Finite Group Databasebeta

← #30748569

Subgroups — #30748569

Order 17152.

614 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22560189C211yesmaximal, characteristic
2C2×#16863133C212yesmaximal
3#22558436C211yesmaximal, characteristic
4#22558087C212yesmaximal
5C4×C22⋊Q8×C67C211yesmaximal, characteristic
6C2×C4⋊(C67⋊C8)—21no—
7C2×C22⋊(C67⋊C8)C2212yes—
8C2×C67⋊#95—21no—
9C2×C67⋊#95—21no—
10C2×C67⋊#95C2212yes—
11#16863151—21no—
12#16863133—24no—
13C2×C22⋊Q8×C67C2211yescharacteristic
14C2×C4×Q8×C67C411yescharacteristic, complemented
15C22×C42×C67C2211yescharacteristic
16#391×C67C411yescharacteristic, complemented
17#389×C67C411yescharacteristic, complemented
18#387×C67—21no—
19#385×C67—21no—
20#380×C67C2211yescharacteristic
21C4×C4⋊C4×C67—21no—
22C4×C4⋊C4×C67C411yescharacteristic, complemented
23C4×C22⋊C4×C67—21no—
24C23×Dic67—22no—
25C22×C67⋊C8—22no—
26C4⋊(C67⋊C8)—42no—
27C22⋊(C67⋊C8)—22no—
28C22⋊(C67⋊C8)—22no—
29C67⋊#95—24no—
30C67⋊#95—22no—
31C67⋊#95—22no—
32C67⋊#95—42no—
33C22×Q8×C67C2×C411yescharacteristic
34C23×C4×C67D411yescharacteristic
35C23×C4×C67D411yescharacteristic
36C23×C4×C67C2311yescharacteristic
37C22⋊Q8×C67—21no—
38C22⋊Q8×C67—21no—
39C22⋊Q8×C67D412yes—
40C22⋊Q8×C67Q812yes—
41C4×Q8×C67—42no—
42C2×C4⋊C4×C67—21no—
43C2×C4⋊C4×C67—21no—
44C2×C4⋊C4×C67—21no—
45C2×C4⋊C4×C67C2×C411yescharacteristic
46C2×C4⋊C4×C67C2×C411yescharacteristic
47C2×C4⋊C4×C67C2×C411yescharacteristic
48C2×C22⋊C4×C67—21no—
49C2×C22⋊C4×C67—21no—
50C2×C42×C67—21no—
51C2×C42×C67—21no—
52C2×C42×C67—42no—
53C2×C42×C67C2×C411yescharacteristic
54C2×C42×C67C2×C411yescharacteristic
55C2×C42×C67D411yescharacteristic
56C2×C42×C67D411yescharacteristic
57#95×C67—21no—
58#95×C67—21no—
59#95×C67—21no—
60#95×C67—21no—
61#95×C67—21no—
62C22×Dic67—24no—
63C22×Dic67—42no—
64C22×Dic67—42no—
65C22×Dic67—42no—
66C22×Dic67—42no—
67C22×Dic67—42no—
68C22×Dic67—42no—
69C2×C67⋊C8—42no—
70C2×C67⋊C8—42no—
71C2×C67⋊C8—42no—
72C24×C67C2×D411yescharacteristic
73C2×Q8×C67—21no—
74C2×Q8×C67—41no—
75C2×Q8×C67—41no—
76C2×Q8×C67C4⋊C412yes—
77C22×C4×C67—21no—
78C22×C4×C67—21no—
79C22×C4×C67—22no—
80C22×C4×C67—22no—
81C22×C4×C67—22no—
82C22×C4×C67—22no—
83C22×C4×C67—21no—
84C22×C4×C67—21no—
85C22×C4×C67—41no—
86C22×C4×C67—41no—
87C22×C4×C67—41no—
88C22×C4×C67—41no—
89C22×C4×C67C22⋊C411yescharacteristic
90C22×C4×C67C22⋊C411yescharacteristic
91C22×C4×C67C22⋊C411yescharacteristic
92C22×C4×C67C22⋊C411yescharacteristic
93C22×C4×C67C22×C411yescharacteristic
94C22×C4×C67C2×D411yescharacteristic
95C22×C4×C67C2×D411yescharacteristic
96C22×C4×C67C2×Q811yescharacteristic
97C22×C4×C67#4111yescharacteristic
98C22×C4×C67#4111yescharacteristic
99C4⋊C4×C67—21no—
100C4⋊C4×C67—22no—
101C4⋊C4×C67—22no—
102C4⋊C4×C67—41no—
103C4⋊C4×C67—41no—
104C4⋊C4×C67—42no—
105C4⋊C4×C67—42no—
106C4⋊C4×C67C4⋊C412yes—
107C22⋊C4×C67—22no—
108C22⋊C4×C67—22no—
109C22⋊C4×C67—42no—
110C42×C67—22no—
111C42×C67—22no—
112C42×C67—41no—
113C42×C67—41no—
114C42×C67—42no—
115C42×C67—42no—
116C23×C67—21no—
117C23×C67—21no—
118C23×C67—21no—
119C23×C67—21no—
120C23×C67C2×C22⋊C411yescharacteristic
121C23×C67#12111yescharacteristic
122C23×C67C22⋊Q811yescharacteristic
123C23×C67#13512yes—
