Finite Group Databasebeta

← #34034844

Subgroups — #34034844

Order 18688.

648 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22627186C211yesmaximal, characteristic, complemented
2#22627184C211yesmaximal, characteristic, complemented
3#22627151C214yesmaximal, complemented
4#22627149C214yesmaximal, complemented
5#22627125C211yesmaximal, characteristic, complemented
6#22627115C211yesmaximal, characteristic, complemented
7#22627113C211yesmaximal, characteristic, complemented
8#64408⋊M32C211yesmaximal, characteristic, complemented
9C2×#17989630C211yesmaximal, characteristic
10#17989709—22no—
11#17989709C2214yescomplemented
12#17989708—22no—
13#17989708C2214yescomplemented
14#17989706—22no—
15#17989706C2211yescharacteristic
16#17989706C2211yescharacteristic
17#17989706C2211yescharacteristic
18#17989706C2211yescharacteristic
19C2×#14556687—22no—
20#17989692C2212yescomplemented
21#17989691C2212yescomplemented
22#17989688C2214yescomplemented
23#17989687C2214yescomplemented
24#17989682C2212yescomplemented
25#64408⋊C16C2212yescomplemented
26C2×#14556679C2211yescharacteristic
27C2×#14556678C2211yescharacteristic
28#17989630C412yes—
29#17989630C412yes—
30#17989630C2212yes—
31#17989630C2212yes—
32C2×#14556661C411yescharacteristic
33C2×#14556656C411yescharacteristic
34C2×#14556655C411yescharacteristic
35C2×C8×#64408C2211yescharacteristic
36C2×#14556612C411yescharacteristic
37#14556688—24no—
38#14556688—21no—
39#14556688D412yes—
40#14556688C2312yes—
41#14556688C2312yes—
42#14556687—24no—
43#14556687—21no—
44#14556687—21no—
45#14556687—21no—
46#14556687D412yes—
47#14556686—24no—
48#14556686—21no—
49#14556686—21no—
50#14556686C2312yes—
51#14556686C2312yes—
52C2×#11924040—24no—
53#14556679—21no—
54#14556679—21no—
55#14556679C2×C412yes—
56#14556679C2312yes—
57#14556678—21no—
58#14556678—21no—
59#14556678C2×C412yes—
60#14556678C2312yes—
61C2×#11924038—21no—
62C2×#11924038C2×C411yescharacteristic
63C2×#11924038C2×C411yescharacteristic
64C2×#11924037—21no—
65C2×#11924036—21no—
66C2×#11924036C2311yescharacteristic
67C22×#10596833—21no—
68C2×C4×#64408C2×C411yescharacteristic
69#14556661C2×C412yes—
70#14556661C2×C412yes—
71#14556656C2×C412yes—
72#14556656C2×C412yes—
73#14556655C2×C412yes—
74#14556655C2×C412yes—
75C8×#64408C2×C411yescharacteristic
76C8×#64408C2×C411yescharacteristic
77C8×#64408C2311yescharacteristic
78C8×#64408C2311yescharacteristic
79C2×#11924029C2×C411yescharacteristic
80C2×#11924027C2×C411yescharacteristic
81#14556612C812yes—
82#14556612C812yes—
83#14556612C2×C412yes—
84#14556612C2×C412yes—
85C2×#11924018C811yescharacteristic
86C2×#11924017C811yescharacteristic
87C2×#11924013C811yescharacteristic
88C2×C8×D73C2×C411yescharacteristic
89C2×SD16×C73C811yescharacteristic
90#11924040—22no—
91#11924040—22no—
92#11924040—22no—
93#11924040C2×D414yes—
94C22×#64408—21no—
95#11924038—21no—
96#11924038—21no—
97#11924038—22no—
98#11924038—22no—
99#11924038C22×C412yes—
100#11924038C22×C412yes—
101#11924037—22no—
102#11924037—22no—
103#11924036—22no—
104#11924036—22no—
105#11924036C22×C411yescharacteristic
106#11924036C2×D411yescharacteristic
107#11924036#4111yescharacteristic
108#11924036C2411yescharacteristic
109C2×#10596833—21no—
110C2×#10596833—21no—
111C2×#10596833—22no—
112C2×#10596833—22no—
113C2×#10596833C2×D411yescharacteristic
114C2×#10596833#4111yescharacteristic
115C4×#64408—21no—
116C4×#64408C22×C411yescharacteristic
117C4×#64408C22×C411yescharacteristic
118#11924029—21no—
