Finite Group Databasebeta

← #34019620

Subgroups — #34019620

Order 18688.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626762C212yesmaximal, complemented
2#22625570C211yesmaximal, characteristic, complemented
3#22625564C211yesmaximal, characteristic, complemented
4#22625541C212yesmaximal, complemented
5#22625534C212yesmaximal, complemented
6#22625519C211yesmaximal, characteristic, complemented
7D8×#10596833C211yesmaximal, characteristic, complemented
8#22625478C211yesmaximal, characteristic, complemented
9#22625242C212yesmaximal, complemented
10C8×#11924037C211yesmaximal, characteristic, complemented
11#22620496C211yesmaximal, characteristic
12#17989637C2212yes—
13#17989636C2212yes—
14#17989622—22no—
15#17989622C2212yes—
16#17989618—22no—
17D4×#10596833C2212yes—
18#17989611—22no—
19#17989604C2211yescharacteristic
20#17989603C2212yes—
21#17989551—22no—
22C4×#11924037C2211yescharacteristic
23#17989450C2211yescharacteristic
24#17989450C2211yescharacteristic
25#17989448C2212yes—
26#17989437C2212yes—
27#17989437C2212yes—
28D4⋊#10596833C2212yes—
29D4⋊#10596833C2212yes—
30#17989429C2211yescharacteristic
31#17989429C2211yescharacteristic
32C8×#10596833C2211yescharacteristic
33C8×#10596833C2211yescharacteristic
34#17989118C2212yes—
35#17988988C411yescharacteristic
36#17988970C411yescharacteristic
37#17988940C412yes—
38#17988828C411yescharacteristic
39#17988814C412yes—
40#17988681C2211yescharacteristic
41#17988678C2212yes—
42D8×Dic73C2211yescharacteristic
43C2×C8×Dic73C2211yescharacteristic
44#466×C73C411yescharacteristic
45C2×#11924037—22no—
46C2×#11924037D411yescharacteristic
47C2×#11924037D411yescharacteristic
48C22×#10596833—22no—
49#14556658—22no—
50#14556658—21no—
51C4⋊#10596833—22no—
52C4⋊#10596833C2312yes—
53C4⋊#10596833C2312yes—
54C22⋊#10596833—22no—
55C22⋊#10596833—22no—
56C22⋊#10596833—22no—
57C22⋊#10596833—21no—
58#14556646—22no—
59#14556646D411yescharacteristic
60#14556646D411yescharacteristic
61C4×#10596833—22no—
62C4×#10596833—22no—
63C4×#10596833C2311yescharacteristic
64C4×#10596833C2311yescharacteristic
65C4×#10596833C2311yescharacteristic
66C4×#10596833C2311yescharacteristic
67C2×Dic292C2×C411yescharacteristic
68C2×#11924016C2×C412yes—
69C22×#10596831C2×C411yescharacteristic
70#14556584—22no—
71#14556582C2×C412yes—
72#14556578C2×C412yes—
73#14556572—22no—
74#14556570C2×C412yes—
75#14556556C2312yes—
76D4×Dic73—22no—
77D4×Dic73C2312yes—
78#14556547—22no—
79C2×C4×Dic73C2311yescharacteristic
80#14556525C2×C411yescharacteristic
81#14556522C2×C412yes—
82#14556516C2×C412yes—
83#14556514C2×C412yes—
84#14556512C2×C412yes—
85C8×Dic73C2×C411yescharacteristic
86C8×Dic73C2×C411yescharacteristic
87C8×Dic73C2311yescharacteristic
88C8×Dic73C2311yescharacteristic
89C2×D8×C73C2×C411yescharacteristic
90C22×C8×C73C2×C411yescharacteristic
91#121×C73C2×C412yes—
92#107×C73C2×C411yescharacteristic
93#102×C73C2×C412yes—
94#11924037—22no—
95#11924037—22no—
96#11924037—44no—
97C2×#10596833—22no—
98C2×#10596833—22no—
99C2×#10596833—21no—
100C2×#10596833—21no—
101C2×#10596833—44no—
102C2×#10596833C2×D411yescharacteristic
103C2×#10596833C2×D411yescharacteristic
104C2×#10596833C2×D411yescharacteristic
