Finite Group Databasebeta

← #30774498

Subgroups — #30774498

Order 17152.

488 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22560201C211yesmaximal, characteristic
2C4×#14523958C211yesmaximal, characteristic
3#22558327C212yesmaximal
4#22558327C212yesmaximal
5C67×#2579C211yesmaximal, characteristic
6#16863420—21no—
7C2×#14523960C2212yes—
8C2×#14523958C2212yes—
9C4×C22⋊Dic67—21no—
10#16863165—21no—
11#16863163—21no—
12C2×C4×D4×C67C2211yescharacteristic
13C22×C4⋊C4×C67C411yescharacteristic, complemented
14#398×C67—21no—
15#396×C67—21no—
16#387×C67—21no—
17#386×C67C2211yescharacteristic
18#386×C67C2211yescharacteristic
19#385×C67—21no—
20#385×C67C411yescharacteristic, complemented
21#385×C67C411yescharacteristic, complemented
22#380×C67C411yescharacteristic, complemented
23C2×#11761612—22no—
24C2×C22⋊Dic67—22no—
25C2×C4×Dic67—41no—
26C4⋊(C67⋊C8)—41no—
27C4⋊(C67⋊C8)—41no—
28#14523960—24no—
29#14523958—22no—
30#14523958Q814yes—
31C67⋊#95—41no—
32C22×D4×C67C2311yescharacteristic
33C23×C4×C67C2×C411yescharacteristic
34C23×C4×C67C2×C411yescharacteristic
35C23×C4×C67C2×C411yescharacteristic
36C23×C4×C67D411yescharacteristic
37C4×D4×C67—41no—
38C4×D4×C67—41no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—21no—
41C2×C4⋊C4×C67—21no—
42C2×C4⋊C4×C67—21no—
43C2×C4⋊C4×C67—21no—
44C2×C4⋊C4×C67—41no—
45C2×C4⋊C4×C67—41no—
46C2×C4⋊C4×C67D411yescharacteristic
47C2×C22⋊C4×C67—21no—
48C2×C22⋊C4×C67—21no—
49C2×C22⋊C4×C67C2×C411yescharacteristic
50C2×C22⋊C4×C67C2×C411yescharacteristic
51C2×C22⋊C4×C67C2×C411yescharacteristic
52C2×C22⋊C4×C67D411yescharacteristic
53C2×C42×C67D411yescharacteristic
54#95×C67—21no—
55#95×C67—21no—
56#95×C67—21no—
57#95×C67—21no—
58#95×C67—21no—
59C22×Dic67—42no—
60#11761612—42no—
61#11761612—42no—
62C2×C67⋊C8—42no—
63C22⋊Dic67—24no—
64C22⋊Dic67—42no—
65C4×Dic67—82no—
66C24×C67C22×C411yescharacteristic
67C24×C67C2×D411yescharacteristic
68C2×D4×C67—41no—
69C2×D4×C67—41no—
70C2×D4×C67C2×Q812yes—
71C2×D4×C67#4112yes—
72C22×C4×C67—21no—
73C22×C4×C67—21no—
74C22×C4×C67—21no—
75C22×C4×C67—21no—
76C22×C4×C67—21no—
77C22×C4×C67—21no—
78C22×C4×C67—21no—
79C22×C4×C67—21no—
80C22×C4×C67—42no—
81C22×C4×C67—41no—
82C22×C4×C67—41no—
83C22×C4×C67—81no—
84C22×C4×C67—81no—
85C22×C4×C67—81no—
86C22×C4×C67C22⋊C411yescharacteristic
87C22×C4×C67C22⋊C411yescharacteristic
88C22×C4×C67C22⋊C411yescharacteristic
89C22×C4×C67C22⋊C411yescharacteristic
90C22×C4×C67C2×D411yescharacteristic
91C4⋊C4×C67—41no—
92C4⋊C4×C67—42no—
93C4⋊C4×C67—42no—
94C4⋊C4×C67—41no—
95C4⋊C4×C67—41no—
96C4⋊C4×C67—81no—
97C4⋊C4×C67—81no—
98C22⋊C4×C67—41no—
99C22⋊C4×C67—41no—
100C22⋊C4×C67—42no—
101C22⋊C4×C67—41no—
102C22⋊C4×C67—41no—
103C22⋊C4×C67—81no—
104C42×C67—21no—
105C42×C67—21no—
106C23×C67—21no—
107C23×C67—21no—
108C23×C67—21no—
109C23×C67—21no—
110C23×C67—41no—
111C23×C67—41no—
112C23×C67—41no—
113C23×C67—41no—
114C23×C67C2×C22⋊C411yescharacteristic
115C23×C67C4×Q812yes—
116C23×C67C22⋊Q812yes—
117D4×C67—42no—
118D4×C67—42no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—22no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
125C2×C4×C67—41no—
126C2×C4×C67—42no—
127C2×C4×C67—41no—
128C2×C4×C67—42no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67—41no—
132C2×C4×C67—41no—
133C2×C4×C67—42no—
134C2×C4×C67—41no—
135C2×C4×C67—41no—
136C2×C4×C67—41no—
137C2×C4×C67—41no—
138C2×C4×C67—41no—
139C2×C4×C67—41no—
140C2×C4×C67—41no—
141C2×C4×C67—41no—
142C2×C4×C67—41no—
143C2×C4×C67—81no—
144C2×C4×C67—81no—
145C2×C4×C67—81no—
146C2×C4×C67—82no—
147C2×C4×C67#9911yescharacteristic
148C2×C4×C67#9911yescharacteristic
149C2×C4×C67#12411yescharacteristic
150C2×C4×C67C4⋊Q811yescharacteristic
151C2×Dic67—44no—
152C2×Dic67—82no—
153C2×Dic67—82no—
154C67⋊C8—82no—
155C22×C67—21no—
156C22×C67—21no—
157C22×C67—22no—
158C22×C67—22no—
159C22×C67—42no—
160C22×C67—42no—
161C22×C67—42no—
162C22×C67—41no—
163C22×C67—41no—
164C22×C67—41no—
165C22×C67—41no—
166C22×C67—81no—
