Finite Group Databasebeta

← #34019558

Subgroups — #34019558

Order 18688.

656 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626863C212yesmaximal, complemented
2#22625564C212yesmaximal, complemented
3#22625560C214yesmaximal, complemented
4Q16×#10596833C212yesmaximal, complemented
5#22625488C211yesmaximal, characteristic, complemented
6#22625266C212yesmaximal, complemented
7C8×#11924037C211yesmaximal, characteristic, complemented
8#135×Dic73C211yesmaximal, characteristic
9Q8×#10596833C2214yes—
10#17989622—24no—
11#17989622C2214yes—
12#17989604—22no—
13#17989604C2211yescharacteristic
14#17989590—22no—
15#17989590C2212yes—
16C4×#11924037C2211yescharacteristic
17#17989451C2212yes—
18#17989450C2212yes—
19Q8⋊#10596833C2214yes—
20D4⋊#10596833C2214yes—
21#17989429C2212yes—
22C8×#10596833C2212yes—
23#41×Dic73C2212yes—
24Q16×Dic73C411yescharacteristic
25Q16×Dic73C2211yescharacteristic
26SD16×Dic73C412yes—
27SD16×Dic73C2212yes—
28D8×Dic73C411yescharacteristic
29D8×Dic73C2211yescharacteristic
30#17988625C411yescharacteristic
31#17988583C412yes—
32C2×C8×Dic73C2211yescharacteristic
33C2×#135×C73C411yescharacteristic
34C2×#11924037—24no—
35C2×#11924037—22no—
36C2×#11924037—21no—
37#14556658—22no—
38#14556658D411yescharacteristic
39#14556658D411yescharacteristic
40C4⋊#10596833—24no—
41C4⋊#10596833C2314yes—
42C22⋊#10596833—24no—
43C22⋊#10596833D412yes—
44#14556646—21no—
45C4×#10596833—24no—
46C4×#10596833C2312yes—
47C4×#10596833C2312yes—
48C22×#10596831C2×C411yescharacteristic
49Q8×Dic73C2×C412yes—
50Q8×Dic73C2312yes—
51D4×Dic73—22no—
52D4×Dic73—22no—
53D4×Dic73C2×C412yes—
54D4×Dic73C2312yes—
55#14556550—22no—
56#14556550C2×C412yes—
57C2×C4×Dic73—22no—
58C2×C4×Dic73C2311yescharacteristic
59#14556525C2×C411yescharacteristic
60#14556525C2×C411yescharacteristic
61#14556523C2×C412yes—
62#14556515C2×C412yes—
63#14556515C2×C412yes—
64#14556512C2×C412yes—
65#14556512C2×C412yes—
66C8×Dic73C2×C411yescharacteristic
67C8×Dic73C2×C411yescharacteristic
68C8×Dic73C2311yescharacteristic
69C8×Dic73C2311yescharacteristic
70C2×#41×C73C2×C412yes—
71#135×C73—24no—
72C2×Q16×C73C2×C411yescharacteristic
73C2×SD16×C73C2×C412yes—
74C2×D8×C73C2×C411yescharacteristic
75C22×C8×C73C2×C411yescharacteristic
76#11924037—24no—
77#11924037—84no—
78#11924037—82no—
79C2×#10596833—22no—
80C2×#10596833—24no—
81C2×#10596833C2×D412yes—
82C2×#10596833C2×D412yes—
83C22×Dic73—22no—
84C22×Dic73—22no—
85C22×Dic73C2×D411yescharacteristic
86C22×Dic73#4111yescharacteristic
87C2×#10596831—41no—
88C2×#10596831C22×C411yescharacteristic
89C2×#10596831C22×C411yescharacteristic
90C4⋊Dic73—22no—
91C4⋊Dic73—22no—
92C4⋊Dic73C22×C412yes—
93C4⋊Dic73C22×C412yes—
94C22⋊Dic73—22no—
95C22⋊Dic73—22no—
96C22⋊Dic73—22no—
97C4×Dic73—22no—
98C4×Dic73—22no—
99C4×Dic73—22no—