124C23×C67#13712yes—
125Q8×C67—42no—
126Q8×C67—42no—
127C2×C4×C67—21no—
128C2×C4×C67—21no—
129C2×C4×C67—21no—
130C2×C4×C67—21no—
131C2×C4×C67—21no—
132C2×C4×C67—21no—
133C2×C4×C67—21no—
134C2×C4×C67—21no—
135C2×C4×C67—22no—
136C2×C4×C67—22no—
137C2×C4×C67—21no—
138C2×C4×C67—21no—
139C2×C4×C67—21no—
140C2×C4×C67—21no—
141C2×C4×C67—21no—
142C2×C4×C67—21no—
143C2×C4×C67—41no—
144C2×C4×C67—41no—
145C2×C4×C67—41no—
146C2×C4×C67—41no—
147C2×C4×C67—44no—
148C2×C4×C67—44no—
149C2×C4×C67—41no—
150C2×C4×C67—41no—
151C2×C4×C67—41no—
152C2×C4×C67—41no—
153C2×C4×C67—41no—
154C2×C4×C67—41no—
155C2×C4×C67—41no—
156C2×C4×C67—41no—
157C2×C4×C67—41no—
158C2×C4×C67—41no—
159C2×C4×C67#9912yes—
160C2×C4×C67#10412yes—
161C2×C4×C67C2×C4⋊C411yescharacteristic
162C2×C4×C67#11711yescharacteristic
163C2×C4×C67#12111yescharacteristic
164C2×C4×C67C22⋊Q811yescharacteristic
165C2×Dic67—44no—
166C2×Dic67—44no—
167C2×Dic67—44no—
168C2×Dic67—82no—
169C2×Dic67—82no—
170C2×Dic67—82no—
171C2×Dic67—82no—
172C67⋊C8—82no—
173C22×C67—21no—
174C22×C67—21no—
175C22×C67—21no—
176C22×C67—21no—
177C22×C67—22no—
178C22×C67—22no—
179C22×C67—22no—
180C22×C67—22no—
181C22×C67#38011yescharacteristic
182C22×C67C2×#9911yescharacteristic
183C22×C67#41011yescharacteristic
184C22×C67#41611yescharacteristic
185C22×C67#41911yescharacteristic
186C22×C67C2×#10411yescharacteristic
187C22×C67#42111yescharacteristic
188C22×C67#46412yes—
189C22×C67#47912yes—
190C4×C67—21no—
191C4×C67—21no—
192C4×C67—21no—
193C4×C67—21no—
194C4×C67—22no—
195C4×C67—22no—
196C4×C67—42no—
197C4×C67—42no—
198C4×C67—41no—
199C4×C67—41no—
200C4×C67—81no—
201C4×C67—81no—
202Dic67—84no—
203#8574—671nomaximal, Sylow
204C2×C67—22no—
205C2×C67—22no—
206C2×C67#150011yescharacteristic
207C2×C67#150811yescharacteristic
208C2×C67#151511yescharacteristic
209C2×C67#151611yescharacteristic
210C2×C67#179911yescharacteristic
211C2×C67#180511yescharacteristic
212C2×C67#180611yescharacteristic
213C4×C22⋊Q8D6711yescharacteristic
214#1797—671no—
215C2×#328—672no—
216#1434—671no—
217#1294—672no—
218C67#857411yescharacteristic, complemented, Sylow
219C2×C22⋊Q8C2×D6711yescharacteristic
220C2×C4×Q8Dic6711yescharacteristic, complemented
221C22×C42C2×D6711yescharacteristic
222C2×C4⋊C8—1341no—
223C2×C22⋊C8—672no—
224#391Dic6711yescharacteristic, complemented
225#389Dic6711yescharacteristic, complemented
226#387—21no—
227#385—21no—
228#380C2×D6711yescharacteristic
229C4×C4⋊C4—21no—
230C4×C4⋊C4Dic6711yescharacteristic, complemented
231C4×C22⋊C4—21no—
232C2×#95—672no—
233C2×#95—1341no—
234C2×#95—1341no—
235#336—1341no—
236#328—1344no—
237C22×Q8C2×Dic6711yescharacteristic
238C23×C4—1342no—
239C23×C4C22⋊D6711yescharacteristic
240C23×C4C22⋊D6711yescharacteristic
241C23×C4C22×D6711yescharacteristic
242C22×C8—1342no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C22⋊Q8Dic13412yes—
246C22⋊Q8D26812yes—
247C4×Q8—42no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C2×Dic6711yescharacteristic
252C2×C4⋊C4C2×Dic6711yescharacteristic
253C2×C4⋊C4C2×Dic6711yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—42no—
259C2×C42C2×Dic6711yescharacteristic
260C2×C42C2×Dic6711yescharacteristic
