Finite Group Databasebeta

← #10691626

Subgroups — #10691626

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×#4739C211yesmaximal, characteristic
2C22⋊C4×Dic6C211yesmaximal, characteristic
3#69963C211yesmaximal, characteristic
4#69946C211yesmaximal, characteristic
5#69457C211yesmaximal, characteristic
6C3×#1282C211yesmaximal, characteristic
7#67676C211yesmaximal, characteristic
8#9258—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×#924C2211yescharacteristic
11C22×Dic12C412yescomplemented
12C2×C4×Dic6—21no—
13C2×C3×C22⋊C8C2211yescharacteristic
14C2×C3×C22⋊C8C2211yescharacteristic
15C3×C4×C22⋊C4C2211yescharacteristic
16#5212C2211yescharacteristic
17#5212C2211yescharacteristic
18C2×#816—22no—
19C2×#816C412yescomplemented
20#4942—21no—
21Dic3×C22⋊C4—21no—
22#4929—21no—
23#4739—21no—
24#4739—22no—
25#4739—21no—
26#4646—22no—
27#4527—21no—
28#4527—21no—
29#4467—22no—
30C2×#451—31no—
31Q8×C22⋊C4—31no—
32#1996—31no—
33#1787—31no—
34#1520—31no—
35#1289—31no—
36#1282D311yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C23×Dic3—21no—
39C22×Dic6—21no—
40C22×Dic6—21no—
41C22×Dic6—21no—
42C22×Dic6—44no—
43C22×Dic6C2×C412yes—
44C22×Dic6C2×C411yescharacteristic
45C22×Dic6D411yescharacteristic
46C22×Dic6D412yes—
47C22×Dic6D411yescharacteristic
48C22×C3×C8—21no—
49C22×C3×C8C2×C412yes—
50C2×C3×C22⋊C4D411yescharacteristic
51C2×C3×C22⋊C4D411yescharacteristic
52C2×C3×C42—21no—
53C2×#217—21no—
54C2×#217—21no—
55C2×#217—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3C2×C411yescharacteristic
59#924—21no—
60#924—21no—
61#924—21no—
62#924—21no—
63C2×Dic3⋊C4—22no—
64C2×Dic3⋊C4—21no—
65C2×C4×Dic3—22no—
66C2×Dic12—22no—
67C2×Dic12—22no—
68C2×Dic12—44no—
69C4×Dic6—44no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×C22⋊C8—21no—
73C3×C22⋊C8—21no—
74C3×#95—21no—
75C3⋊#95—22no—
76C3⋊#95—22no—
77C3⋊#95—21no—
78C3⋊#95D412yes—
79#816—22no—
80#816—22no—
81#816—44no—
82C23×Q8—31no—
83C22×Q16—32no—
84C2×C22⋊Q8—31no—
85C2×C4×Q8—61no—
86#451—61no—
87#451—62no—
88#451—61no—
89C2×#103—32no—
90C2×#103—62no—
91C2×C22⋊C8C2×D311yescharacteristic
92C2×C22⋊C8C2×D311yescharacteristic
93#391—62no—
94#391—61no—
95#380—31no—
96#380—31no—
97#380—61no—
98C4×C22⋊C4—61no—
99C4×C22⋊C4C2×D311yescharacteristic
100#342—61no—
101#342—61no—
102#328—62no—
103C24×C3C2×D411yescharacteristic
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—42no—
107C22×C3×C4C22×C411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C22×C3×C4#4111yescharacteristic
113C22×Dic3—21no—
114C22×Dic3—22no—
115C22×Dic3—21no—
116C22×Dic3—22no—
117C22×Dic3—22no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—44no—
121C22×Dic3—41no—
122C2×Dic6—21no—
123C2×Dic6—24no—
124C2×Dic6—24no—
125C2×Dic6—24no—
126C2×Dic6—22no—
127C2×Dic6—22no—
128C2×Dic6—21no—
129C2×Dic6—21no—
130C2×Dic6—21no—
131C2×Dic6—42no—
132C2×Dic6—42no—
133C2×Dic6—42no—
134C2×Dic6—44no—
135C2×Dic6—44no—
136C2×Dic6—44no—
137C2×Dic6—42no—
138C2×Dic6C22⋊C412yes—
139C2×Dic6C22⋊C412yes—
140C2×C3×C8—22no—
141C2×C3×C8—42no—
142C2×C3×C8—42no—
143C2×C3×C8—41no—
144C3×C22⋊C4—21no—
145C3×C22⋊C4—21no—
146C3×C22⋊C4—21no—
147C3×C22⋊C4Q1612yescomplemented
148C3×C42—42no—
149#217—41no—
150#217—41no—
151#217—41no—
152#217—41no—
153#217—41no—
154#217—41no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—42no—
158C4⋊Dic3—42no—
159Dic3⋊C4—44no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162C4×Dic3—44no—
163Dic12—44no—
164Dic12—44no—
165C22×Q8—32no—
166C22×Q8—31no—
167C22×Q8—31no—
168C22×Q8—32no—
169C22×Q8—31no—
170C22×Q8—61no—
171C22×Q8—61no—
172C22×Q8—61no—
173C22×Q8—124no—
174C23×C4—61no—
175C23×C4C22×D311yescharacteristic
176C2×Q16—62no—
177C2×Q16—62no—
178C2×Q16—124no—
179C22×C8—21no—
180C22×C8C4×D312yes—
181C22⋊Q8—61no—
182C22⋊Q8—61no—
183C22⋊Q8—61no—
184C22⋊Q8—61no—
185C4×Q8—124no—
186C2×C4⋊C4—31no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4—61no—
190C2×C4⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C22⋊C4—61no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4D1211yescharacteristic
195C2×C22⋊C4D1211yescharacteristic
196C2×C42—21no—
197C2×C42—62no—
198#103—62no—
199#103—62no—
200#103—124no—
201C22⋊C8—21no—
202C22⋊C8—21no—
