Finite Group Databasebeta

← #34030240

Subgroups — #34030240

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989646C211yesmaximal, characteristic, complemented
2C2×#17989645C211yesmaximal, characteristic, complemented
3#22626322C214yesmaximal, complemented
4#22626319C214yesmaximal, complemented
5#22626274C211yesmaximal, characteristic, complemented
6#22626175C211yesmaximal, characteristic, complemented
7#22626173C211yesmaximal, characteristic, complemented
8#22626140C211yesmaximal, characteristic, complemented
9#22618830C211yesmaximal, characteristic
10#17989646—21no—
11#17989646—21no—
12#17989646C2212yescomplemented
13#17989646C2212yescomplemented
14#17989645—21no—
15#17989645—21no—
16#17989645C2212yescomplemented
17#17989645C2212yescomplemented
18C2×#14556666—21no—
19C2×#14556665—21no—
20C2×#14556665C2211yescharacteristic
21C2×#14556665C2211yescharacteristic
22C2×#14556665C2211yescharacteristic
23C2×#14556664—21no—
24C22×#11924031—21no—
25C22×#11924031C2211yescharacteristic
26#17989510C2214yescomplemented
27#17989486C2214yescomplemented
28#17989485C2214yescomplemented
29#17989475C2212yescomplemented
30C4×#11924031C2212yescomplemented
31#17989385C411yescharacteristic
32C2×#14556599C2211yescharacteristic
33C2×#14556598C2211yescharacteristic
34#17988706C414yes—
35#17988701C2214yes—
36#17988635C411yescharacteristic
37#17988605C411yescharacteristic
38#17988543C2211yescharacteristic
39#421×C73C411yescharacteristic
40#14556666—22no—
41#14556666—22no—
42#14556665—22no—
43#14556665—22no—
44#14556665—21no—
45#14556665—21no—
46#14556665—21no—
47#14556665—21no—
48#14556665C2312yes—
49#14556665C2312yes—
50#14556664—22no—
51#14556664—22no—
52C2×#11924031—22no—
53C2×#11924031—22no—
54C2×#11924031—21no—
55C2×#11924031—21no—
56C2×#11924031D411yescharacteristic
57C2×#11924031D411yescharacteristic
58C2×#11924031D411yescharacteristic
59C2×#11924031D411yescharacteristic
60C2×#11924031C2312yes—
61C2×#11924031C2311yescharacteristic
62C2×#11924031C2311yescharacteristic
63C2×#11924028C2×C411yescharacteristic
64#14556622C812yes—
65#14556621C812yes—
66#14556617C812yes—
67#14556616C812yes—
68#14556611C2×C414yes—
69#14556599—21no—
70#14556599—21no—
71#14556599C2×C412yes—
72#14556599C2312yes—
73#14556598—21no—
74#14556598—21no—
75#14556598C2×C412yes—
76#14556598C2312yes—
77C2×#11924012—21no—
78C2×#11924012C2×C411yescharacteristic
79C2×#11924012C2×C411yescharacteristic
80C2×#11924011—21no—
81C22×#10596831—21no—
82C22×#10596831C2311yescharacteristic
83C2×C8×D73—21no—
84#14556526C2×C414yes—
85#14556519C2×C414yes—
86#14556502C2×C412yes—
87C4×#10596831C2312yes—
88C2×#41×C73C2×C411yescharacteristic
89C2×M16×C73C2×C411yescharacteristic
90#117×C73C2×C411yescharacteristic
91#104×C73C2×C414yes—
92#11924031—22no—
93#11924031—22no—
94#11924031—22no—
95#11924031C2×D411yescharacteristic
96#11924031C2×D411yescharacteristic
97#11924031C2×D411yescharacteristic
98#11924031C2×D411yescharacteristic
99#11924028—21no—
100#11924028—21no—
101#11924028C2×C812yes—
102#11924028C22×C412yes—
103C2×Dic146C2×C811yescharacteristic
104C2×D292C2×C811yescharacteristic
105C2×C22⋊D73—21no—
106C2×C4×D73—21no—
107#11924018C2×C814yes—
108#11924016C2×C814yes—
109#11924015C2×C814yes—
110#11924013C2×C814yes—
111#11924012—21no—
112#11924012—22no—
113#11924012—22no—
114#11924012C2×C812yes—
115#11924012C2×C812yes—
116#11924012C22×C412yes—
117#11924011—22no—