100C4×Dic73C22×C411yescharacteristic
101C4×Dic73C2×D411yescharacteristic
102C4×Dic73#4111yescharacteristic
103C4×Dic73C2411yescharacteristic
104#41×C73—24no—
105#41×C73—42no—
106C2×Q8×C73C22×C412yes—
107C2×D4×C73—22no—
108C2×D4×C73C22×C412yes—
109C22×C4×C73—22no—
110C22×C4×C73C22×C411yescharacteristic
111Q16×C73M1614yescomplemented
112SD16×C73—24no—
113D8×C73—22no—
114C2×C8×C73—22no—
115C2×C8×C73C22×C411yescharacteristic
116C2×C8×C73C22×C411yescharacteristic
117C23×C73—22no—
118C23×C73C4×D411yescharacteristic
119Q8×C73—22no—
120Q8×C73C2×M1614yes—
121D4×C73—22no—
122D4×C73—22no—
123D4×C73—44no—
124C2×C4×C73—22no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—44no—
128C2×C4×C73C4×D411yescharacteristic
129C2×C4×C73C23×C411yescharacteristic
130C2×Dic73—21no—
131C2×Dic73—21no—
132C2×Dic73—22no—
133C2×Dic73—22no—
134C2×Dic73—22no—
135C2×Dic73—42no—
136C2×Dic73—42no—
137C2×Dic73—44no—
138C2×Dic73C4×D411yescharacteristic
139C2×Dic73C4×D411yescharacteristic
140C2×Dic73C22×D411yescharacteristic
141C2×Dic73C2×#4111yescharacteristic
142#10596833—24no—
143#10596833—22no—
144#10596833—22no—
145#10596833—44no—
146C8×C73—21no—
147C8×C73C2×M1612yes—
148#10596831—21no—
149#10596831—21no—
150C22×C73—21no—
151C22×C73—21no—
152C22×C73—21no—
153C22×C73—42no—
154C22×C73—42no—
155C22×C73—42no—
156C22×C73C2×C4×D411yescharacteristic
157C4×C73—21no—
158C4×C73—24no—
159C4×C73C22×M1611yescharacteristic
160C4×C73#56811yescharacteristic
161Dic73—21no—
162Dic73—21no—
163Dic73—42no—
164Dic73—42no—
165Dic73#44312yes—
166Dic73#57611yescharacteristic
167Dic73#57811yescharacteristic
168#18059—731nomaximal, Sylow
169C2×C73—41no—
170C2×C73—82no—
171C2×C73D4×M1611yescharacteristic
172C2×C73#295111yescharacteristic
173C2×C73C2×#44311yescharacteristic
174#2991—732no—
175C4×#135—731no—
176#1589—732no—
177#1588—734no—
178C8×Q16—732no—
179#1567—731no—
180#1489—732no—
181C8×M16—731no—
182C73#1805911yescharacteristic, complemented, Sylow
183C2×#135#6440811yescharacteristic
184C4×#41—732no—
185C8×Q8—734no—
186C4×Q16—731no—
187C4×Q16—731no—
188C4×SD16—732no—
189C4×SD16—732no—
190C4×D8—731no—
191C4×D8—731no—
192#436—734no—
193#436—1464no—
194#431—731no—
195#431—1462no—
196#427—731no—
197#423—732no—
198#423—1462no—
199#416—732no—
200C4×M16—731no—
201C2×C4×C8—731no—
202#335—732no—
203#335—732no—
204Q8⋊C8—734no—
205D4⋊C8—734no—
206#322—732no—
207C82—732no—
208C2×#41C2×#6440812yes—
209#135—24no—
210C2×Q16C2×#6440811yescharacteristic
211C2×SD16C2×#6440812yes—
212C2×D8C2×#6440811yescharacteristic
213C2×M16—1464no—
214C2×M16—1462no—
215C2×M16—1461no—
216C22×C8—731no—
217C22×C8C2×#6440811yescharacteristic
218C4×Q8—732no—
219C4×Q8—732no—
220C4×D4—732no—
221C4×D4—732no—
222C4×D4—1462no—
223C4×D4—1462no—
224#117—732no—
225#117—1462no—
226C2×C42—731no—