105C2×#10596833C2×D411yescharacteristic
106C2×Dic146C22×C412yes—
107C22×Dic73—22no—
108C22×Dic73C2×D411yescharacteristic
109C22×Dic73#4111yescharacteristic
110Dic292—22no—
111#11924016—42no—
112C2×#10596831—41no—
113C2×#10596831C22×C411yescharacteristic
114C2×#10596831C22×C411yescharacteristic
115C4⋊Dic73—22no—
116C4⋊Dic73C22×C412yes—
117#11924007—22no—
118#11924007—22no—
119#11924007—22no—
120C22⋊Dic73—22no—
121C22⋊Dic73—22no—
122C22⋊Dic73—22no—
123C22⋊Dic73—22no—
124C4×Dic73—22no—
125C4×Dic73C22×C411yescharacteristic
126C4×Dic73C2×D411yescharacteristic
127C4×Dic73#4111yescharacteristic
128C4×Dic73C2411yescharacteristic
129C2×D4×C73—22no—
130C2×D4×C73C22×C412yes—
131C22×C4×C73C22×C411yescharacteristic
132D8×C73—22no—
133C2×C8×C73—22no—
134C2×C8×C73C22×C411yescharacteristic
135C2×C8×C73C22×C411yescharacteristic
136C4⋊C4×C73C22×C412yes—
137C22⋊C4×C73—22no—
138C23×C73—22no—
139C23×C73C4×D411yescharacteristic
140D4×C73—22no—
141D4×C73—22no—
142D4×C73—82no—
143C2×C4×C73—21no—
144C2×C4×C73—21no—
145C2×C4×C73—22no—
146C2×C4×C73C4×D411yescharacteristic
147C2×C4×C73C23×C411yescharacteristic
148C2×Dic73—21no—
149C2×Dic73—21no—
150C2×Dic73—22no—
151C2×Dic73—22no—
152C2×Dic73—41no—
153C2×Dic73—42no—
154C2×Dic73—42no—
155C2×Dic73C4×D411yescharacteristic
156C2×Dic73C4×D411yescharacteristic
157C2×Dic73C22×D411yescharacteristic
158C2×Dic73C2×#4111yescharacteristic
159Dic146—22no—
160Dic146—22no—
161#10596833—21no—
162#10596833—21no—
163#10596833—41no—
164#10596833—41no—
165#10596833—42no—
166#10596833—42no—
167#10596833#13512yes—
168#10596833#13512yes—
169C8×C73—21no—
170C8×C73#13112yes—
171#10596831—21no—
172#10596831—21no—
173C22×C73—21no—
174C22×C73—21no—
175C22×C73—21no—
176C22×C73—42no—
177C22×C73—42no—
178C22×C73—42no—
179C22×C73C2×C4×D411yescharacteristic
180C4×C73—21no—
181C4×C73—42no—
182C4×C73C2×#13111yescharacteristic
183C4×C73#56811yescharacteristic
184Dic73—21no—
185Dic73—21no—
186Dic73—21no—
187Dic73—42no—
188Dic73—42no—
189Dic73C2×#13511yescharacteristic
190Dic73C2×#13511yescharacteristic
191#18069—731nomaximal, Sylow
192C2×C73—41no—
193C2×C73—82no—
194C2×C73#293511yescharacteristic
195C2×C73C4×#13511yescharacteristic
196C2×C73#295711yescharacteristic
197#3168—731no—
198#2969—732no—
199#1591—731no—
200#1589—731no—
201#1583—732no—
202#1582—732no—
203C8×Q16—731no—
204C8×D8—731no—
205#1564—731no—
206#1482—732no—
207C8×M16—731no—
208C73#1806911yescharacteristic, complemented, Sylow
209#529—732no—
210#499—731no—
211#495—731no—
212#488—732no—
213#467—731no—
214#466#6440811yescharacteristic
215#465—732no—
216#446—732no—
217C8×Q8—732no—
218C4×Q16—731no—
219C4×SD16—732no—
220C4×D8—731no—
221#436—732no—
222#436—1462no—
223#435—1462no—
224C8×D4—732no—
225#433—1462no—
226#431—731no—
227#431—732no—
228#405—1462no—
229C4×M16—731no—
230C2×C4×C8—731no—
231#335—731no—
232#335—731no—
233#334—732no—
234Q8⋊C8—732no—
235Q8⋊C8—732no—
236D4⋊C8—732no—
237D4⋊C8—732no—
238#322—731no—
239#322—731no—
240C82—731no—
241C82—731no—
242C2×Q16—731no—
243C2×SD16—732no—
244C2×D8C2×#6440811yescharacteristic
245C2×M16—731no—
246C2×M16—731no—
247C2×M16—1462no—
248C22×C8—731no—
249C22×C8—1462no—
250C22×C8C2×#6440811yescharacteristic
251#126—1462no—