118#11924011—22no—
119C2×#10596831—21no—
120C2×#10596831—22no—
121C2×#10596831—22no—
122C2×#10596831C22×C411yescharacteristic
123C2×#10596831C22×C412yes—
124C2×#10596831C2×D411yescharacteristic
125C2×#10596831C2×D411yescharacteristic
126C2×#10596831#4111yescharacteristic
127C2×#10596831#4111yescharacteristic
128C2×#10596831C2411yescharacteristic
129C8×D73—22no—
130C8×D73—22no—
131#41×C73—21no—
132#41×C73—21no—
133#41×C73C2×C812yes—
134#41×C73C22×C412yes—
135C2×Q8×C73C2×C811yescharacteristic
136C2×D4×C73—21no—
137C2×D4×C73C2×C811yescharacteristic
138C22×C4×C73—21no—
139C22×C4×C73C22×C411yescharacteristic
140M16×C73—21no—
141M16×C73C22×C412yes—
142C2×C8×C73—21no—
143C4⋊C4×C73—21no—
144C22⋊C4×C73—21no—
145C42×C73C22×C412yes—
146C23×C73—21no—
147C23×C73#13111yescharacteristic
148C22×D73—21no—
149Q8×C73—21no—
150Q8×C73C22×C812yes—
151D4×C73—22no—
152D4×C73—22no—
153D4×C73—21no—
154D4×C73C22×C812yes—
155C2×C4×C73—21no—
156C2×C4×C73—22no—
157C2×C4×C73—22no—
158C2×C4×C73—22no—
159C2×C4×C73C4×D411yescharacteristic
160C2×C4×C73C4×D411yescharacteristic
161C2×C4×C73C22×C811yescharacteristic
162C2×C4×C73C22×C811yescharacteristic
163C2×C4×C73#13111yescharacteristic
164C2×C4×C73C23×C411yescharacteristic
165C22⋊D73—22no—
166C22⋊D73—22no—
167C2×Dic73—21no—
168D292—21no—
169D292C22×C812yes—
170C4×D73—22no—
171C4×D73—22no—
172Dic146—21no—
173Dic146C22×C812yes—
174C8×C73—22no—
175#10596831—22no—
176#10596831C4×D411yescharacteristic
177#10596831C4×D411yescharacteristic
178#10596831C22×C814yes—
179#10596831C22×D411yescharacteristic
180#10596831C2×#4111yescharacteristic
181C22×C73—22no—
182C22×C73—41no—
183C22×C73—42no—
184C22×C73C8×D411yescharacteristic
185C22×C73C8×D411yescharacteristic
186C22×C73C2×#13111yescharacteristic
187C2×D73—22no—
188C2×D73—42no—
189C4×C73—22no—
190C4×C73—42no—
191C4×C73C8×D411yescharacteristic
192C4×C73C8×D411yescharacteristic
193C4×C73C2×C4×D411yescharacteristic
194C4×C73C23×C811yescharacteristic
195Dic73—22no—
196#33441—731nomaximal, Sylow
197C2×C73—21no—
198C2×C73—21no—
199C2×C73—21no—
200C2×C73—42no—
201C2×C73C2×C8×D411yescharacteristic
202D73—42no—
203C2×#504—731no—
204C2×#504—731no—
205#2955—731no—
206#2171—734no—
207#2170—734no—
208#2153—731no—
209#2117—731no—
210#2117—731no—
211#2108—731no—
212C73#3344111yescharacteristic, complemented, Sylow
213#575—731no—
214C2×#131—731no—
215C2×#131—731no—
216#504—732no—
217#504—732no—
218#504—732no—
219#504—732no—
220#504—1461no—
221#504—1461no—
222#504—1461no—
223#504—1461no—
224C2×M32—731no—
225C2×M32—731no—
226C2×M32—731no—
227C2×M32—1461no—
228C2×M32—1461no—
229C22×C16—731no—
230C22×C16—1461no—
231C22×C16—1461no—
232#444—734no—
233#443—734no—
234#430—731no—
235#421—731no—
236#421#6440811yescharacteristic
237#405—731no—
238#364—734no—
239#350—734no—
240#350—734no—
241#346—732no—
242C4×C16—732no—
243C2×#41—731no—
244C2×#41C2×#6440811yescharacteristic
245#137—732no—
246#137—732no—
247AGL(1,ℤ/8)—732no—
248AGL(1,ℤ/8)—732no—
249#135—734no—
250#131—732no—
251#131—732no—
252#131—732no—
253#131—732no—
254#131—1461no—
255#131—1461no—
256#131—1461no—
257#131—1461no—
258C2×M16—731no—
259C2×M16—731no—
260C2×M16—1461no—
261C2×M16—1461no—
262C2×M16C2×#6440811yescharacteristic
263C22×C8—731no—
264C22×C8—1461no—
265C22×C8—1461no—
266#117C2×#6440811yescharacteristic
267M32—732no—
268M32—732no—
269M32—1462no—
270M32—1462no—
271M32—1462no—