119#11924029—21no—
120#11924029C2×C812yes—
121#11924029C22×C412yes—
122#11924027—21no—
123#11924027—21no—
124#11924027C2×C812yes—
125#11924027C22×C412yes—
126C2×Dic146C2×C811yescharacteristic
127C2×D292—21no—
128C2×D292C2×C811yescharacteristic
129C2×C22⋊D73—21no—
130C22×Dic73—21no—
131C2×C4×D73—21no—
132C2×C4×D73C22×C411yescharacteristic
133#11924018C2×C812yes—
134#11924018C2×C812yes—
135#11924017C2×C812yes—
136#11924017C2×C812yes—
137#11924013C2×C812yes—
138#11924013C2×C812yes—
139C2×#10596831C2×C811yescharacteristic
140C8×D73C2×C811yescharacteristic
141C8×D73C2×C811yescharacteristic
142C8×D73C22×C411yescharacteristic
143C8×D73C22×C411yescharacteristic
144C2×Q8×C73C2×C811yescharacteristic
145C2×D4×C73C2×C811yescharacteristic
146SD16×C73C2×C812yes—
147SD16×C73C2×C812yes—
148C2×C8×C73C2×C811yescharacteristic
149C23×C73—21no—
150C22×D73—21no—
151C22×D73#13111yescharacteristic
152C2×#64408—21no—
153C2×#64408—21no—
154C2×#64408—42no—
155Q8×C73—21no—
156Q8×C73C22×C812yes—
157D4×C73—21no—
158D4×C73C22×C812yes—
159C2×C4×C73—21no—
160C2×C4×C73C22×C811yescharacteristic
161C22⋊D73—22no—
162C22⋊D73—22no—
163C2×Dic73—21no—
164C2×Dic73—22no—
165C2×Dic73—22no—
166C2×Dic73C4×D411yescharacteristic
167D292—22no—
168D292—22no—
169D292—21no—
170D292C22×C812yes—
171C4×D73—22no—
172C4×D73—22no—
173C4×D73C4×D411yescharacteristic
174C4×D73C22×C811yescharacteristic
175C4×D73#13111yescharacteristic
176C4×D73C23×C411yescharacteristic
177Dic146—21no—
178Dic146C22×C812yes—
179#10596833—22no—
180#10596833—22no—
181#10596833C4×D411yescharacteristic
182#10596833C4×D411yescharacteristic
183#10596833C22×D411yescharacteristic
184#10596833C2×#4111yescharacteristic
185C8×C73C22×C811yescharacteristic
186C8×C73C22×C811yescharacteristic
187#10596831C22×C811yescharacteristic
188#10596831C22×C811yescharacteristic
189C22×C73—22no—
190C22×C73—81no—
191C22×C73C8×D411yescharacteristic
192C2×D73—22no—
193C2×D73—41no—
194C2×D73—81no—
195C2×D73C8×D411yescharacteristic
196C2×D73C2×#13111yescharacteristic
197#64408—22no—
198#64408—41no—
199C4×C73—22no—
200C4×C73C8×D411yescharacteristic
201C4×C73C23×C811yescharacteristic
202Dic73—22no—
203Dic73C8×D411yescharacteristic
204Dic73C2×C4×D411yescharacteristic
205#33443—731nomaximal, Sylow
206C2×C73—21no—
207C2×C73—42no—
208C2×C73C2×C8×D411yescharacteristic
209D73—21no—
210D73—21no—
211D73—42no—
212#3407—731no—
213#3407—731no—
214C2×#443—731no—
215#2171—734no—
216#2170—734no—
217C2×#364—731no—
218#2117—731no—
219#2117—731no—
220C4×M32—731no—
221C73#3344311yescharacteristic, complemented, Sylow
222C2×#135—731no—
223C2×#131—731no—
224C2×#131—731no—
225#504—734no—
226#504—734no—
227#504—1462no—
228#504—1462no—
229C2×M32—731no—
230C2×M32—731no—
231C2×M32—731no—
232C2×M32—731no—
233C2×M32—1462no—
234C2×M32—1462no—
235#443—732no—
236#443—732no—
237#443—732no—
238#443—732no—
239C2×#108—731no—
240C2×#104—731no—
241C2×#104—731no—
242C2×C4×C8—731no—
243#364—732no—
244#364—732no—
245#350—734no—
246#350—734no—
247#346—732no—
248C4×C16—732no—
249C2×#41—731no—
250C2×#41—731no—
251#135—732no—
252#135—732no—
253#135—732no—
254#135—732no—
255C2×Q16—731no—
256C2×SD16—731no—
257C2×SD16#1059684211yescharacteristic
258C2×D8—731no—
259#131—732no—
260#131—732no—
261#131—732no—
262#131—732no—
263#131—1461no—
264#131—1461no—
265#131—1461no—
266#131—1461no—
267C2×M16—731no—
268C2×M16—731no—
269C2×M16—1461no—
270C2×M16—1461no—
271C22×C8—731no—
272C22×C8—731no—
273C22×C8—1461no—
274C22×C8—1461no—
