Finite Group Databasebeta

← #10691629

Subgroups — #10691629

Order 768.

708 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#451C211yesmaximal, characteristic
2Q8×#217C211yesmaximal, characteristic
3#72132C211yesmaximal, characteristic
4#71701C211yesmaximal, characteristic
5#71690C211yesmaximal, characteristic
6#67677C211yesmaximal, characteristic
7#67676C211yesmaximal, characteristic
8#9258—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×C3×Q16C412yescomplemented
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×Q8×Dic3—21no—
13C3×#451—21no—
14C3×#451—22no—
15C3×#451—21no—
16C2×C3×#103C412yescomplemented
17C2×C3×C22⋊C8C2211yescharacteristic
18#5229—21no—
19#5229—22no—
20C2×#835—22no—
21#5212—21no—
22#5212C2211yescharacteristic
23#5212C2211yescharacteristic
24C4×#217—21no—
25C4×#217C2211yescharacteristic
26C2×#830C2211yescharacteristic
27#4538—22no—
28#4526—21no—
29#4526—21no—
30C2×#451D311yescharacteristic
31Q8×C22⋊C4—31no—
32#1996—31no—
33#1787—31no—
34#1520—31no—
35#1289—31no—
36#1282—31no—
37C22×C3×Q8—21no—
38C22×C3×Q8—21no—
39C22×C3×Q8—21no—
40C22×C3×Q8—44no—
41C22×C3×Q8C2×C412yes—
42C22×C3×Q8C2×C411yescharacteristic
43C22×C3×Q8D411yescharacteristic
44C22×C3×Q8D412yes—
45C22×C3×Q8D411yescharacteristic
46C23×C3×C4—21no—
47C23×C3×C4C2311yescharacteristic
48C2×C3×Q16—22no—
49C2×C3×Q16—22no—
50C2×C3×Q16—44no—
51C22×C3×C8C2×C412yes—
52C3×C22⋊Q8—21no—
53C3×C22⋊Q8—21no—
54C3×C22⋊Q8—21no—
55C3×C22⋊Q8—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4C2×C411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×#217—21no—
60C2×#217—21no—
61C2×#217D411yescharacteristic
62C2×#217D411yescharacteristic
63Q8×Dic3—44no—
64C2×C4⋊Dic3—22no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4×Dic3—22no—
68C2×C4×Dic3—21no—
69C22×C3⋊C8—21no—
70C3×#103—22no—
71C3×#103—22no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74#835—44no—
75C3⋊#95—22no—
76C3⋊#95—22no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79C3⋊#95D412yes—
80#830—21no—
81#830—21no—
82C23×Q8C2×D311yescharacteristic
83C22×Q16Dic312yescomplemented
84C2×C22⋊Q8C2×D311yescharacteristic
85C2×C4×Q8—61no—
86#451—21no—
87#451—22no—
88#451—21no—
89C2×#103—62no—
90C2×#103Dic312yescomplemented
91C2×C22⋊C8—31no—
92C2×C22⋊C8C2×D311yescharacteristic
93#391—62no—
94#391—61no—
95#380—31no—
96#380—31no—
97#380—61no—
98C4×C22⋊C4—31no—
99C4×C22⋊C4—61no—
100#342—61no—
101#342—61no—
102#328—62no—
103C24×C3C2×D411yescharacteristic
104C2×C3×Q8—24no—
105C2×C3×Q8—22no—
106C2×C3×Q8—22no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C2×C3×Q8—24no—
110C2×C3×Q8—24no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—42no—
114C2×C3×Q8—42no—
115C2×C3×Q8—42no—
116C2×C3×Q8—44no—
117C2×C3×Q8—42no—
118C2×C3×Q8—44no—
119C2×C3×Q8—44no—
120C2×C3×Q8C22⋊C412yes—
121C2×C3×Q8C22⋊C412yes—
122C22×C3×C4—21no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—42no—
129C22×C3×C4—44no—
130C22×C3×C4C22×C411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—22no—
139C22×Dic3—41no—
140C3×Q16—44no—
141C3×Q16—44no—
142C2×C3×C8—22no—
143C2×C3×C8—42no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150#217—21no—
151#217—21no—
152#217—21no—
153#217—41no—
154#217—41no—
155#217—41no—
156#217—41no—
157#217Q1612yescomplemented
158C4⋊Dic3—44no—
159C4⋊Dic3—42no—
160C4⋊Dic3—42no—
161C4×Dic3—44no—
162C4×Dic3—42no—
163C2×C3⋊C8—42no—
164C2×C3⋊C8—41no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—44no—
169C22×Q8C2×Dic312yes—
170C22×Q8C2×Dic311yescharacteristic
171C22×Q8#6711yescharacteristic
172C22×Q8#6712yes—
173C22×Q8#6711yescharacteristic
174C23×C4—21no—
175C23×C4C22×D311yescharacteristic
176C2×Q16—22no—
177C2×Q16—22no—
178C2×Q16—44no—
179C22×C8—61no—
180C22×C8C2×Dic312yes—
181C22⋊Q8—21no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C22⋊Q8—21no—
185C4×Q8—124no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4C2×Dic311yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—31no—
193C2×C22⋊C4—31no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C42—62no—
197C2×C42—61no—
198#103—22no—
199#103—22no—
200#103—124no—
201C22⋊C8—21no—
202C22⋊C8—21no—
203C22⋊C8—61no—
204C22⋊C8—61no—
205#95—32no—
206#95—62no—
207#95—62no—
208#95—61no—
209#95—61no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3C4×D411yescharacteristic