203C22⋊C8—21no—
204C22⋊C8—21no—
205#95—21no—
206#95—32no—
207#95—62no—
208#95—62no—
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
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4C2×C22⋊C411yescharacteristic
232C2×C3×C4C4×D411yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12311yescharacteristic
235C2×Dic3—24no—
236C2×Dic3—22no—
237C2×Dic3—22no—
238C2×Dic3—44no—
239C2×Dic3—44no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—42no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—42no—
246C2×Dic3—81no—
247C2×Dic3—81no—
248C2×Dic3—81no—
249C2×Dic3—81no—
250C2×Dic3—81no—
251Dic6—24no—
252Dic6—22no—
253Dic6—22no—
254Dic6—22no—
255Dic6—44no—
256Dic6—44no—
257Dic6—42no—
258Dic6—42no—
259Dic6—81no—
260Dic6#9914yes—
261C3×C8—42no—
262C3×C8—81no—
263C24C2×D1211yescharacteristic
264C2×Q8—32no—
265C2×Q8—32no—
266C2×Q8—64no—
267C2×Q8—64no—
268C2×Q8—64no—
269C2×Q8—62no—
270C2×Q8—62no—
271C2×Q8—61no—
272C2×Q8—61no—
273C2×Q8—61no—
274C2×Q8—61no—
275C2×Q8—122no—
276C2×Q8—122no—
277C2×Q8—122no—
278C2×Q8—122no—
279C2×Q8—124no—
280C2×Q8—124no—
281C2×Q8—124no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—61no—
286C22×C4—62no—
287C22×C4—61no—
288C22×C4—62no—
289C22×C4—62no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4—124no—
293C22×C4—121no—
294C22×C4C2×C4×D311yescharacteristic
295C22×C4#23511yescharacteristic
296C22×C4D3×D411yescharacteristic
297C22×C4D3×D411yescharacteristic
298C22×C4D3×D411yescharacteristic
299C22×C4#23911yescharacteristic
300Q16—124no—
301Q16—124no—
302C2×C8—22no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—41no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—122no—
309C4⋊C4—122no—
310C4⋊C4—124no—
311C4⋊C4—121no—
312C4⋊C4—121no—
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—121no—
317C22⋊C4—121no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—121no—
321C22⋊C4—121no—
322C22⋊C4Dic1212yescomplemented
323C42—42no—
324C42—124no—
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—21no—
344C3×C4—21no—
345C3×C4—42no—
346C3×C4—81no—
347C3×C4C2×#9912yes—
348Dic3—24no—
349Dic3—42no—
350Dic3—82no—
351Dic3—81no—
352Dic3—81no—
353Dic3—161no—
354C23—41no—
355C23—41no—
356C23C4×D1211yescharacteristic
357C23#88211yescharacteristic
358C23#89511yescharacteristic
359C23#90411yescharacteristic
360C23C2×Dic1211yescharacteristic
361C23#90911yescharacteristic
362C23#90911yescharacteristic
363Q8—34no—
364Q8—64no—
365Q8—62no—
366Q8—62no—
367Q8—62no—
368Q8—124no—
369Q8—124no—
370Q8—122no—
371Q8—122no—
372Q8—241no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—64no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—124no—
389C2×C4—124no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—122no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—122no—
396C2×C4—241no—
397C2×C4—241no—
398C2×C4—241no—
399C2×C4—241no—
400C2×C4—241no—
401C2×C4D3×C22⋊C411yescharacteristic
402C2×C4#88411yescharacteristic
403C2×C4#89311yescharacteristic
404C2×C4#89411yescharacteristic
405C8—42no—
406C8—81no—
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—
426C22#466511yescharacteristic
427C22C4×Dic1211yescharacteristic
428C22#471611yescharacteristic
429C22#473311yescharacteristic
430C22#473911yescharacteristic
431C22#484511yescharacteristic
432C22#484911yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—42no—
437C4—64no—
438C4—81no—
439C4—122no—
440C4—242no—
441C4—241no—
442C4—241no—
443C4—481no—
444C4D3×#9912yes—
445C3#925811yescharacteristic, complemented, Sylow
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2#6868511yescharacteristic
453C2#6897011yescharacteristic
454C2#6903011yescharacteristic
subgroupG/H
centerC22#4665
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1282C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1282D3
hypocenterC3#9258
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
11. C22×Dic12C4semidirect
19. C2×#816C4semidirect
147. C3×C22⋊C4Q16semidirect
322. C22⋊C4Dic12semidirect
445. C3#9258semidirect

Maximal subgroups

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

subgroupindex
C2×#47392
C22⋊C4×Dic62
#699632
#699462
#694572
C3×#12822
#676762
#92583