252#125—732no—
253#124—732no—
254#123—1462no—
255C22⋊Q8—732no—
256#121C2×#6440812yes—
257C4×Q8—732no—
258C4×D4—732no—
259C4×D4—1462no—
260#117—1462no—
261C2×C42—731no—
262#107—731no—
263#107C2×#6440811yescharacteristic
264#106—732no—
265C4⋊C8—732no—
266C4⋊C8—732no—
267C4⋊C8—1461no—
268C4⋊C8—1462no—
269C4⋊C8—1462no—
270#103—732no—
271#103—732no—
272#102—732no—
273#102C2×#6440812yes—
274C22⋊C8—1462no—
275C22⋊C8—1462no—
276C22⋊C8—1462no—
277C22⋊C8—1461no—
278#97—731no—
279#97—731no—
280#97—1462no—
281C4×C8—731no—
282C4×C8—731no—
283C4×C8—731no—
284C4×C8—731no—
285C4×C8—731no—
286C4×C8—731no—
287C4×C8—731no—
288C4×C8—731no—
289C4×C8—1462no—
290C4×C8—1462no—
291C2×Q8—732no—
292C2×D4—22no—
293C2×D4C22×#6440812yes—
294C22×C4—731no—
295C22×C4—731no—
296C22×C4—1462no—
297C22×C4C22×#6440811yescharacteristic
298Q16—1462no—
299SD16—2922no—
300D8—22no—
301M16—2924no—
302M16—1462no—
303M16—1462no—
304C2×C8—22no—
305C2×C8—2924no—
306C2×C8—2921no—
307C2×C8—731no—
308C2×C8—731no—
309C2×C8—731no—
310C2×C8—731no—
311C2×C8—731no—
312C2×C8—731no—
313C2×C8—1461no—
314C2×C8—1461no—
315C2×C8—1462no—
316C2×C8—1462no—
317C2×C8C22×#6440811yescharacteristic
318C2×C8C22×#6440811yescharacteristic
319C4⋊C4—732no—
320C4⋊C4—1462no—
321C4⋊C4—1462no—
322C4⋊C4—1462no—
323C4⋊C4—1462no—
324C4⋊C4C22×#6440812yes—
325C22⋊C4—22no—
326C22⋊C4—1462no—
327C22⋊C4—1462no—
328C22⋊C4—1462no—
329C22⋊C4—1462no—
330C42—731no—
331C42—731no—
332C42—731no—
333C42—731no—
334C42—1462no—
335C23—22no—
336C23D4×#6440811yescharacteristic
337Q8—1462no—
338Q8—1462no—
339D4—22no—
340D4—22no—
341D4—82no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—2922no—
346C2×C4—2922no—
347C2×C4—2921no—
348C2×C4—731no—
349C2×C4—731no—
350C2×C4—731no—
351C2×C4—731no—
352C2×C4—1462no—
353C2×C4—1462no—
354C2×C4—1461no—
355C2×C4—1461no—
356C2×C4D4×#6440811yescharacteristic
357C2×C4C23×#6440811yescharacteristic
358C8—21no—
359C8—2922no—
360C8—2922no—
361C8—2921no—
362C8—2921no—
363C8—732no—
364C8—732no—
365C8—1461no—
366C8—1461no—
367C8—1461no—
368C8—1461no—
369C8#1455667812yes—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22C2×D4×#6440811yescharacteristic
377C4—21no—
378C4—42no—
379C4—2922no—
380C4—2922no—
381C4—731no—
382C4—731no—
383C4—1461no—
384C4—1461no—
385C4—1461no—
386C4C2×#1455667811yescharacteristic
387C4#1798966611yescharacteristic
388C2—41no—
389C2—82no—
390C2#2262667311yescharacteristic
391C2#135×#6440811yescharacteristic
392C2#2262670711yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting, prime radical#466×C73C4
socleC22×C73C2×C4×D4
cosocleC4×Dic73C24
hypercenter#466#64408
hypocenterC73#18069
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #22626762C2semidirect
2. #22625570C2semidirect
3. #22625564C2semidirect
4. #22625541C2semidirect
5. #22625534C2semidirect
6. #22625519C2semidirect
7. D8×#10596833C2semidirect
8. #22625478C2semidirect
9. #22625242C2semidirect
10. C8×#11924037C2semidirect
208. C73#18069semidirect

Maximal subgroups

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

subgroupindex
#226267622
#226255702
#226255642
#226255412
#226255342
#226255192
D8×#105968332
#226254782
#226252422
C8×#119240372
#226204962
#1806973