213C23×C3#12011yescharacteristic
214C23×C3#12111yescharacteristic
215C23×C3C2×Q1611yescharacteristic
216C23×C3#13511yescharacteristic
217C23×C3#13711yescharacteristic
218C23×C3#13711yescharacteristic
219C3×Q8—24no—
220C3×Q8—22no—
221C3×Q8—22no—
222C3×Q8—22no—
223C3×Q8—44no—
224C3×Q8—44no—
225C3×Q8—42no—
226C3×Q8—42no—
227C3×Q8—81no—
228C3×Q8#9914yes—
229C2×C3×C4—24no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—44no—
239C2×C3×C4—44no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4C2×C22⋊C411yescharacteristic
246C2×C3×C4C4×D411yescharacteristic
247C2×C3×C4#12111yescharacteristic
248C2×C3×C4#12311yescharacteristic
249C2×Dic3—22no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—42no—
254C2×Dic3—41no—
255C2×Dic3—42no—
256C2×Dic3—41no—
257C2×Dic3—42no—
258C2×Dic3—81no—
259C2×Dic3—81no—
260C2×Dic3—81no—
261C3×C8—42no—
262C3⋊C8—81no—
263C24D3×D411yescharacteristic
264C2×Q8—24no—
265C2×Q8—22no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—44no—
277C2×Q8—42no—
278C2×Q8—44no—
279C2×Q8—44no—
280C2×Q8#21712yes—
281C2×Q8#21712yes—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—61no—
291C22×C4—62no—
292C22×C4—61no—
293C22×C4—121no—
294C22×C4D3×D411yescharacteristic
295C22×C4#23711yescharacteristic
296C22×C4#23711yescharacteristic
297C22×C4C22×Dic311yescharacteristic
298C22×C4C2×#6711yescharacteristic
299C22×C4C2×#6711yescharacteristic
300Q16—44no—
301Q16—44no—
302C2×C8—22no—
303C2×C8—42no—
304C2×C8—122no—
305C2×C8—121no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—124no—
311C4⋊C4—122no—
312C4⋊C4—122no—
313C22⋊C4—32no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—61no—
317C22⋊C4—61no—
318C22⋊C4—61no—
319C22⋊C4—121no—
320C22⋊C4—121no—
321C22⋊C4—121no—
322C22⋊C4—121no—
323C42—124no—
324C42—122no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3#38611yescharacteristic
336C22×C3C4×Q1611yescharacteristic
337C22×C3#44111yescharacteristic
338C22×C3#45111yescharacteristic
339C22×C3#45211yescharacteristic
340C22×C3#46211yescharacteristic
341C22×C3#46411yescharacteristic
342C3×C4—21no—
343C3×C4—24no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—42no—
347C3×C4—81no—
348C3×C4—81no—
349C3×C4C2×#9912yes—
350Dic3—42no—
351Dic3—81no—
352Dic3—82no—
353Dic3—161no—
354C23—41no—
355C23—41no—
356C23D3×Q1611yescharacteristic
357C23#91811yescharacteristic
358C23#91811yescharacteristic
359C23#91911yescharacteristic
360C23D4×Dic311yescharacteristic
361C23#93711yescharacteristic
362C23#93911yescharacteristic
363Q8—24no—
364Q8—22no—
365Q8—22no—
366Q8—22no—
367Q8—44no—
368Q8—44no—
369Q8—42no—
370Q8—42no—
371Q8—81no—
372Q8#83414yes—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—62no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—122no—
393C2×C4—121no—
394C2×C4—122no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—122no—
398C2×C4—241no—
399C2×C4—241no—
400C2×C4—241no—
401C2×C4D4×Dic311yescharacteristic
402C2×C4#93511yescharacteristic
403C2×C4#93911yescharacteristic
404C2×C4C2×#21711yescharacteristic
405C8—42no—
406C8—241no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3#187811yescharacteristic
414C2×C3#189811yescharacteristic
415C2×C3#190511yescharacteristic
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22Dic3×Q1611yescharacteristic
427C22#518211yescharacteristic
428C22#518411yescharacteristic
429C22#518511yescharacteristic
430C22#518611yescharacteristic
431C22#518711yescharacteristic
432C22#522011yescharacteristic
433C4—21no—
434C4—24no—
435C4—21no—
436C4—21no—
437C4—42no—
438C4—81no—
439C4—81no—
440C4—122no—
441C4—241no—
442C4—242no—
443C4—481no—
444C4C2×#83412yes—
445C3#925811yescharacteristic, complemented, Sylow
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2#7167011yescharacteristic
453C2#7206511yescharacteristic
454C2#7210111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#451C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#451D3
hypocenterC3#9258
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
10. C22×C3×Q16C4semidirect
16. C2×C3×#103C4semidirect
83. C22×Q16Dic3semidirect
90. C2×#103Dic3semidirect
157. #217Q16semidirect
445. C3#9258semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4512
Q8×#2172
#721322
#717012
#716902
#676772
#676762
#92583