275C2×C42—731no—
276M32—732no—
277M32—732no—
278M32—732no—
279M32—732no—
280M32—1464no—
281M32—1464no—
282M32—1461no—
283M32—1461no—
284M32—1461no—
285M32—1461no—
286C2×C16—732no—
287C2×C16—732no—
288C2×C16—1464no—
289C2×C16—1464no—
290C2×C16—1461no—
291C2×C16—1461no—
292#108—732no—
293#108—732no—
294#104—732no—
295#104—732no—
296#104—732no—
297#104—732no—
298C4×C8—731no—
299C4×C8—731no—
300C4×C8—731no—
301C4×C8—731no—
302#41—732no—
303#41—732no—
304#41—732no—
305#41—732no—
306#41—1461no—
307#41—1461no—
308#41—1461no—
309#41—1461no—
310C2×Q8—731no—
311C2×Q8C2×#1059684211yescharacteristic
312C2×D4—731no—
313C2×D4—1461no—
314C2×D4—1461no—
315C2×D4C2×#1059684211yescharacteristic
316C22×C4—731no—
317C22×C4—1461no—
318C22×C4—1461no—
319C22×C4—1461no—
320Q16—732no—
321Q16—732no—
322SD16—732no—
323SD16—732no—
324SD16C2×#1059684212yes—
325SD16C2×#1059684212yes—
326D8—732no—
327D8—732no—
328M16—732no—
329M16—732no—
330M16—1462no—
331M16—1462no—
332M16—1462no—
333M16—1462no—
334M16—1461no—
335M16—1461no—
336C2×C8—731no—
337C2×C8—731no—
338C2×C8—731no—
339C2×C8—731no—
340C2×C8—731no—
341C2×C8—731no—
342C2×C8—731no—
343C2×C8—731no—
344C2×C8—731no—
345C2×C8—731no—
346C2×C8—731no—
347C2×C8—1462no—
348C2×C8—1462no—
349C2×C8—1462no—
350C2×C8—1462no—
351C2×C8—1461no—
352C2×C8—1461no—
353C2×C8C2×#1059684211yescharacteristic
354C42—731no—
355C42—731no—
356C42—1461no—
357C16—734no—
358C16—1462no—
359C16—1462no—
360C16—1462no—
361C23—21no—
362C23—731no—
363C23—1461no—
364Q8—21no—
365Q8—732no—
366Q8—1461no—
367Q8C22×#1059684212yes—
368D4—21no—
369D4—732no—
370D4—1462no—
371D4—1462no—
372D4—1462no—
373D4—1462no—
374D4—1461no—
375D4C22×#1059684212yes—
376C2×C4—21no—
377C2×C4—2922no—
378C2×C4—731no—
379C2×C4—731no—
380C2×C4—731no—
381C2×C4—731no—
382C2×C4—731no—
383C2×C4—1462no—
384C2×C4—1462no—
385C2×C4—1461no—
386C2×C4—1462no—
387C2×C4—1462no—
388C2×C4—1461no—
389C2×C4—1461no—
390C2×C4C22×#1059684211yescharacteristic
391C8—731no—
392C8—731no—
393C8—731no—
394C8—731no—
395C8—731no—
396C8—731no—
397C8—1462no—
398C8—1462no—
399C8C22×#1059684211yescharacteristic
400C8C22×#1059684211yescharacteristic
401C22—22no—
402C22—81no—
403C22—2921no—
404C22—5841no—
405C22—731no—
406C22—731no—
407C22—1462no—
408C22D4×#1059684211yescharacteristic
409C4—22no—
410C4—2921no—
411C4—731no—
412C4—731no—
413C4—1462no—
414C4—1462no—
415C4D4×#1059684211yescharacteristic
416C4C23×#1059684211yescharacteristic
417C2—21no—
418C2—42no—
419C2—2922no—
420C2—1461no—
421C2—1461no—
422C2C2×D4×#1059684211yescharacteristic
subgroupG/H
centerC2C2×D4×#10596842
commutatorC4×C73C23×C8
FrattiniC4C23×#10596842
FittingC2×SD16×C73C8
socleC2×C73C2×C8×D4
prime radicalC2×#14556612C4
cosocle#11924036C24
hypercenterC2×SD16#10596842
hypocenterC73#33443
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. #22627186C2semidirect
2. #22627184C2semidirect
3. #22627151C2semidirect
4. #22627149C2semidirect
5. #22627125C2semidirect
6. #22627115C2semidirect
7. #22627113C2semidirect
8. #64408⋊M32C2semidirect
11. #17989709C22semidirect
13. #17989708C22semidirect
20. #17989692C22semidirect
21. #17989691C22semidirect
22. #17989688C22semidirect
23. #17989687C22semidirect
24. #17989682C22semidirect
25. #64408⋊C16C22semidirect
221. C73#33443semidirect

Maximal subgroups

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

subgroupindex
#226271862
#226271842
#226271512
#226271492
#226271252
#226271152
#226271132
#64408⋊M322
C2×#179896302
#3344373