261C2×C42C22⋊D6711yescharacteristic
262C2×C42C22⋊D6711yescharacteristic
263C4⋊C8—2682no—
264C22⋊C8—1342no—
265C22⋊C8—1342no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—2682no—
272#95—1344no—
273#95—1342no—
274#95—1342no—
275C24C2×C22⋊D6711yescharacteristic
276C2×Q8—21no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8C4⋊Dic6712yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—2682no—
293C22×C4—2682no—
294C22×C4—2682no—
295C22×C4—2682no—
296C22×C4—2682no—
297C22×C4—2682no—
298C22×C4—1344no—
299C22×C4C22⋊Dic6711yescharacteristic
300C22×C4C22⋊Dic6711yescharacteristic
301C22×C4C22⋊Dic6711yescharacteristic
302C22×C4C22⋊Dic6711yescharacteristic
303C22×C4C22×Dic6711yescharacteristic
304C22×C4C2×C22⋊D6711yescharacteristic
305C22×C4C2×D26811yescharacteristic
306C22×C4C2×Dic13411yescharacteristic
307C22×C4#1176162811yescharacteristic
308C22×C4#1176162811yescharacteristic
309C2×C8—2682no—
310C2×C8—2682no—
311C2×C8—2682no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4C4⋊Dic6712yes—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—42no—
323C42—22no—
324C42—22no—
325C42—41no—
326C42—41no—
327C42—42no—
328C42—42no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23C2×C22⋊Dic6711yescharacteristic
334C23#1452401711yescharacteristic
335C23#1452402211yescharacteristic
336C23#1452406312yes—
337C23#1452407412yes—
338Q8—42no—
339Q8—42no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—2684no—
373C2×C4—2684no—
374C2×C4—2684no—
375C2×C4—5362no—
376C2×C4—5362no—
377C2×C4—5362no—
378C2×C4—5362no—
379C2×C4#1452395812yes—
380C2×C4#1452397012yes—
381C2×C4C2×C4⋊Dic6711yescharacteristic
382C2×C4#1452400211yescharacteristic
383C2×C4#1452401711yescharacteristic
384C2×C4#1452402211yescharacteristic
385C8—5362no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22#1686325611yescharacteristic
395C22C2×#1452395811yescharacteristic
396C22#1686337011yescharacteristic
397C22#1686339211yescharacteristic
398C22#1686340511yescharacteristic
399C22C2×#1452397011yescharacteristic
400C22#1686341311yescharacteristic
401C22#1686361812yes—
402C22#1686370312yes—
403C4—22no—
404C4—22no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—42no—
412C4—42no—
413C4—81no—
414C4—81no—
415C4—5364no—
416C2—22no—
417C2—22no—
418C2#2255869511yescharacteristic
419C2#2255872911yescharacteristic
420C2#2255876011yescharacteristic
421C2#2255876511yescharacteristic
422C2#2256019711yescharacteristic
423C2#2256022411yescharacteristic
424C2#2256022911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic67
commutatorC22×C4×C67C22×C4
FrattiniC23×C4C22×D67
FittingC4×C22⋊Q8×C67C2
socleC23×C67C2×C22⋊C4
prime radicalC24×C67C2×D4
cosocleC23×C4×C67C23
hypercenterC4×C22⋊Q8D67
hypocenterC67#8574
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
14. C2×C4×Q8×C67C4semidirect
16. #391×C67C4semidirect
17. #389×C67C4semidirect
22. C4×C4⋊C4×C67C4semidirect
218. C67#8574semidirect
220. C2×C4×Q8Dic67semidirect
224. #391Dic67semidirect
225. #389Dic67semidirect
230. C4×C4⋊C4Dic67semidirect

Maximal subgroups

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

subgroupindex
#225601892
C2×#168631332
#225584362
#225580872
C4×C22⋊Q8×C672
#857467