167C22×C67#39111yescharacteristic
168C22×C67C2×#9911yescharacteristic
169C22×C67#41011yescharacteristic
170C4×C67—21no—
171C4×C67—21no—
172C4×C67—21no—
173C4×C67—21no—
174C4×C67—41no—
175C4×C67—42no—
176C4×C67—41no—
177C4×C67—41no—
178C4×C67—81no—
179C4×C67—81no—
180C4×C67—81no—
181Dic67—84no—
182#13297—671nomaximal, Sylow
183C2×C67—21no—
184C2×C67—21no—
185C2×C67—42no—
186C2×C67—42no—
187C2×C67#199611yescharacteristic
188C2×C67#212011yescharacteristic
189C2×C67#212111yescharacteristic
190#2579D6711yescharacteristic
191#1800—671no—
192C4×#99—671no—
193#1390—672no—
194#1390—672no—
195C67#1329711yescharacteristic, complemented, Sylow
196C2×C4×D4C2×D6711yescharacteristic
197C22×C4⋊C4Dic6711yescharacteristic, complemented
198#423—1341no—
199C2×#100—672no—
200C2×#99—672no—
201#398—21no—
202#396—21no—
203#387—21no—
204#386C2×D6711yescharacteristic
205#386C2×D6711yescharacteristic
206#385—21no—
207#385Dic6711yescharacteristic, complemented
208#385Dic6711yescharacteristic, complemented
209#380Dic6711yescharacteristic, complemented
210C4×C22⋊C4—1341no—
211#343—1341no—
212#342—1341no—
213C22×D4C22×D6711yescharacteristic
214C23×C4C2×Dic6711yescharacteristic
215C23×C4C2×Dic6711yescharacteristic
216C23×C4C2×Dic6711yescharacteristic
217C23×C4C22⋊D6711yescharacteristic
218C2×M16—1342no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C22⋊D6711yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—1342no—
232C2×C22⋊C4C2×Dic6711yescharacteristic
233C2×C22⋊C4C2×Dic6711yescharacteristic
234C2×C22⋊C4C2×Dic6711yescharacteristic
235C2×C22⋊C4C22⋊D6711yescharacteristic
236C2×C42—2681no—
237C2×C42C22⋊D6711yescharacteristic
238C4⋊C8—2681no—
239C4⋊C8—2681no—
240#100—1344no—
241#99—674no—
242#99—1342no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—2681no—
249C24C22×Dic6711yescharacteristic
250C24C2×C22⋊D6711yescharacteristic
251C2×D4—41no—
252C2×D4—41no—
253C2×D4Q8×D6712yes—
254C2×D4#1176162912yes—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—2682no—
270C22×C4C22⋊Dic6711yescharacteristic
271C22×C4C22⋊Dic6711yescharacteristic
272C22×C4C22⋊Dic6711yescharacteristic
273C22×C4C22⋊Dic6711yescharacteristic
274C22×C4C2×C22⋊D6711yescharacteristic
275M16—2682no—
276M16—2682no—
277C2×C8—2682no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—81no—
284C4⋊C4—81no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—81no—
291C22⋊C4—2682no—
292C22⋊C4—1344no—
293C42—21no—
294C42—21no—
295C42—5362no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23C2×C22⋊Dic6711yescharacteristic
305C23Q8×Dic6712yes—
306C23#1452401912yes—
307D4—42no—
308D4—42no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—2684no—
338C2×C4—5362no—
339C2×C4—5362no—
340C2×C4#1452395811yescharacteristic
341C2×C4#1452395811yescharacteristic
342C2×C4#1452403211yescharacteristic
343C2×C4#1452404111yescharacteristic
344C8—5362no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—81no—
357C22#1686330311yescharacteristic
358C22C2×#1452395811yescharacteristic
359C22#1686337011yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—5364no—
372C2—21no—
373C2—21no—
374C2—42no—
375C2—42no—
376C2#2256119511yescharacteristic
377C2#2256185011yescharacteristic
378C2#2256185511yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC24×C67C22×C4
FrattiniC22×D4C22×D67
FittingC67×#2579C2
socleC22×C67C2×#99
prime radical, cosocleC22×D4×C67C23
hypercenter#2579D67
hypocenterC67#13297
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2579

Complements

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

NH ≅ G/Ntype
13. C22×C4⋊C4×C67C4semidirect
20. #385×C67C4semidirect
21. #385×C67C4semidirect
22. #380×C67C4semidirect
195. C67#13297semidirect
197. C22×C4⋊C4Dic67semidirect
207. #385Dic67semidirect
208. #385Dic67semidirect
209. #380Dic67semidirect

Maximal subgroups

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

subgroupindex
#225602012
C4×#145239582
#225583272
C67×#25792
#1329767