272M32—1462no—
273M32—1461no—
274M32—1461no—
275M32—1461no—
276M32—1461no—
277C2×C16—731no—
278C2×C16—731no—
279C2×C16—731no—
280C2×C16—731no—
281C2×C16—731no—
282C2×C16—731no—
283C2×C16—732no—
284C2×C16—1462no—
285C2×C16—1462no—
286C2×C16—1462no—
287C2×C16—1462no—
288C2×C16—1461no—
289C2×C16—1461no—
290#108—734no—
291#104—734no—
292#104C2×#6440814yes—
293#97—732no—
294C4×C8—732no—
295#41—21no—
296#41—21no—
297#41—732no—
298#41—732no—
299#41—1461no—
300#41—1461no—
301#41#1192403612yes—
302#41C22×#6440812yes—
303C2×Q8—731no—
304C2×Q8#1192403611yescharacteristic
305C2×D4—21no—
306C2×D4—731no—
307C2×D4—1461no—
308C2×D4#1192403611yescharacteristic
309C22×C4—21no—
310C22×C4—1461no—
311C22×C4C22×#6440811yescharacteristic
312Q16—734no—
313SD16—734no—
314SD16—734no—
315D8—734no—
316M16—21no—
317M16—732no—
318M16—732no—
319M16—732no—
320M16—1462no—
321M16—1462no—
322M16—1462no—
323M16—1462no—
324M16—1461no—
325M16C22×#6440812yes—
326C2×C8—21no—
327C2×C8—732no—
328C2×C8—731no—
329C2×C8—731no—
330C2×C8—731no—
331C2×C8—731no—
332C2×C8—731no—
333C2×C8—731no—
334C2×C8—1462no—
335C2×C8—1462no—
336C2×C8—1462no—
337C2×C8—1462no—
338C2×C8—1461no—
339C4⋊C4—21no—
340C22⋊C4—21no—
341C42C22×#6440812yes—
342C16—731no—
343C16—731no—
344C16—731no—
345C16—731no—
346C16—1462no—
347C16—1462no—
348C16—1462no—
349C23—21no—
350C23—1461no—
351C23#1455667811yescharacteristic
352Q8—21no—
353Q8—732no—
354Q8—1461no—
355Q8C2×#1192403612yes—
356D4—22no—
357D4—22no—
358D4—21no—
359D4—732no—
360D4—1462no—
361D4—1462no—
362D4—1461no—
363D4C2×#1192403612yes—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—1461no—
369C2×C4—1462no—
370C2×C4—1462no—
371C2×C4D4×#6440811yescharacteristic
372C2×C4D4×#6440811yescharacteristic
373C2×C4C2×#1192403611yescharacteristic
374C2×C4C2×#1192403611yescharacteristic
375C2×C4#1455667811yescharacteristic
376C2×C4C23×#6440811yescharacteristic
377C8—22no—
378C8—731no—
379C8—731no—
380C8—731no—
381C8—731no—
382C8—734no—
383C8—1462no—
384C22—22no—
385C22—41no—
386C22—42no—
387C22—2922no—
388C22—1462no—
389C22#1798961411yescharacteristic
390C22#1798961411yescharacteristic
391C22C2×#1455667811yescharacteristic
392C4—22no—
393C4—42no—
394C4—1462no—
395C4#1798961411yescharacteristic
396C4#1798961411yescharacteristic
397C4C2×D4×#6440811yescharacteristic
398C4C22×#1192403611yescharacteristic
399C2—21no—
400C2—21no—
401C2—21no—
402C2—42no—
403C2—2922no—
404C2C2×#1798961411yescharacteristic
subgroupG/H
centerC4C2×D4×#64408
commutatorC4×C73C23×C8
FrattiniC2×C4C23×#64408
Fitting#421×C73C4
socleC2×C73C2×C8×D4
prime radical#17989385C4
cosocleC2×#10596831C24
hypercenter#421#64408
hypocenterC73#33441
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#17989646C2semidirect
2. C2×#17989645C2semidirect
3. #22626322C2semidirect
4. #22626319C2semidirect
5. #22626274C2semidirect
6. #22626175C2semidirect
7. #22626173C2semidirect
8. #22626140C2semidirect
12. #17989646C22semidirect
13. #17989646C22semidirect
16. #17989645C22semidirect
17. #17989645C22semidirect
26. #17989510C22semidirect
27. #17989486C22semidirect
28. #17989485C22semidirect
29. #17989475C22semidirect
30. C4×#11924031C22semidirect
212. C73#33441semidirect

Maximal subgroups

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

subgroupindex
C2×#179896462
C2×#179896452
#226263222
#226263192
#226262742
#226261752
#226261732
#226261402
#226188302
#3344173