—2921no—
293C22×C4—2921no—
294C22×C4#1192401311yescharacteristic
295C22×C4#1192401311yescharacteristic
296C22×C4C2×C22⋊D7311yescharacteristic
297Q16—732no—
298Q16—1461no—
299SD16—1464no—
300D8—732no—
301D8—1461no—
302C2×C8—22no—
303C2×C8C2×C22⋊D7311yescharacteristic
304C2×C8C2×C22⋊D7311yescharacteristic
305C2×C8C2×C22⋊D7311yescharacteristic
306C2×C8Q8×D7312yes—
307C2×C8#1192402912yes—
308C2×C8C23×D7311yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—2921no—
316C4⋊C4—2921no—
317C4⋊C4—1461no—
318C4⋊C4—1461no—
319C22⋊C4—42no—
320C22⋊C4—2921no—
321C22⋊C4—2921no—
322C22⋊C4—2921no—
323C42—42no—
324C42—2921no—
325C42—2921no—
326C42—2921no—
327C42#1192401311yescharacteristic
328C42#1192401311yescharacteristic
329C42C2×C22⋊D7311yescharacteristic
330C42C2×C22⋊D7311yescharacteristic
331C16—1462no—
332C16—1462no—
333C23—2921no—
334C23C2×#1192401311yescharacteristic
335Q8—82no—
336Q8—82no—
337Q8—2921no—
338Q8—1462no—
339D4—2924no—
340D4—2921no—
341D4—1462no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—2921no—
347C2×C4—2924no—
348C2×C4—2921no—
349C2×C4—2921no—
350C2×C4—5842no—
351C2×C4—5842no—
352C2×C4#1455656712yes—
353C2×C4C2×#1192401311yescharacteristic
354C2×C4C2×#1192401311yescharacteristic
355C2×C4C2×#1192401311yescharacteristic
356C2×C4C2×#1192401311yescharacteristic
357C2×C4C2×#1192401311yescharacteristic
358C2×C4#1455662212yes—
359C2×C4C22×C22⋊D7311yescharacteristic
360C8—21no—
361C8—21no—
362C8#1455656712yes—
363C8C2×Q8×D7311yescharacteristic
364C8C2×#1192402911yescharacteristic
365C22—21no—
366C22—21no—
367C22—2922no—
368C22—5841no—
369C22—5841no—
370C22#1798886412yes—
371C22C22×#1192401311yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—82no—
376C4—82no—
377C4—2922no—
378C4—5841no—
379C4—5841no—
380C4#1798886412yes—
381C4#1798901712yes—
382C4C2×#1455656711yescharacteristic
383C4C2×#1455662211yescharacteristic
384C2—22no—
385C2—5842no—
386C2C2×#1798886411yescharacteristic
387C2C2×#1798901711yescharacteristic
388C2#2262306511yescharacteristic
subgroupG/H
centerC2×C4C2×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3160C2
socleC22×C73C22×D8
prime radicalC2×#14556613C22
hypercenter#3160D73
hypocenterC73#33626
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3160

Complements

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

NH ≅ G/Ntype
2. #22612092C2semidirect
3. #22612085C2semidirect
4. #22612080C2semidirect
5. #22612075C2semidirect
6. #22612070C2semidirect
7. #22612065C2semidirect
8. #22611713C2semidirect
9. C2×C4⋊#11924003C2semidirect
11. C73×#3160C2semidirect
45. #14556613Q8semidirect
195. #3160D73semidirect
206. C73#33626semidirect

Maximal subgroups

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

subgroupindex
C4×#145566132
#226120922
#226120852
#226120802
#226120752
#226120702
#226120652
#226117132
C2×C4⋊#119240032
#226116362
C73×#31602
#3362673