227C2×C42—1462no—
228#107—731no—
229#107—731no—
230#106—732no—
231C4⋊C8—731no—
232C4⋊C8—731no—
233C4⋊C8—734no—
234C4⋊C8—1462no—
235C4⋊C8—1464no—
236#103—732no—
237#103—732no—
238#102—732no—
239#102—732no—
240C22⋊C8—732no—
241C22⋊C8—1464no—
242#97—1461no—
243C4×C8—732no—
244C4×C8—731no—
245C4×C8—731no—
246C4×C8—731no—
247C4×C8—731no—
248C4×C8—732no—
249C4×C8—1464no—
250#41—24no—
251#41—42no—
252C2×Q8C22×#6440812yes—
253C2×D4—22no—
254C2×D4C22×#6440812yes—
255C22×C4—22no—
256C22×C4—731no—
257C22×C4—731no—
258C22×C4—1462no—
259C22×C4—1462no—
260C22×C4C22×#6440811yescharacteristic
261Q16#1192403714yescomplemented
262SD16—24no—
263D8—22no—
264M16—5842no—
265M16—5844no—
266M16—1464no—
267C2×C8—22no—
268C2×C8—2921no—
269C2×C8—732no—
270C2×C8—732no—
271C2×C8—731no—
272C2×C8—731no—
273C2×C8—1464no—
274C2×C8—1462no—
275C2×C8C22×#6440811yescharacteristic
276C2×C8C22×#6440811yescharacteristic
277C4⋊C4—732no—
278C4⋊C4—732no—
279C4⋊C4—1462no—
280C4⋊C4—1462no—
281C22⋊C4—1462no—
282C22⋊C4—1462no—
283C22⋊C4—1462no—
284C42—731no—
285C42—731no—
286C42—731no—
287C42—731no—
288C42—1462no—
289C42—1462no—
290C42—1462no—
291C23—22no—
292C23D4×#6440811yescharacteristic
293Q8—22no—
294Q8C2×#1192403714yes—
295D4—22no—
296D4—22no—
297D4—44no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—44no—
302C2×C4—2922no—
303C2×C4—2922no—
304C2×C4—2924no—
305C2×C4—731no—
306C2×C4—731no—
307C2×C4—731no—
308C2×C4—731no—
309C2×C4—1462no—
310C2×C4—1462no—
311C2×C4—1461no—
312C2×C4—1462no—
313C2×C4—1461no—
314C2×C4D4×#6440811yescharacteristic
315C2×C4C23×#6440811yescharacteristic
316C8—21no—
317C8—2924no—
318C8—1462no—
319C8—1462no—
320C8—1464no—
321C8—1461no—
322C8—1461no—
323C8C2×#1192403712yes—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22C2×D4×#6440811yescharacteristic
331C4—21no—
332C4—24no—
333C4—2922no—
334C4—2922no—
335C4—731no—
336C4—731no—
337C4—732no—
338C4—1461no—
339C4—1461no—
340C4C22×#1192403711yescharacteristic
341C4#1798966511yescharacteristic
342C2—41no—
343C2—82no—
344C2D4×#1192403711yescharacteristic
345C2#2262669311yescharacteristic
346C2D73⋊#44311yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting, prime radicalC2×#135×C73C4
socleC22×C73C2×C4×D4
cosocleC4×Dic73C24
hypercenterC2×#135#64408
hypocenterC73#18059
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #22626863C2semidirect
2. #22625564C2semidirect
3. #22625560C2semidirect
4. Q16×#10596833C2semidirect
5. #22625488C2semidirect
6. #22625266C2semidirect
7. C8×#11924037C2semidirect
111. Q16×C73M16semidirect
182. C73#18059semidirect
261. Q16#11924037semidirect

Maximal subgroups

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

subgroupindex
#226268632
#226255642
#226255602
Q16×#105968332
#226254882
#226252662
C8×#119240372
#135×Dic732
#1805973